Vector Bundles over Multipullback Quantum Complex Projective Spaces

TL;DR

This paper classifies vector bundles over quantum projective spaces and spheres, using groupoid C*-algebras and Rieffel's stable rank results to determine when modules are free, and explicitly identifies quantum line bundles.

Contribution

It applies groupoid C*-algebra techniques and stable rank theory to classify projective modules over quantum spaces, providing explicit descriptions of quantum line bundles.

Findings

Modules of rank higher than loor(n/2)+3 are free over quantum spheres.

Identifies a large portion of the positive cone of the K0-group.

Explicitly represents quantum line bundles as elementary projections.

Abstract

We work on the classification of isomorphism classes of finitely generated projective modules over the C*-algebras $C\left( \mathbb{P}^{n}\left( \mathcal{T}\right) \right) $ and $C\left( \mathbb{S}_{H}^{2n+1}\right) $ of the quantum complex projective spaces $\mathbb{P}^{n}\left( \mathcal{T} \right) $ and the quantum spheres $\mathbb{S}_{H}^{2n+1}$, and the quantum line bundles $L_{k}$ over $\mathbb{P}^{n}\left( \mathcal{T}\right) $, studied by Hajac and collaborators. Motivated by the groupoid approach of Curto, Muhly, and Renault to the study of C*-algebraic structure, we analyze $C\left( \mathbb{P}^{n}\left( \mathcal{T}\right) \right) $, $C\left( \mathbb{S}_{H}^{2n+1}\right) $, and $L_{k}$ in the context of groupoid C*-algebras, and then apply Rieffel's stable rank results to show that all finitely generated projective modules over $C\left( \mathbb{S}_{H} ^{2n+1}\right) $ of rank higher than $\left\lfloor \frac{n}{2}\right\rfloor +3$ are free modules. Furthermore, besides identifying a large portion of the positive cone of the $K_{0}$-group of $C\left( \mathbb{P}^{n}\left( \mathcal{T}\right) \right) $, we also explicitly identify $L_{k}$ with concrete representative elementary projections over $C\left( \mathbb{P} ^{n}\left( \mathcal{T}\right) \right) $.

Equations

$$P_m:=\sum _{i=1}^m{e}_{ii}\in M_m\left(\mathbb{C}\right)\subset \mathcal{K}$$

$$P_{-m}:=I-P_m\in {\mathcal{K}}^{+}$$

$${\left({\mathbb{Z}}^n⋉{\overline{\mathbb{Z}}}^n\right)|}_{{\overline{{\mathbb{Z}}_{\ge }}}^n}\cong {\times }^n\left({\left(\mathbb{Z}⋉\overline{\mathbb{Z}}\right)|}_{\overline{{\mathbb{Z}}_{\ge }}}\right)$$

$$C^{\ast }\left({\mathfrak{T}}_n\right)\equiv C^{\ast }\left({\left({\mathbb{Z}}^n⋉{\overline{\mathbb{Z}}}^n\right)|}_{{\overline{{\mathbb{Z}}_{\ge }}}^n}\right)\cong {\mathcal{T}}^{\otimes n}\equiv {\otimes }^n\mathcal{T}.$$

$$0\to C^{\ast }\left({\left({\mathbb{Z}}^n⋉{\overline{\mathbb{Z}}}^n\right)|}_{{\mathbb{Z}}_{\ge }^n}\right)\cong \mathcal{K}\left({\ell }^2\left({\mathbb{Z}}_{\ge }^n\right)\right)\to C^{\ast }\left({\mathfrak{T}}_n\right)\equiv {\mathcal{T}}^{\otimes n}\to C^{\ast }\left({\mathfrak{G}}_n\right)\to 0$$

$${\mathfrak{G}}_n:={\left({\mathbb{Z}}^n⋉{\overline{\mathbb{Z}}}^n\right)|}_{{\overline{{\mathbb{Z}}_{\ge }}}^n\backslash {\mathbb{Z}}_{\ge }^n}$$

$$0\to C^{\ast }\left({\left({\mathbb{Z}}^n⋉{\overline{\mathbb{Z}}}^n\right)|}_{{\overline{{\mathbb{Z}}_{\ge }}}^n\backslash \left\{{\infty }^n\right\}}\right)\to C^{\ast }\left({\mathfrak{T}}_n\right)\equiv {\mathcal{T}}^{\otimes n}\stackrel{{\sigma }_n}{\to }{C}^{\ast }\left({\left({\mathbb{Z}}^n⋉{\overline{\mathbb{Z}}}^n\right)|}_{\left\{{\infty }^n\right\}}\right)\cong C^{\ast }\left({\mathbb{Z}}^n\right)\cong C\left({\mathbb{T}}^n\right)\to 0$$

$$C\left({\mathbb{S}}_H^{2n-1}\right)\cong \left({\otimes }^n\mathcal{T}\right)/\left({\otimes }^n\mathcal{K}\right)$$

$$C\left({\mathbb{S}}_H^{2n-1}\right)\cong C^{\ast }\left({\mathfrak{G}}_n\right)$$

$$0\to C^{\ast }\left({\left({\mathbb{Z}}^n⋉{\overline{\mathbb{Z}}}^n\right)|}_{{\overline{{\mathbb{Z}}_{\ge }}}^n\backslash \left({\mathbb{Z}}_{\ge }^n\cup \left\{{\infty }^n\right\}\right)}\right)\to C\left({\mathbb{S}}_H^{2n-1}\right)\cong \left({\otimes }^n\mathcal{T}\right)/\left({\otimes }^n\mathcal{K}\right)\stackrel{{\tau }_n}{\to }C\left({\mathbb{T}}^n\right)\to 0.$$

$${\mathcal{T}}^{\otimes n}\equiv C^{\ast }\left({\mathfrak{T}}_n\right)\equiv {\mathcal{I}}_n⊳{\mathcal{I}}_{n-1}⊳\cdots ⊳{\mathcal{I}}_1⊳{\mathcal{I}}_0⊳{\mathcal{I}}_{-1}\equiv \left\{0\right\}$$

$${\mathcal{I}}_j/{\mathcal{I}}_{j-1}\cong {\oplus }^{\frac{n!}{j!\left(n-j\right)!}}\left(\mathcal{K}\left({\ell }^2\left({\mathbb{Z}}_{\ge }^{n-j}\right)\right)\otimes C\left({\mathbb{T}}^j\right)\right),$$

$$Y_j:=X_0\bigsqcup X_1\bigsqcup \cdots \bigsqcup X_j$$

$${\mathbb{Z}}_{\ge }^n=Y_0\subset Y_1\subset \cdots \subset Y_n={\overline{{\mathbb{Z}}_{\ge }}}^n$$

$${\mathcal{I}}_j:=C^{\ast }\left({\left({\mathbb{Z}}^n⋉{\overline{\mathbb{Z}}}^n\right)|}_{Y_j}\right)\equiv C^{\ast }\left({{\mathfrak{T}}_n|}_{Y_j}\right).$$

$$C^{\ast }\left({{\mathfrak{T}}_n|}_{Y_j\backslash Y_{j-1}}\right)={\oplus }^{\frac{n!}{j!\left(n-j\right)!}}{C}^{\ast }\left({\left({\mathbb{Z}}^n⋉{\overline{\mathbb{Z}}}^n\right)|}_{{\mathbb{Z}}_{\ge }^{n-j}\times \left\{{\infty }^j\right\}}\right)$$

$$={\oplus }^{\frac{n!}{j!\left(n-j\right)!}}{C}^{\ast }\left(\left({\left({\mathbb{Z}}^{n-j}⋉{\mathbb{Z}}^{n-j}\right)|}_{{\mathbb{Z}}_{\ge }^{n-j}}\right)\times {\mathbb{Z}}^j\right)={\oplus }^{\frac{n!}{j!\left(n-j\right)!}}\left(\mathcal{K}\left({\ell }^2\left({\mathbb{Z}}_{\ge }^{n-j}\right)\right)\otimes C\left({\mathbb{T}}^j\right)\right).$$

$${\mathcal{I}}_j/{\mathcal{I}}_{j-1}\cong C^{\ast }\left({{\mathfrak{T}}_n|}_{Y_j\backslash Y_{j-1}}\right)\cong {\oplus }^{\frac{n!}{j!\left(n-j\right)!}}\left(\mathcal{K}\left({\ell }^2\left({\mathbb{Z}}_{\ge }^{n-j}\right)\right)\otimes C\left({\mathbb{T}}^j\right)\right).$$

$$C\left({\mathbb{S}}_H^{2n-1}\right)\equiv C^{\ast }\left({\mathfrak{G}}_n\right)\equiv {\mathcal{J}}_n⊳{\mathcal{J}}_{n-1}⊳\cdots ⊳{\mathcal{J}}_1⊳{\mathcal{J}}_0\equiv \left\{0\right\}$$

$${\mathcal{J}}_j/{\mathcal{J}}_{j-1}\cong {\oplus }^{\frac{n!}{j!\left(n-j\right)!}}\left(\mathcal{K}\left({\ell }^2\left({\mathbb{Z}}_{\ge }^{n-j}\right)\right)\otimes C\left({\mathbb{T}}^j\right)\right).$$

$$\lfloor \frac{n}{2}\rfloor +1\le \mathrm{sr}\left(C\left({\mathbb{S}}_H^{2n-1}\right)\right)\le \mathrm{sr}\left({\mathcal{T}}^{\otimes n}\right)\le \lfloor \frac{n+1}{2}\rfloor +1$$

$$\mathrm{csr}\left({\mathcal{T}}^{\otimes n}\right)\le \mathrm{csr}\left(C\left({\mathbb{S}}_H^{2n-1}\right)\right)\le \lfloor \frac{n+1}{2}\rfloor +1$$

$$\max \left\{\mathrm{sr}\left(\mathcal{A}/\mathcal{I}\right),\mathrm{sr}\left(\mathcal{I}\right)\right\}\le \mathrm{sr}\left(\mathcal{A}\right)\le \max \left\{\mathrm{sr}\left(\mathcal{A}/\mathcal{I}\right),\mathrm{sr}\left(\mathcal{I}\right),\mathrm{csr}\left(\mathcal{A}/\mathcal{I}\right)\right\},$$

$$\mathrm{csr}\left(\mathcal{A}\right)\le \max \left\{\mathrm{csr}\left(\mathcal{A}/\mathcal{I}\right),\mathrm{csr}\left(\mathcal{I}\right)\right\}.$$

$$0\to {\mathcal{I}}_{j-1}\to {\mathcal{I}}_j\to {\mathcal{I}}_j/{\mathcal{I}}_{j-1}\cong {\oplus }^{\frac{n!}{j!\left(n-j\right)!}}\left(\mathcal{K}\left({\ell }^2\left({\mathbb{Z}}_{\ge }^{n-j}\right)\right)\otimes C\left({\mathbb{T}}^j\right)\right)\to 0$$

$$0\to \mathcal{K}\left({\ell }^2\left({\mathbb{Z}}_{\ge }^n\right)\right)\cong {\mathcal{I}}_0\to {\mathcal{I}}_1\to {\mathcal{I}}_1/{\mathcal{I}}_0\cong {\oplus }^n\left(\mathcal{K}\left({\ell }^2\left({\mathbb{Z}}_{\ge }^{n-1}\right)\right)\otimes C\left(\mathbb{T}\right)\right)\to 0$$

$$\lfloor \frac{n}{2}\rfloor +1\le \mathrm{sr}\left({\mathcal{T}}^{\otimes n}\right)\le \lfloor \frac{n+1}{2}\rfloor +1$$

$$0\to {\mathcal{I}}_{n-1}\to {\mathcal{I}}_n\equiv {\mathcal{T}}^{\otimes n}\to {\mathcal{I}}_n/{\mathcal{I}}_{n-1}\cong C\left({\mathbb{T}}^n\right)\to 0.$$

$$\mathrm{sr}\left({\mathcal{T}}^{\otimes n}\right)=\lfloor \frac{n}{2}\rfloor +1\text{ (and hence }\mathrm{sr}\left(C\left({\mathbb{S}}_H^{2n-1}\right)\right)=\lfloor \frac{n}{2}\rfloor +1\text{)}$$

$$\mathrm{sr}\left(C\left({\mathbb{S}}_H^{2n-1}\right)\right)=\mathrm{sr}\left({\mathcal{T}}^{\otimes n}\right)=\lfloor \frac{n}{2}\rfloor +1$$

Full text

Keywords:

multipullback quantum projective space; multipullback quantum sphere; quantum line bundle; finitely generated projective module; cancellation problem; Toeplitz algebra of polydisk; groupoid C*-algebra; stable rank; noncommutative vector bundle

AMS 2010 Mathematics Subject Classification:

46L80; 46L85

1 Introduction

Since the concept of noncommutative geometry first popularized by Connes [5], many interesting examples of a C*-algebra $\mathcal{A}$ viewed as the algebra $C\left(X_{q}\right)$ of continuous functions on a virtual quantum space $X_{q}$ have been constructed with a topological or geometrical motivation, and analyzed in comparison with their classical counterpart. For example, quantum odd-dimensional spheres and associated complex projective spaces have been introduced and studied by Soibelman, Vaksman, Meyer, and others [32, 14] as $\mathbb{S}_{q}^{2n+1}$ and $\mathbb{C}P_{q}^{n}$ via a quantum universal enveloping algebra approach, and by Hajac and his collaborators including Baum, Kaygun, Matthes, Nest, Pask, Sims, Szymański, Zieliński, and others [2, 10, 9, 12] as $\mathbb{S}_{H}^{2n+1}$ and $\mathbb{P}^{n}\left(\mathcal{T}\right)$ via a multi-pullback and Toeplitz algebra approach. Actually $\mathbb{S}_{H}^{2n+1}$ is the untwisted special case of the more general version of $\theta$-twisted spheres $\mathbb{S}_{H,\theta}^{2n+1}$ introduced in [12].

Motivated by Swan’s work [30], the concept of a noncommutative vector bundle $E_{q}$ over a quantum space $X_{q}$ can be reformulated as a finitely generated projective (left) module $\Gamma\left(E_{q}\right)$ over $C\left(X_{q}\right)$. Based on the strong connection approach to quantum principal bundles [8] for compact quantum groups [34, 35], Hajac and his collaborators introduced quantum line bundles $L_{k}$ of degree $k$ over $\mathbb{P}^{n}\left(\mathcal{T}\right)$ as some rank-one projective modules realized as spectral subspaces $C\left(\mathbb{S}_{H}^{2n+1}\right)_{k}$ of $C\left(\mathbb{S}_{H}^{2n+1}\right)$ under a $U\left(1\right)$-action [12]. Besides having the $K_{0}$-group of $C\left(\mathbb{P}^{n}\left(\mathcal{T}\right)\right)$ computed, they found that $L_{k}$ is not stably free unless $k=0$, extending earlier results for the case of $n=1$ [10, 11].

It has always been an interesting but challenging task to classify finitely generated projective modules over an algebra up to isomorphism, which goes beyond their classification up to stable isomorphism by $K_{0}$-group and appears in the form of so-called cancellation problem. Classically it is known that the cancellation law holds for complex vector bundles of rank no less than $\frac{d}{2}$ over a $d$-dimensional CW-complex, which implies that all complex vector bundles over $\mathbb{S}^{2n+1}$ of rank $n+1$ or above are trivial.

The study of such classification problem for C*-algebras was popularized by Rieffel [21, 22] who introduced useful versions of stable ranks for C*-algebras to facilitate the analysis involved. Some successes have been achieved for certain quantum algebras [22, 23, 25, 1, 19]. In particular, Peterka showed that all finitely generated projective modules over the $\theta$-deformed $3$-spheres $S_{\theta}^{3}$ are free, and constructed all those over $S_{\theta}^{4}$ up to isomorphism [19]. With more effort, the result of Bach [1] on the cancellation law for $\mathbb{S}_{q}^{2n+1}$ and $\mathbb{C}P_{q}^{n}$ can be strengthened to a complete classification of finitely generated projective modules over them, which we will address elsewhere.

With the $K_{0}$-group of $C\left(\mathbb{P}^{n}\left(\mathcal{T}\right)\right)$ known [12], it is natural to try to classify finitely generated projective modules over $C\left(\mathbb{P}^{n}\left(\mathcal{T}\right)\right)$ and identify the line bundles $L_{k}$ among them. In [29], a complete solution was obtained for the special case of $n=1$.

In this paper, we use the powerful groupoid approach to C*-algebras initiated by Renault [20] and popularized by Curto, Muhly, and Renault [6, 15] to study multi-variable Toeplitz C*-algebras $\mathcal{T}^{\otimes n}$, quantum spheres $C\left(\mathbb{S}_{H}^{2n+1}\right)$, and quantum complex projective spaces $C\left(\mathbb{P}^{n}\left(\mathcal{T}\right)\right)$. Utilizing results on stable ranks of C*-algebras obtained by Rieffel [21], we analyze finitely generated projective modules over $\mathcal{T}^{\otimes n+1}$ and $C\left(\mathbb{S}_{H}^{2n+1}\right)$, and get those of rank higher than $\left\lfloor\frac{n}{2}\right\rfloor+3$ and also a large class of “standard” modules classified up to isomorphism. Furthermore, besides identifying a large portion of the positive cone of the $K_{0}$-group $K_{0}\left(C\left(\mathbb{P}^{n}\left(\mathcal{T}\right)\right)\right)$, we explicitly identify the quantum line bundles $L_{k}$ with concrete representative elementary projections.

On the other hand, there are still a lot of questions to be further investigated, e.g. whether the cancellation law holds for low-ranked finitely generated projective modules, and whether the more general case of $\theta$-twisted multipullback quantum sphere $\mathbb{S}_{H,\theta}^{2n+1}$ brings in new phenomena. Finally it is of interest to note the recent work of Farsi, Hajac, Maszczyk, and Zieliński [7] on $K_{0}\left(C\left(\mathbb{P}^{2}\left(\mathcal{T}\right)\right)\right)$, identifying its free generators arising from Milnor modules as sums of $L_{k}$, which are also expressed in terms of elementary projections, showing a perfect consistency with our result.

2 Notations

Taking the groupoid approach to C*-algebras initiated by Renault [20] and popularized by the work of Curto, Muhly, and Renault [6, 15], we give a description of the C*-algebras $C\left(\mathbb{S}_{H}^{2n-1}\right)$ and $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)$ of [12] as some concrete groupoid C*-algebras. We refer to [20, 15] for the concepts and theory of groupoid C*-algebras used freely in the following discussion.

By abuse of notation, for any C*-algebra homomorphism $\phi:\mathcal{A}\rightarrow\mathcal{B}$, we denote the C*-algebra homomorphism $M_{k}\left(\phi\right):M_{k}\left(\mathcal{A}\right)\rightarrow M_{k}\left(\mathcal{B}\right)$ for $k\in\mathbb{N}\equiv\left\{1,2,3,...\right\}$ also by $\phi$. We use $\mathcal{A}^{\times}$ to denote the set of all invertible elements of an algebra $\mathcal{A}$, and use $\mathcal{A}^{+}$ to denote the minimal unitization of $\mathcal{A}$. For any topological group $G$, we use $G^{0}$ to denote the identity component of $G$, i.e. the connected component that contains the identity element of $G$.

We denote by $M_{\infty}\left(\mathcal{A}\right)$ the direct limit (or the union as sets) of the increasing sequence of matrix algebras $M_{n}\left(\mathcal{A}\right)$ over $\mathcal{A}$ with the canonical inclusion $M_{n}\left(\mathcal{A}\right)\subset M_{n+1}\left(\mathcal{A}\right)$ identifying $x\in M_{n}\left(\mathcal{A}\right)$ with $x\boxplus 0\in M_{n+1}\left(\mathcal{A}\right)$ for any algebra $\mathcal{A}$, where $\boxplus$ denotes the standard diagonal concatenation (sum) of two matrices. So the size of an element in $M_{\infty}\left(\mathcal{A}\right)$ can be taken arbitrarily large. We also use $GL_{\infty}\left(\mathcal{A}\right)$ to denote the direct limit of the general linear groups $GL_{n}\left(\mathcal{A}\right)$ over a unital C*-algebra $\mathcal{A}$ with $GL_{n}\left(\mathcal{A}\right)$ embedded in $GL_{n+1}\left(\mathcal{A}\right)$ by identifying $x\in GL_{n}\left(\mathcal{A}\right)$ with $x\boxplus 1\in GL_{n+1}\left(\mathcal{A}\right)$.

By an idempotent $P$ over a unital C*-algebra $\mathcal{A}$, we mean an element $P\in M_{\infty}\left(\mathcal{A}\right)$ with $P^{2}=P$, and a self-adjoint idempotent in $M_{\infty}\left(\mathcal{A}\right)$ is called a projection over $\mathcal{A}$. Two idempotents $P,Q\in M_{\infty}\left(\mathcal{A}\right)$ are called equivalent, denoted as $P\sim Q$, if there exists $U\in GL_{\infty}\left(\mathcal{A}\right)$ such that $UPU^{-1}=Q$. Each idempotent $P\in M_{n}\left(\mathcal{A}\right)$ over $\mathcal{A}$ defines a finitely generated left projective module $E:=\mathcal{A}^{n}P$ over $\mathcal{A}$ where elements of $\mathcal{A}^{n}$ are viewed as row vectors. The mapping $P\mapsto\mathcal{A}^{n}P$ induces a bijective correspondence between the equivalence classes of idempotents over $\mathcal{A}$ and the isomorphism classes of finitely generated left projective modules over $\mathcal{A}$ [3]. From now on, by a module over $\mathcal{A}$, we mean a left $\mathcal{A}$-module, unless otherwise specified.

Two finitely generated projective modules $E,F$ over $\mathcal{A}$ are called stably isomorphic if they become isomorphic after being augmented by the same finitely generated free $\mathcal{A}$-module, i.e. $E\oplus\mathcal{A}^{k}\cong F\oplus\mathcal{A}^{k}$ for some $k\geq 0$. Correspondingly, two idempotents $P$ and $Q$ are called stably equivalent if $P\boxplus I_{k}$ and $Q\boxplus I_{k}$ are equivalent for some identity matrix $I_{k}$. The $K_{0}$-group $K_{0}\left(\mathcal{A}\right)$ classifies idempotents over $\mathcal{A}$ up to stable equivalence. The classification of idempotents over a C*-algebra up to equivalence, appearing as the so-called cancellation problem, was popularized by Rieffel’s pioneering work [21, 22] and is in general an interesting but difficult question.

The set of all equivalence classes of idempotents over a C*-algebra $\mathcal{A}$ is an abelian monoid $\mathfrak{P}\left(\mathcal{A}\right)$ with its binary operation provided by the diagonal sum $\boxplus$. The image of the canonical homomorphism from $\mathfrak{P}\left(\mathcal{A}\right)$ into $K_{0}\left(\mathcal{A}\right)$ is the so-called positive cone of $K_{0}\left(\mathcal{A}\right)$.

Furthermore, it is well-known [3] that in the above descriptions of $\mathfrak{P}\left(\mathcal{A}\right)$ and $K_{0}\left(\mathcal{A}\right)$, one can restrict to the self-adjoint idempotents, called projections over $\mathcal{A}$, and their unitary equivalence classes, which faithfully represent the elements of $\mathfrak{P}\left(\mathcal{A}\right)$ and $K_{0}\left(\mathcal{A}\right)$.

In this paper, we use freely the basic techniques and manipulations for $K$-theory found in [3, 31].

For a Hilbert space $\mathcal{H}$, we denote the C*-algebra consisting of all compact linear operators on $\mathcal{H}$ by $\mathcal{K}\left(\mathcal{H}\right)$, or simply by $\mathcal{K}$ if $\mathcal{H}$ is the essentially unique separable infinite-dimensional Hilbert space.

In the following, we use the notations $\mathbb{Z}_{\geq k}:=\left\{n\in\mathbb{Z}|n\geq k\right\}$ and $\mathbb{Z}_{\geq}:=\mathbb{Z}_{\geq 0}$. In particular, $\mathbb{N}=\mathbb{Z}_{\geq 1}$. We use $I$ to denote the identity operator canonically contained in $\mathcal{K}^{+}\subset\mathcal{B}\left(\ell^{2}\left(\mathbb{Z}_{\geq}\right)\right)$, and

[TABLE]

to denote the standard $m\times m$ identity matrix in $M_{m}\left(\mathbb{C}\right)\subset\mathcal{K}$ for any integer $m\geq 0$ (with $M_{0}\left(\mathbb{C}\right)=0$ and $P_{0}=0$ understood). We also use the notation

[TABLE]

for integers $m>0$, and take symbolically $P_{-0}\equiv I-P_{0}=I\neq P_{0}$.

3 Quantum spaces as groupoid C*-algebras

Let $\mathfrak{T}_{n}:=\left.\left(\mathbb{Z}^{n}\ltimes\overline{\mathbb{Z}}^{n}\right)\right|_{\overline{\mathbb{Z}_{\geq}}^{n}}$ with $n\geq 1$ be the transformation group groupoid $\mathbb{Z}^{n}\ltimes\overline{\mathbb{Z}}^{n}$ restricted to the positive “cone” $\overline{\mathbb{Z}_{\geq}}^{n}$ where $\overline{\mathbb{Z}}:=\mathbb{Z}\cup\left\{+\infty\right\}$ containing $\mathbb{Z}_{\geq}\equiv\left\{n\in\mathbb{Z}|n\geq 0\right\}$ carries the standard topology, and $\mathbb{Z}^{n}$ acts on $\overline{\mathbb{Z}}^{n}$ componentwise in the canonical way. From the groupoid isomorphism

[TABLE]

and the well-known C*-algebra isomorphism $C^{\ast}\left(\left.\left(\mathbb{Z}\ltimes\overline{\mathbb{Z}}\right)\right|_{\overline{\mathbb{Z}_{\geq}}}\right)\cong\mathcal{T}$ for the Toeplitz C*-algebra $\mathcal{T}$, we get the groupoid C*-algebra

[TABLE]

We consider two important nontrivial invariant open subsets of the unit space $\overline{\mathbb{Z}_{\geq}}^{n}$ of $\mathfrak{T}_{n}$, namely, $\mathbb{Z}_{\geq}^{n}$ the smallest one and $\overline{\mathbb{Z}_{\geq}}^{n}\backslash\left\{\infty^{n}\right\}$ the largest one, where $\infty^{n}:=\left(\infty,...,\infty\right)\in\overline{\mathbb{Z}_{\geq}}^{n}$. By the theory of groupoid C*-algebras developed in Renault’s book [20], they give rise to two short exact sequences of C*-algebras

[TABLE]

with $\mathcal{K}\left(\ell^{2}\left(\mathbb{Z}_{\geq}^{n}\right)\right)\cong\otimes^{n}\mathcal{K}\equiv\mathcal{K}^{\otimes n}$ where

[TABLE]

is $\mathfrak{T}_{n}$ restricted to the “limit boundary” of its unit space, and

[TABLE]

where the quotient map $\sigma_{n}$ extends the notion of the well-known symbol map $\sigma$ on $\mathcal{T}$ in the case of $n=1$.

Note that the open invariant set $\mathbb{Z}_{\geq}^{n}$ being dense in the unit space $\overline{\mathbb{Z}_{\geq}}^{n}$ of $\mathfrak{T}_{n}$ induces a faithful representation $\pi_{n}$ of $C^{\ast}\left(\mathfrak{T}_{n}\right)$ on $\ell^{2}\left(\mathbb{Z}_{\geq}^{n}\right)$ that realizes the groupoid C*-algebra $C^{\ast}\left(\mathfrak{T}_{n}\right)$ and its closed ideal $C^{\ast}\left(\left.\left(\mathbb{Z}^{n}\ltimes\overline{\mathbb{Z}}^{n}\right)\right|_{\mathbb{Z}_{\geq}^{n}}\right)$ respectively as a C*-subalgebra of $\mathcal{B}\left(\ell^{2}\left(\mathbb{Z}_{\geq}^{n}\right)\right)$ and the closed ideal $\mathcal{K}\left(\ell^{2}\left(\mathbb{Z}_{\geq}^{n}\right)\right)$ consisting of all compact operators on $\ell^{2}\left(\mathbb{Z}_{\geq}^{n}\right)$.

In this paper, we freely identify elements of $C^{\ast}\left(\mathfrak{T}_{n}\right)\equiv\mathcal{T}^{\otimes n}$ with operators on $\ell^{2}\left(\mathbb{Z}_{\geq}^{n}\right)$ via the faithful representation $\pi_{n}$ and use these two conceptually different notions interchangeably.

In [12], Hajac, Nest, Pask, Sims, and Zieliński defined the (untwisted) multipullback or *Heegaard quantum odd-dimensional sphere $S_{H}^{2n-1}$ as the quantum space of the multipullback C-algebra [18] determined by homomorphisms of the form $\mathrm{id}^{\otimes j}\otimes\sigma\otimes\mathrm{id}^{\otimes n-j-1}$ from $\mathcal{T}^{\otimes i}\otimes C\left(\mathbb{T}\right)\otimes\mathcal{T}^{\otimes n-i-1}$ with $i\neq j$ to some $\mathcal{T}^{\otimes m}\otimes C\left(\mathbb{T}\right)\otimes\mathcal{T}^{\otimes k}\otimes C\left(\mathbb{T}\right)\otimes\mathcal{T}^{\otimes n-m-k-2}$. (Actually more general $\theta$-twisted quantum spheres $S_{H,\theta}^{2n-1}$ are studied there.) They showed that

[TABLE]

and hence we have

[TABLE]

identified as a groupoid C*-algebra.

With the ideal $C^{\ast}\left(\left.\left(\mathbb{Z}^{n}\ltimes\overline{\mathbb{Z}}^{n}\right)\right|_{\overline{\mathbb{Z}_{\geq}}^{n}\backslash\left\{\infty^{n}\right\}}\right)$ containing the ideal $C^{\ast}\left(\left.\left(\mathbb{Z}^{n}\ltimes\overline{\mathbb{Z}}^{n}\right)\right|_{\mathbb{Z}_{\geq}^{n}}\right)$, the quotient map $\sigma_{n}$ induces a well-defined quotient map $\tau_{n}$ in the short exact sequence

[TABLE]

4 Stable ranks of quantum spaces

In his seminal paper [21], Rieffel introduced and popularized the notions of topological stable rank $\operatorname{tsr}\left(\mathcal{A}\right)$ and connected stable rank $\operatorname{csr}\left(\mathcal{A}\right)$ of a C*-algebra $\mathcal{A}$, which are useful tools in the study of cancellation problems for finitely generated projective modules. Later, Herman and Vaserstein [13] showed that for C*-algebras $\mathcal{A}$, Rieffel’s topological stable rank coincides with the Bass stable rank used in algebraic K-theory. So we will denote $\operatorname{tsr}\left(\mathcal{A}\right)$ simply as $\operatorname{sr}\left(\mathcal{A}\right)$ in our discussion.

In this section, we review an estimate of the stable ranks of the Toeplitz algebras $\mathcal{T}^{\otimes n}$ and quantum spheres $C\left(\mathbb{S}_{H}^{2n-1}\right)$, which will be used in our study of their finitely generated projective modules. For the case of $n=1$, it is known [21] that $\operatorname{sr}\left(\mathcal{T}\right)=\operatorname{csr}\left(C\left(\mathbb{T}\right)\right)=2$.

As an illustration of the groupoid approach to C*-algebras, we first establish some composition sequence structure for $\mathcal{T}^{\otimes n}$ and $C\left(\mathbb{S}_{H}^{2n-1}\right)$, which leads to an easy estimate of their stable ranks.

Proposition 1. There is a finite composition sequence of closed ideals

[TABLE]

such that $\mathcal{T}^{\otimes n}/\mathcal{I}_{0}\cong C\left(\mathbb{S}_{H}^{2n-1}\right)$, and for $0\leq j\leq n$,

[TABLE]

where $\mathbb{T}^{0}$ and $\mathbb{Z}_{\geq}^{0}$ denote a singleton.

Proof. For $0\leq j\leq n$, let $X_{j}$ be the set consisting of $z\in\overline{\mathbb{Z}_{\geq}}^{n}$ with exactly $j$ of the components $z_{1},z_{2},...,z_{n}$ being equal to $\infty$, and hence $X_{n}=\left\{\infty^{n}\right\}$. Then the sets

[TABLE]

are open invariant subsets of the unit space $\overline{\mathbb{Z}_{\geq}}^{n}$ of $\mathfrak{T}_{n}$ with

[TABLE]

which determines an increasing chain of closed ideals $\mathcal{I}_{0}\vartriangleleft\mathcal{I}_{1}\vartriangleleft\cdots\vartriangleleft\mathcal{I}_{n}$ of $C^{\ast}\left(\mathfrak{T}_{n}\right)$ defined by

[TABLE]

Note that $Y_{j}\backslash Y_{j-1}=X_{j}$ with $Y_{-1}:=\emptyset$ is a disjoint union of $\frac{n!}{j!\left(n-j\right)!}$ copies of $\mathbb{Z}_{\geq}^{n-j}\times\left\{\infty^{j}\right\}$ each of which is gotten from one of the $\frac{n!}{j!\left(n-j\right)!}$ possible selections of exactly $j$ of the $n$ components of $\overline{\mathbb{Z}_{\geq}}^{n}$. With each such copy of $\mathbb{Z}_{\geq}^{n-j}\times\left\{\infty^{j}\right\}$ clearly a closed invariant subset of $Y_{j}\backslash Y_{j-1}$, these $\frac{n!}{j!\left(n-j\right)!}$ copies of $\mathbb{Z}_{\geq}^{n-j}\times\left\{\infty^{j}\right\}$ are open invariant subsets of $Y_{j}\backslash Y_{j-1}$, and hence

[TABLE]

[TABLE]

Thus with $\mathcal{I}_{j}=C^{\ast}\left(\left.\mathfrak{T}_{n}\right|_{Y_{j}}\right)$ and $\mathcal{I}_{j-1}=C^{\ast}\left(\left.\mathfrak{T}_{n}\right|_{Y_{j-1}}\right)$, we get

[TABLE]

$\square$

Corollary 1. There is a finite composition sequence of closed ideals

[TABLE]

such that for $1\leq j\leq n$,

[TABLE]

Proof. With $\mathcal{I}_{0}=\mathcal{K}\left(\ell^{2}\left(\mathbb{Z}_{\geq}^{n}\right)\right)$ and hence $C^{\ast}\left(\mathfrak{T}_{n}\right)/\mathcal{I}_{0}\cong C\left(\mathbb{S}_{H}^{2n-1}\right)$, we simply take $\mathcal{J}_{j}:=\mathcal{I}_{j}/\mathcal{I}_{0}$. $\square$

The above composition sequences lead to the straightforward estimates

[TABLE]

and

[TABLE]

for all $n\geq 1$, based on the general rules established in [21] that (i) $\operatorname{sr}\left(\mathcal{A}\otimes\mathcal{K}\right)=\min\left\{2,\operatorname{sr}\left(\mathcal{A}\right)\right\}$, (ii) for any closed ideal $\mathcal{I}$ of a C*-algebra $\mathcal{A}$,

[TABLE]

and (iii) $\operatorname{sr}\left(C\left(X\right)\right)=\left\lfloor\frac{n}{2}\right\rfloor+1$ for any $n$-dimensional CW-complex $X$, and the rule [25, 16, 17] that for any closed ideal $\mathcal{I}$ of a C*-algebra $\mathcal{A}$, (iv) $\operatorname{csr}\left(\mathcal{A}\otimes\mathcal{K}\right)\leq 2$ (with $\operatorname{csr}\left(\mathcal{K}\right)=1$) and (v)

[TABLE]

Indeed, for $n>1$, applying (i)-(ii) and (iv)-(v) to the short exact sequences

[TABLE]

inductively for $j$ increasing from $1$ to $n-1$, starting with the exact sequence

[TABLE]

for $j=1$, we get $\operatorname{csr}\left(\mathcal{I}_{j}\right),\operatorname{sr}\left(\mathcal{I}_{j}\right)\leq 2$ for all $1\leq j\leq n-1$. In particular, $\operatorname{csr}\left(\mathcal{I}_{n-1}\right),\operatorname{sr}\left(\mathcal{I}_{n-1}\right)\leq 2$, which is also valid for $n=1$ since $\mathcal{I}_{0}\cong\mathcal{K}\left(\ell^{2}\left(\mathbb{Z}_{\geq}^{n}\right)\right)$. Then with $\operatorname{csr}\left(C\left(\mathbb{T}^{n}\right)\right)\leq\left\lfloor\frac{n+1}{2}\right\rfloor+1$ by homotopy theory [33], we get $\operatorname{csr}\left(\mathcal{T}^{\otimes n}\right)\leq\left\lfloor\frac{n+1}{2}\right\rfloor+1$ and

[TABLE]

by further applying (ii)-(iii) and (v) to the short exact sequence

[TABLE]

Similar argument yields $\operatorname{csr}\left(C\left(\mathbb{S}_{H}^{2n-1}\right)\right)\leq\left\lfloor\frac{n+1}{2}\right\rfloor+1$ and $\left\lfloor\frac{n}{2}\right\rfloor+1\leq\operatorname{sr}\left(C\left(\mathbb{S}_{H}^{2n-1}\right)\right)\leq\left\lfloor\frac{n+1}{2}\right\rfloor+1$, with the inequality $\operatorname{sr}\left(C\left(\mathbb{S}_{H}^{2n-1}\right)\right)\leq\operatorname{sr}\left(\mathcal{T}^{\otimes n}\right)$ obviously valid by (ii). Also $\operatorname{csr}\left(\mathcal{T}^{\otimes n}\right)\leq\operatorname{csr}\left(C\left(\mathbb{S}_{H}^{2n-1}\right)\right)$ by (iv)-(v).

Such an estimate determining $\operatorname{sr}\left(\mathcal{T}^{\otimes n}\right)$ sharply for even $n$ and up to an error of $1$ for odd $n>1$ as stated above was first obtained by G. Nagy in [16] and then sharpened to the exact value

[TABLE]

for general $n>1$ by Nistor in [17] which also gives $\operatorname{csr}\left(\mathcal{T}^{\otimes n}\right)\leq\left\lfloor\frac{n+1}{2}\right\rfloor+1$. We summarize these results as follows.

Proposition 2. For all $n>1$,

[TABLE]

and

[TABLE]

Corollary 2. For any $n>1$ and any $k\geq\left\lfloor\frac{n}{2}\right\rfloor+3$, the topological group $GL_{k}\left(\mathcal{T}^{\otimes n}\right)$ is connected.

Proof. By the Künneth formula [3] for $K$-groups, we get $K_{1}\left(\mathcal{T}^{\otimes n}\right)=0$ since $K_{1}\left(\mathcal{T}\right)=0$ is well known. So by the theorem [21] that $K_{1}\left(\mathcal{A}\right)\cong GL_{k}\left(\mathcal{A}\right)/GL_{k}^{0}\left(\mathcal{A}\right)$ for any unital C*-algebra $\mathcal{A}$ with $k\geq\operatorname{sr}\left(\mathcal{A}\right)+2$, we get $GL_{k}\left(\mathcal{T}^{\otimes n}\right)=GL_{k}^{0}\left(\mathcal{T}^{\otimes n}\right)$ for any $k\geq\left\lfloor\frac{n}{2}\right\rfloor+3\geq\operatorname{sr}\left(\mathcal{T}^{\otimes n}\right)+2$.

$\square$

Note that the above statement holds for the case of $n=1$, since $GL_{k}\left(\mathcal{T}\right)$ is connected for all $k\geq 1$ in the case of $n=1$ by the index theory of Toeplitz operators for the unit disk $\mathbb{D}$.

5 Projective modules over $\mathcal{T}^{\otimes n}$

Before proceeding to study finitely generated projective modules over $\mathcal{T}^{\otimes n}$, we now point out a structure of $\mathcal{T}^{\otimes n}$ which facilitates some inductive procedures for the study of such modules.

For all $n\in\mathbb{N}$, the topological groupoid $\left.\mathfrak{T}_{n}\right|_{\overline{\mathbb{Z}_{\geq}}^{n-1}\times\left\{\infty\right\}}$ is isomorphic to the product topological groupoid $\mathfrak{T}_{n-1}\times\mathbb{Z}$, while the topological groupoid $\left.\mathfrak{T}_{n}\right|_{\overline{\mathbb{Z}_{\geq}}^{n-1}\times\mathbb{Z}_{\geq}}$ is isomorphic to the product topological groupoid $\mathfrak{T}_{n-1}\times\left.\left(\mathbb{Z}\ltimes\mathbb{Z}\right)\right|_{\mathbb{Z}_{\geq}}$, where the closed subset $\overline{\mathbb{Z}_{\geq}}^{n-1}\times\left\{\infty\right\}$ and its open complement $\overline{\mathbb{Z}_{\geq}}^{n-1}\times\mathbb{Z}_{\geq}$ in the unit space $\overline{\mathbb{Z}_{\geq}}^{n}$ of $\mathfrak{T}_{n}$ are invariant. (Here it is understood that when $n-1=0$, the first factor $\overline{\mathbb{Z}_{\geq}}^{n-1}$ is dropped.) Hence we get the short exact sequence of C*-algebras

[TABLE]

with $\mathcal{T}^{\otimes 0}:=\mathbb{C}$. Furthermore the quotient maps $\kappa_{n}$ for $n\in\mathbb{N}$ resulting from a groupoid restriction satisfy the commuting diagram

[TABLE]

where $\equiv$ stands for a canonical isomorphism and $\sigma_{0}:=\operatorname{id}_{\mathbb{C}}$.

To classify the isomorphism classes of finitely generated projective $\mathcal{T}^{\otimes n}$-modules $E$ or equivalently the equivalence classes of idempotents $P\in M_{\infty}\left(\mathcal{T}^{\otimes n}\right)$ over $\mathcal{T}^{\otimes n}$, we first define the rank of (the class of) $E$ or $P$ as the classical rank of (the isomorphism class of) the vector bundle corresponding to (the class of) the $C\left(\mathbb{T}^{n}\right)$-module $C\left(\mathbb{T}^{n}\right)\otimes_{\mathcal{T}^{\otimes n}}E$ or the projection $\sigma_{n}\left(P\right)$ over $C\left(\mathbb{T}^{n}\right)$.

The set of equivalence classes of idempotents $P\in M_{\infty}\left(\mathcal{T}^{\otimes n}\right)$ equipped with the binary operation $\boxplus$ becomes an abelian graded monoid

[TABLE]

where $\mathfrak{P}_{m}\left(\mathcal{T}^{\otimes n}\right)$ is the set of all (equivalence classes of) idempotents over $\mathcal{T}^{\otimes n}$ of rank $m$, and

[TABLE]

for $m,l\geq 0$. Clearly $\mathfrak{P}_{0}\left(\mathcal{T}^{\otimes n}\right)$ is a submonoid of $\mathfrak{P}\left(\mathcal{T}^{\otimes n}\right)$.

Next we define a submonoid of $\mathfrak{P}\left(\mathcal{T}^{\otimes n}\right)$ generated by “standard” type of idempotents, which turns out to contain (equivalence classes of) all idempotents of sufficiently high ranks, and then classify its elements.

Note that each permutation $\Theta$ on $\left\{1,2,...,n\right\}$ induces canonically a C*-algebra automorphism, still denoted as $\Theta$ by abuse of notation, on $\mathcal{T}^{\otimes n}$ by permuting the indices of the factors in $a_{1}\otimes a_{2}\otimes\cdots\otimes a_{n}\in\mathcal{T}^{\otimes n}$ for $a_{i}\in\mathcal{T}$. A permutation $\Theta$ on $\left\{1,2,...,n\right\}$ is called a $\left(j,n-j\right)$-shuffle on $\left\{1,2,...,n\right\}$ if $\Theta\left(1\right)<\Theta\left(2\right)<\cdots<\Theta\left(j\right)$ and $\Theta\left(j+1\right)<\Theta\left(j+2\right)<\cdots<\Theta\left(n\right)$.

Some basic projections over $\mathcal{T}^{\otimes n}$ are given by $\Theta\left(P_{j,l}\right)$ where

[TABLE]

for $l\geq 0$ and $0\leq j\leq n$ (in particular, $P_{n,m}\equiv\boxplus^{m}\left(\otimes^{n}I\right)\equiv\boxplus^{m}\tilde{I}$ for the unit $\tilde{I}$ of $\mathcal{T}^{\otimes n}$), and $\Theta$ is (the automorphism defined by) a $\left(j,n-j\right)$-shuffle on $\left\{1,2,...,n\right\}$. Note that $\Theta\left(P_{j,l}\right)=\Theta\left(\boxplus^{l}P_{j,1}\right)=\boxplus^{l}\Theta\left(P_{j,1}\right)$,

[TABLE]

and $\left(\otimes^{j}I\right)\otimes\left(\otimes^{n-j-1}P_{1}\right)\otimes P_{l}\sim P_{j,l}$ over $\mathcal{T}^{\otimes n}$ since $P_{l}\sim\boxplus^{l}P_{1}$ over $\mathcal{K}^{+}\subset\mathcal{T}$. Furthermore

[TABLE]

and hence $\Theta\left(P_{j,l}\right)\in\mathfrak{P}_{0}\left(\mathcal{T}^{\otimes n}\right)$ if $j<n$ and $\Theta\left(P_{n,l}\right)=P_{n,l}\in\mathfrak{P}_{l}\left(\mathcal{T}^{\otimes n}\right)$, where $1\in C\left(\mathbb{T}^{n}\right)$ is the constant function $1$ on $\mathbb{T}^{n}$. So the set $\mathfrak{P}_{0}^{\prime}\left(\mathcal{T}^{\otimes n}\right)\subset\mathfrak{P}_{0}\left(\mathcal{T}^{\otimes n}\right)$ consisting of (the equivalence classes of) all possible $\boxplus$-sums of $\Theta\left(P_{j,l}\right)$ with $l\geq 0$ and $\Theta$ a $\left(j,n-j\right)$-shuffle on $\left\{1,2,...,n\right\}$ for $0\leq j\leq n-1$ is a submonoid of $\mathfrak{P}_{0}\left(\mathcal{T}^{\otimes n}\right)$. For $m\geq 1$, we define a singleton

[TABLE]

where $\tilde{I}$ denotes the identity element of $\mathcal{T}^{\otimes n}$. Clearly $\sqcup_{m=1}^{\infty}\mathfrak{P}_{m}^{\prime}\left(\mathcal{T}^{\otimes n}\right)$ is also a submonoid of $\mathfrak{P}\left(\mathcal{T}^{\otimes n}\right)$.

We define a partial ordering $\prec$ on the collection

[TABLE]

by the condition that $\left(j^{\prime},\Theta^{\prime}\right)\prec\left(j,\Theta\right)$ if and only if $\Theta\left(\left\{1,2,...,j\right\}\right)\supsetneqq\Theta^{\prime}\left(\left\{1,2,...,j^{\prime}\right\}\right)$ (and hence $j>j^{\prime}$). Here $\left\{1,2,...,0\right\}\equiv\emptyset$ is understood. Note that $\operatorname{id}_{\left\{1,2,...,n\right\}}$ is a $\left(j,n-j\right)$-shuffle for every $j$, and $\left(n,\operatorname{id}_{\left\{1,2,...,n\right\}}\right)$ is the greatest element while $\left(0,\operatorname{id}_{\left\{1,2,...,n\right\}}\right)$ is the smallest element in $\Omega$ with respect to $\prec$.

Proposition 3. $\mathfrak{P}^{\prime}\left(\mathcal{T}^{\otimes n}\right)=\sqcup_{m=0}^{\infty}\mathfrak{P}_{m}^{\prime}\left(\mathcal{T}^{\otimes n}\right)$ is a graded submonoid of $\mathfrak{P}\left(\mathcal{T}^{\otimes n}\right)$ and its monoid structure is explicitly determined by that for any $l,l^{\prime}>0$ and any $\left(j^{\prime},\Theta^{\prime}\right)\prec\left(j,\Theta\right)$ in $\Omega$,

[TABLE]

Proof. Note that since $\mathfrak{P}_{0}^{\prime}\left(\mathcal{T}^{\otimes n}\right)$ and $\boxplus_{m=1}^{\infty}\mathfrak{P}_{m}^{\prime}\left(\mathcal{T}^{\otimes n}\right)$ are submonoids of $\mathfrak{P}\left(\mathcal{T}^{\otimes n}\right)$, the set $\mathfrak{P}^{\prime}\left(\mathcal{T}^{\otimes n}\right)$ is a submonoid if $\Theta\left(P_{n,m}\right)\boxplus\Theta^{\prime}\left(P_{j^{\prime},l^{\prime}}\right)\sim\Theta\left(P_{n,m}\right)$ holds for all $m>0$ and all $\Theta^{\prime}\left(P_{j^{\prime},l^{\prime}}\right)$ with $j^{\prime}\leq n-1$. Since $\left(n,\operatorname{id}_{\left\{1,2,...,n\right\}}\right)$ is the greatest element in $\Omega$, it remains to show that $\Theta\left(P_{j,l}\right)\boxplus\Theta^{\prime}\left(P_{j^{\prime},l^{\prime}}\right)\sim\Theta\left(P_{j,l}\right)$ for $n\geq j>j^{\prime}\geq 0$ with $\Theta\left(\left\{1,2,...,j\right\}\right)\supset\Theta^{\prime}\left(\left\{1,2,...,j^{\prime}\right\}\right)$ and $l,l^{\prime}>0$.

Note that for $\Theta\left(\left\{1,2,...,j\right\}\right)\supset\Theta^{\prime}\left(\left\{1,2,...,j^{\prime}\right\}\right)$, there exists a permutation $\Theta^{\prime\prime}$ (not necessarily a shuffle) on $\left\{1,2,...,n\right\}$ such that $\Theta^{\prime\prime}\left(\Theta\left(P_{j,l}\right)\right)=P_{j,l}$ and $\Theta^{\prime\prime}\left(\Theta^{\prime}\left(P_{j^{\prime},l^{\prime}}\right)\right)=P_{j^{\prime},l^{\prime}}$. (In fact, one can find a permutation $\Theta^{\prime\prime}$ such that $\Theta^{\prime\prime}\Theta$ fixes each of $j+1,..,n$, and $\Theta^{\prime\prime}\Theta^{\prime}$ is each of $1,2,...,j^{\prime}$.) So it suffices to prove that

[TABLE]

whenever $j>j^{\prime}$ and $l,l^{\prime}>0$. Furthermore since $P_{j,l}=\boxplus^{l}P_{j,1}$, we only need to show that $P_{j,1}\boxplus P_{j^{\prime},1}\sim P_{j,1}$ for $j>j^{\prime}$.

Note that $U\left(P_{1}\boxplus I\right)U^{\ast}=0\boxplus I$ in $M_{2}\left(\mathcal{T}\right)$ for the unitary

[TABLE]

where $\mathcal{S}\in\mathcal{T}$ is the (forward) unilateral shift on $\ell^{2}\left(\mathbb{Z}_{\geq}\right)$. So

[TABLE]

[TABLE]

Thus by iteration of this result, we can “expand” $P_{j,1}$ to get for any $0\leq k<j$,

[TABLE]

and hence

[TABLE]

$\square$

For each $\left(j,\Theta\right)\in\Omega$, let $X_{\Theta}\subset\overline{\mathbb{Z}_{\geq}}^{n}$ be the invariant closed subset of the unit space of $\mathfrak{T}_{n}$ consisting of $z\in\overline{\mathbb{Z}_{\geq}}^{n}$ with $z_{k}=\infty$ for all $k\in\Theta\left(\left\{1,2,...,j\right\}\right)$, and let

[TABLE]

be the canonical quotient map, where the isomorphism implicitly involves a rearrangement of factors by the inverse permutation $\Theta^{-1}$. Here as before, $\mathbb{T}^{0}$ is a singleton. Defining $\rho_{\left(j,\Theta\right)}\left(P\right)$ for an idempotent $P$ over $C^{\ast}\left(\mathfrak{T}_{n}\right)$ as the rank of the projection operator $\sigma_{\left(j,\Theta\right)}\left(P\right)\left(t\right)\in\mathcal{B}\left(\ell^{2}\left(\mathbb{Z}_{\geq}^{n-j}\right)\right)$ for any $t\in\mathbb{T}^{j}$, which depends only on the equivalence class of $P$, we get a well-defined monoid homomorphism

[TABLE]

A (finite) $\boxplus$-sum of (the equivalence classes of) projections $\Theta\left(P_{j,l}\right)$ indexed by some $\left(j,\Theta\right)\in\Omega$ that are mutually unrelated by $\prec$ with $l\equiv l_{\left(j,\Theta\right)}>0$ depending on $\left(j,\Theta\right)$ is called a reduced $\boxplus$-sum of standard projections over $\mathcal{T}^{\otimes n}$. It is understood that an “empty” $\boxplus$-sum represents the zero projection and is a reduced $\boxplus$-sum. Two reduced $\boxplus$-sums are called different when they have different sets of (mutually $\prec$-unrelated) indices $\left(j,\Theta\right)\in\Omega$ or have different weight functions $l$ of $\left(j,\Theta\right)$. We are going to show that different reduced $\boxplus$-sums are inequivalent projections. Clearly each projection $\Theta\left(P_{j,l}\right)$ with $\left(j,\Theta\right)\in\Omega$ and $l>0$ is a reduced $\boxplus$-sum.

Theorem 1. The submonoid $\mathfrak{P}^{\prime}\left(\mathcal{T}^{\otimes n}\right)=\sqcup_{m=0}^{\infty}\mathfrak{P}_{m}^{\prime}\left(\mathcal{T}^{\otimes n}\right)$ of $\mathfrak{P}\left(\mathcal{T}^{\otimes n}\right)$ consists exactly of reduced $\boxplus$-sums of standard projections over $\mathcal{T}^{\otimes n}$, and different reduced $\boxplus$-sums are mutually inequivalent projections. Furthermore the monoid homomorphism

[TABLE]

is injective, with $\rho_{\left(j,\Theta\right)}\left(\Theta\left(P_{j,l}\right)\right)=l\in\mathbb{N}$.

Proof. By definition, $\mathfrak{P}^{\prime}\left(\mathcal{T}^{\otimes n}\right)$ consists of $\boxplus$-sums of (the equivalence classes of) projections $\Theta\left(P_{j,l}\right)$ with $\left(j,\Theta\right)\in\Omega$ and $l>0$. Since $\Theta\left(P_{j,l}\right)+\Theta\left(P_{j,l^{\prime}}\right)\sim\Theta\left(P_{j,l+l^{\prime}}\right)$, we only need to consider in the following those $\boxplus$-sums, in which all summands $\Theta\left(P_{j,l}\right)$ are indexed by distinct $\left(j,\Theta\right)\in\Omega$ with $l$ depending on $\left(j,\Theta\right)$. For any such a $\boxplus$-sum, using the property that $\Theta\left(P_{j,l}\right)\boxplus\Theta^{\prime}\left(P_{j^{\prime},l^{\prime}}\right)\sim\Theta\left(P_{j,l}\right)$ for any $\left(j^{\prime},\Theta^{\prime}\right)\prec\left(j,\Theta\right)$, we can remove one by one those $\boxplus$-summands $\Theta^{\prime}\left(P_{j^{\prime},l^{\prime}}\right)$ with $\left(j^{\prime},\Theta^{\prime}\right)$ dominated by the index of another summand, without changing the equivalence class, until we reach a $\boxplus$-sum of $\Theta\left(P_{j,l}\right)$ with $\left(j,\Theta\right)\in\Omega$ mutually unrelated by $\prec$, i.e. a reduced $\boxplus$-sum. So $\mathfrak{P}^{\prime}\left(\mathcal{T}^{\otimes n}\right)$ consists of the reduced $\boxplus$-sums.

Note that for $\left(j,\Theta\right)\in\Omega$ and $l>0$,

[TABLE]

and hence $\rho_{\left(j,\Theta\right)}\left(\Theta\left(P_{j,l}\right)\right)=l\in\mathbb{N}$ the operator rank of $\boxplus^{l}\left(\otimes^{n-j}P_{1}\right)\in\mathcal{B}\left(\oplus^{l}\ell^{2}\left(\mathbb{Z}_{\geq}^{n-j}\right)\right)$. But for $\left(j^{\prime},\Theta^{\prime}\right)\neq\left(j,\Theta\right)$,

[TABLE]

because either $\sigma_{\left(j,\Theta\right)}\left(\Theta^{\prime}\left(P_{j^{\prime},l^{\prime}}\right)\right)=0$ when $\Theta\left(\left\{1,2,...,j\right\}\right)\backslash\Theta^{\prime}\left(\left\{1,2,...,j^{\prime}\right\}\right)\neq\emptyset$, or $\sigma_{\left(j,\Theta\right)}\left(\Theta^{\prime}\left(P_{j^{\prime},l^{\prime}}\right)\right)$ is an infinite-dimensional projection when $\Theta^{\prime}\left(\left\{1,2,...,j^{\prime}\right\}\right)\supset\Theta\left(\left\{1,2,...,j\right\}\right)$ (but $\Theta\left(\left\{1,2,...,j\right\}\right)\neq\Theta^{\prime}\left(\left\{1,2,...,j^{\prime}\right\}\right)$ since $\left(j^{\prime},\Theta^{\prime}\right)\neq\left(j,\Theta\right)$), i.e. when $\left(j,\Theta\right)\prec\left(j^{\prime},\Theta^{\prime}\right)$.

For a reduced $\boxplus$-sum $P$ of $\Theta^{\prime}\left(P_{j^{\prime},l^{\prime}}\right)$ indexed by $\left(j^{\prime},\Theta^{\prime}\right)$ in some subset $A\subset\Omega$, the $\left(j,\Theta\right)$-component of $\rho\left(P\right)$ is

[TABLE]

for any $\left(j,\Theta\right)\in\Omega$, since if $\left(j,\Theta\right)\in A$ then $\left(j,\Theta\right)$ is $\prec$-unrelated to any other $\left(j^{\prime},\Theta^{\prime}\right)\in A$. So $\rho\left(P\right)$ completely determines the summands of a reduced $\boxplus$-sum $P$, namely, P\is the $\boxplus$-sum of exactly those $\Theta\left(P_{j,l}\right)$ with $l$ equal to the $\left(j,\Theta\right)$-component of $\rho\left(P\right)$ that is a strictly positive integer. Since $\mathfrak{P}^{\prime}\left(\mathcal{T}^{\otimes n}\right)$ consists of reduced $\boxplus$-sums, this also shows that the clearly well-defined monoid homomorphism $\rho$ is injective.

Thus if $P\sim P^{\prime}$ for two reduced $\boxplus$-sums $P$ and $P^{\prime}$ and hence $\rho\left(P\right)=\rho\left(P^{\prime}\right)$, then the summands of $P$ and $P^{\prime}$ are exactly the same, i.e. $P$ and $P^{\prime}$ are the same reduced $\boxplus$-sum. So different reduced $\boxplus$-sums are mutually inequivalent projections.

$\square$

Proposition 4. $\mathfrak{P}\left(\mathcal{T}\right)=\mathfrak{P}^{\prime}\left(\mathcal{T}\right)$. More concretely,

[TABLE]

where $\overline{\mathbf{Z}_{\geq}}^{2}$ is equipped with the canonical monoid structure.

Proof. It suffices to show that any element of $\mathfrak{P}_{0}\left(\mathcal{T}\right)\equiv\mathfrak{P}_{0}\left(\mathcal{T}^{\otimes 1}\right)$ is of the form $P_{0,l}$ (realized as $\left(0,l\right)\in\overline{\mathbf{Z}_{\geq}}^{2}$) and any element of $\mathfrak{P}_{m}\left(\mathcal{T}\right)\equiv\mathfrak{P}_{m}\left(\mathcal{T}^{\otimes 1}\right)$ for $m\in\mathbb{N}$ is of the form $P_{1,m}$ (realized as $\left(m,\infty\right)\in\overline{\mathbf{Z}_{\geq}}^{2}$).

The argument sketched below is similar to one used in [29].

Since any complex vector bundle over $\mathbb{T}$ is trivial, any idempotent over $C\left(\mathbb{T}\right)$ is equivalent to the standard projection $1\otimes P_{m}\in C\left(\mathbb{T}\right)\otimes M_{\infty}\left(\mathbb{C}\right)$ for some $m\in\mathbb{Z}_{\geq}$. So for any idempotent $P\in M_{\infty}\left(\mathcal{T}\right)$ over $\mathcal{T}$, there is some $U\in GL_{\infty}\left(C\left(\mathbb{T}\right)\right)$ such that

[TABLE]

for some $m\in\mathbb{Z}_{\geq}$ where $I$ is the identity of $\mathcal{K}^{+}\subset\mathcal{T}$, and hence $VPV^{-1}-\boxplus^{m}I\in M_{\infty}\left(\mathcal{K}\right)$ for any lift $V\in GL_{\infty}\left(\mathcal{T}\right)$ (which exists) of $U\boxplus U^{-1}\in GL_{\infty}^{0}\left(C\left(\mathbb{T}\right)\right)$ along $\sigma$. Replacing $P$ by the equivalent $VPV^{-1}$, we may assume that $P\in\left(\boxplus^{m}I\right)+M_{k-1}\left(\mathcal{K}\right)\subset M_{k-1}\left(\mathcal{K}^{+}\right)$ for some large $k\geq m+1$. Now since $M_{\infty}\left(\mathbb{C}\right)$ is dense in $\mathcal{K}$, there is an idempotent $Q\in\left(\boxplus^{m}I\right)+M_{k-1}\left(M_{N}\left(\mathbb{C}\right)\right)$ sufficiently close to and hence equivalent to $P$ for some large $N$. So replacing $P$ by $Q$, we may assume that $K:=P-\boxplus^{m}I\in M_{k-1}\left(M_{N}\left(\mathbb{C}\right)\right)$.

Rearranging the entries of $P\equiv K+\boxplus^{m}I\in M_{k-1}\left(\mathcal{T}\right)\subset M_{k}\left(\mathcal{T}\right)$ via conjugation by the unitary

[TABLE]

we get

[TABLE]

for some $R\in M_{\left(k-1\right)N}\left(\mathbb{C}\right)\subset\mathcal{K}\subset\mathcal{T}$ which must be an idempotent. Since any idempotent in $\mathcal{K}$ is equivalent over $\mathcal{K}^{+}$ to a standard projection $P_{l}$, we get

[TABLE]

for some $l\in\mathbb{Z}_{\geq}$.

If $m=0$, then clearly $P\sim P_{l}$. Since it is well known that $P_{l}$ and $\boxplus^{l}P_{1}\equiv P_{0,l}$ are equivalent over $\mathcal{K}^{+}$ and hence over $\mathcal{T}\supset\mathcal{K}^{+}$, we get $P\sim P_{0,l}$.

If $m\in\mathbb{N}$, then we can rearrange entries via conjugation by the unitary

[TABLE]

to get

[TABLE]

$\square$

Theorem 2. For $n>1$ and $m>0$, if $\mathfrak{P}_{m}\left(\mathcal{T}^{\otimes n-1}\right)=\mathfrak{P}_{m}^{\prime}\left(\mathcal{T}^{\otimes n-1}\right)\equiv\left\{\boxplus^{m}\left(\otimes^{n-1}I\right)\right\}$ and $GL_{m}\left(\mathcal{T}^{\otimes n-1}\right)$ is connected, then $\mathfrak{P}_{m}\left(\mathcal{T}^{\otimes n}\right)=\mathfrak{P}_{m}^{\prime}\left(\mathcal{T}^{\otimes n}\right)$.

Proof. In this proof, we use $I$ and $\tilde{I}$ to denote respectively the identity elements of $\mathcal{T}^{\otimes n-1}$ and $\mathcal{T}^{\otimes n}$.

Let $P\in\mathfrak{P}_{m}\left(\mathcal{T}^{\otimes n}\right)$. The idempotent $\kappa_{n}\left(P\right)$ over $\mathcal{T}^{\otimes n-1}\otimes C\left(\mathbb{T}\right)$ satisfies that for any $z\in\mathbb{T}$,

[TABLE]

which is of rank $m$ pointwise, and hence

[TABLE]

i.e. $\kappa_{n}\left(P\right)\left(z\right)\sim\boxplus^{m}I$ over $\mathcal{T}^{\otimes n-1}$. In particular, there is a continuous idempotent-valued path $\gamma:\left[0,1\right]\rightarrow M_{k}\left(\mathcal{T}^{\otimes n-1}\right)$ for $k$ sufficiently large going from the idempotent $\kappa_{n}\left(P\right)\left(1\right)$ to $\left(\boxplus^{m}I\right)\boxplus\left(\boxplus^{k-m}0\right)$. Clearly we may assume that $\gamma$ is locally constant at $1$, say, $\gamma\left(t\right)=\boxplus^{m}I$ for $t\geq 1/2$. The concatenation of the path $\gamma^{-1}$, the loop $\kappa_{n}\left(P\right)$, and the path $\gamma$ defines an idempotent-valued continuous loop $\Gamma:\mathbb{T}\rightarrow M_{k}\left(\mathcal{T}^{\otimes n-1}\right)$ starting and ending at $\boxplus^{m}I$ with $\Gamma\left(e^{i\theta}\right)=\left(\boxplus^{m}I\right)\boxplus\left(\boxplus^{k-m}0\right)$, say, for all $\theta\in\left[3\pi/2,2\pi\right]$ (and $\left[0,\pi/2\right]$), and is homotopic to the loop $\kappa_{n}\left(P\right)$ via idempotents, i.e. there is a path of idempotents in $M_{k}\left(\mathcal{T}^{\otimes n-1}\otimes C\left(\mathbb{T}\right)\right)$ from $\kappa_{n}\left(P\right)$ to $\Gamma$. Consequently, there is a continuous path of invertibles $U_{t}\in GL_{k}\left(\mathcal{T}^{\otimes n-1}\otimes C\left(\mathbb{T}\right)\right)$ with $U_{0}=I_{k}$ such that $U_{1}\kappa_{n}\left(P\right)U_{1}^{-1}=\Gamma$, which can be lifted along $\kappa_{n}$ to a continuous path of invertible $V_{t}\in GL_{k}\left(\mathcal{T}^{\otimes n}\right)$ with $V_{0}=I_{k}$ such that $\kappa_{n}\left(V_{1}PV_{1}^{-1}\right)=\Gamma$.

Replacing $P$ by the equivalent idempotent $V_{1}PV_{1}^{-1}$, we may now assume directly that the idempotent $\kappa_{n}\left(P\right)$ over $\mathcal{T}^{\otimes n-1}\otimes C\left(\mathbb{T}\right)$ is a continuous loop of idempotents in $M_{k}\left(\mathcal{T}^{\otimes n-1}\right)$ such that $\kappa_{n}\left(P\right)\left(e^{i\theta}\right)=\left(\boxplus^{m}I\right)\boxplus\left(\boxplus^{k-m}0\right)$ for all $\theta\in\left[3\pi/2,2\pi\right]$. So there is a continuous path

[TABLE]

with $W_{0}=I_{k}$ such that

[TABLE]

for all $\theta\in\left[0,3\pi/2\right]$. In particular,

[TABLE]

and hence $W_{3\pi/2}=W^{\prime}\boxplus W^{\prime\prime}$ for some invertibles $W^{\prime}\in GL_{m}\left(\mathcal{T}^{\otimes n-1}\right)$ and $W^{\prime\prime}\in GL_{k-m}\left(\mathcal{T}^{\otimes n-1}\right)$.

By the connectedness assumption on $GL_{m}\left(\mathcal{T}^{\otimes n-1}\right)$, there is a continuous path $\alpha:\left[3\pi/2,2\pi\right]\rightarrow GL_{m}\left(\mathcal{T}^{\otimes n-1}\right)$ with $\alpha\left(3\pi/2\right)=W^{\prime}$ and $\alpha\left(2\pi\right)=I_{m}$. Since by Künneth formula, $K_{1}\left(\mathcal{T}^{\otimes n-1}\right)=0$ and hence $GL_{N}\left(\mathcal{T}^{\otimes n-1}\right)$ is connected for $N$ sufficiently large, we may suitably increase the value of $k$ by adding diagonal $\boxplus$-summands [math] to idempotents and diagonal $\boxplus$-summands $I$ to invertibles, so that $GL_{k-m}\left(\mathcal{T}^{\otimes n-1}\right)$ is also connected and hence there is a continuous path $\beta:\left[3\pi/2,2\pi\right]\rightarrow GL_{k-m}\left(\mathcal{T}^{\otimes n-1}\right)$ with $\beta\left(3\pi/2\right)=W^{\prime\prime}$ and $\beta\left(2\pi\right)=I_{k-m}$. Now the function $\theta\mapsto W_{\theta}$ can be continuously extended to the whole interval $\left[0,2\pi\right]$ by setting

[TABLE]

for $\theta\in\left[3\pi/2,2\pi\right]$, giving rise to a well-defined continuous loop

[TABLE]

i.e. $W\in GL_{k}\left(\mathcal{T}^{\otimes n-1}\otimes C\left(\mathbb{T}\right)\right)$, satisfying

[TABLE]

So the idempotent $\kappa_{n}\left(P\right)$ over $\mathcal{T}^{\otimes n-1}\otimes C\left(\mathbb{T}\right)$ is equivalent to the idempotent $\boxplus^{m}I$.

Replacing $P$ by the equivalent idempotent $\tilde{W}\left(P\boxplus\left(\boxplus^{k}0\right)\right)\tilde{W}^{-1}$ for any fixed lifting $\tilde{W}\in GL_{2k}^{0}\left(\mathcal{T}^{\otimes n}\right)$ of $W\boxplus W^{-1}\in GL_{2k}^{0}\left(\mathcal{T}^{\otimes n-1}\otimes C\left(\mathbb{T}\right)\right)$ along $\kappa_{n}$, we may now assume that

[TABLE]

and proceed to show that $P\sim\boxplus^{m}\tilde{I}$.

Note that $P-\left(\left(\boxplus^{m}\tilde{I}\right)\boxplus\left(\boxplus^{2k-m}0\right)\right)\in M_{2k}\left(\mathcal{T}^{\otimes n-1}\otimes\mathcal{K}\right)$. Since $M_{\infty}\left(\mathbb{C}\right)$ is dense in $\mathcal{K}$, we may replace $P$ by a suitable equivalent idempotent and assume that

[TABLE]

for some $N\in\mathbb{N}$.

Rearranging the entries of $P\equiv P\boxplus 0\in M_{2k+1}\left(\mathcal{T}^{\otimes n-1}\otimes M_{N}\left(\mathbb{C}\right)\right)$ via conjugation by the unitary

[TABLE]

we get

[TABLE]

for some

[TABLE]

which must be an idempotent over $\mathcal{T}^{\otimes n-1}$.

Since $K_{0}\left(\mathcal{T}^{\otimes n-1}\right)=\mathbb{Z}$ by Künneth formula, $R\boxplus\left(\boxplus^{r}I\right)\sim\left(\boxplus^{r+\left[R\right]}I\right)$ for a sufficiently large $r\in\mathbb{N}$ where $\left[R\right]\in\mathbb{Z}$ denotes the class of $R$ in $K_{0}\left(\mathcal{T}^{\otimes n-1}\right)$. So there is an invertible $U\in GL_{d}\left(\mathcal{T}^{\otimes n-1}\right)$ for some large $d\geq\max\left\{2kN+r,r+\left[R\right]\right\}$ such that

[TABLE]

With $m>0$, we can rearrange entries via conjugation by the unitary

[TABLE]

to get

[TABLE]

where

[TABLE]

Note that $R^{\prime}$ can be interpreted as $R\boxplus\left(\boxplus^{d-2kN-r}0\right)\boxplus\left(\boxplus^{\infty}I\right)\in\mathcal{T}^{\otimes n-1}\otimes\mathcal{K}^{+}\subset\mathcal{T}^{\otimes n}$, which when conjugated by the invertible $U\equiv U\boxplus\left(\boxplus^{\infty}I\right)\in\mathcal{T}^{\otimes n-1}\otimes\mathcal{K}^{+}\subset\mathcal{T}^{\otimes n}$ becomes

[TABLE]

So we get

[TABLE]

the latter of which when conjugated by $U_{d-r-\left[R\right]}^{-1}$ yields $\tilde{I}\boxplus\left(\boxplus^{m-1}\tilde{I}\right)\boxplus\left(\boxplus^{2k+1-m}0\right)$, where $U_{d-r-\left[R\right]}$ is defined as $U_{d-r}$ by replacing $d-r$ by $d-r-\left[R\right]$. Thus we get $P\sim\left(\boxplus^{m}\tilde{I}\right)\boxplus\left(\boxplus^{2k+1-m}0\right)\equiv\boxplus^{m}\tilde{I}$.

$\square$

Corollary 3. $\mathfrak{P}_{m}\left(\mathcal{T}^{\otimes n}\right)=\mathfrak{P}_{m}^{\prime}\left(\mathcal{T}^{\otimes n}\right)\equiv\left\{\boxplus^{m}\tilde{I}\right\}$ for all $m\geq\left\lfloor\frac{n-1}{2}\right\rfloor+3$ and any $n\in\mathbb{N}$, where $\tilde{I}$ is the identity element of $\mathcal{T}^{\otimes n}$.

Proof. We prove by induction on $n\in\mathbb{N}$. For $n=1$, we already know that $\mathfrak{P}_{m}^{\prime}\left(\mathcal{T}^{\otimes n}\right)\equiv\mathfrak{P}_{m}\left(\mathcal{T}^{\otimes n}\right)$ for all $m>0$.

Now assume as the induction hypothesis that $\mathfrak{P}_{m}^{\prime}\left(\mathcal{T}^{\otimes n}\right)=\mathfrak{P}_{m}\left(\mathcal{T}^{\otimes n}\right)$ for all $m\geq\left\lfloor\frac{n-1}{2}\right\rfloor+3$ for an $n\in\mathbb{N}$.

Since we know that $GL_{m}\left(\mathcal{T}^{\otimes n}\right)$ is connected for all $m\geq\left\lfloor\frac{n}{2}\right\rfloor+3$, the above theorem implies that $\mathfrak{P}_{m}^{\prime}\left(\mathcal{T}^{\otimes n+1}\right)=\mathfrak{P}_{m}\left(\mathcal{T}^{\otimes n+1}\right)$ for all $m\geq\left\lfloor\frac{n}{2}\right\rfloor+3$.

$\square$

It remains open the problem of classification of low-rank idempotents over $\mathcal{T}^{\otimes n}$. In particular, it is not clear whether there are idempotents of non-standard (equivalence) type.

6 Projective modules over $C\left(\mathbb{S}_{H}^{2n-1}\right)$

Most of the arguments and results in the above study of projective modules over $\mathcal{T}^{\otimes n}$ can be adapted to the case of the quantum spheres $C\left(\mathbb{S}_{H}^{2n-1}\right)$.

Let $\partial_{n}:\mathcal{T}^{\otimes n}\rightarrow C\left(\mathbb{S}_{H}^{2n-1}\right)$ be the canonical quotient map by restricting the groupoid $\mathfrak{T}_{n}$ to the closed invariant set $\overline{\mathbb{Z}_{\geq}}^{n}\backslash\mathbb{Z}_{\geq}^{n}$ in its unit space.

First we note that there is a short exact sequence of C*-algebras

[TABLE]

for all $n>1$. Indeed, since $\left(\overline{\mathbb{Z}_{\geq}}^{n-1}\backslash\mathbb{Z}_{\geq}^{n-1}\right)\times\mathbb{Z}_{\geq}$ is an open invariant subset of the unit space $\overline{\mathbb{Z}_{\geq}}^{n}\backslash\mathbb{Z}_{\geq}^{n}$ of the groupoid $\mathfrak{G}_{n}\equiv\left.\left(\mathbb{Z}^{n}\ltimes\overline{\mathbb{Z}}^{n}\right)\right|_{\overline{\mathbb{Z}_{\geq}}^{n}\backslash\mathbb{Z}_{\geq}^{n}}$ with the invariant complement

[TABLE]

the groupoid C*-algebra

[TABLE]

[TABLE]

is a closed ideal of $C^{\ast}\left(\mathfrak{G}_{n}\right)=C\left(\mathbb{S}_{H}^{2n-1}\right)$ with quotient

[TABLE]

[TABLE]

So we get the above short exact sequence with $\lambda_{n}$ being the canonical map from $C^{\ast}\left(\mathfrak{G}_{n}\right)$ to its quotient $\mathcal{T}^{\otimes n-1}\otimes C\left(\mathbb{T}\right)$ resulting from restricting the groupoid $\mathfrak{G}_{n}$ to the closed invariant set $\overline{\mathbb{Z}_{\geq}}^{n-1}\times\left\{\infty\right\}$.

Clearly $\kappa_{n}=\lambda_{n}\circ\partial_{n}$. Furthermore all the quotient maps $\sigma_{\left(j,\Theta\right)}$ on $\mathcal{T}^{\otimes n}$ with $j>0$ factors through $\partial_{n}$ and induces a quotient map

[TABLE]

such that $\sigma_{\left(j,\Theta\right)}=\tau_{\left(j,\Theta\right)}\circ\partial_{n}$.

Note that the quotient maps $\lambda_{n}$ for $n\in\mathbb{N}$ satisfy the commuting diagram

[TABLE]

We define the rank of (the equivalence class of) an idempotent $Q\in M_{\infty}\left(C\left(\mathbb{S}_{H}^{2n-1}\right)\right)$ over $C\left(\mathbb{S}_{H}^{2n-1}\right)$ as the rank of the matrix value $\tau_{n}\left(Q\right)\left(z\right)\in M_{\infty}\left(\mathbb{C}\right)$ at any $z\in\mathbb{T}^{n}$ (independent of $z$ since $\mathbb{T}^{n}$ is connected). Then the set of equivalence classes of idempotents $Q\in M_{\infty}\left(C\left(\mathbb{S}_{H}^{2n-1}\right)\right)$ equipped with the binary operation $\boxplus$ becomes an abelian graded monoid

[TABLE]

where $\mathfrak{P}_{m}\left(C\left(\mathbb{S}_{H}^{2n-1}\right)\right)$ is the set of all (equivalence classes of) idempotents over $C\left(\mathbb{S}_{H}^{2n-1}\right)$ of rank $m$, with clearly

[TABLE]

for $m,l\geq 0$.

Since $\sigma_{n}=\tau_{n}\circ\partial_{n}$, the rank of an idempotent $P$ over $C\left(\mathcal{T}^{\otimes n}\right)$ equals the rank of the idempotent $\partial_{n}P$ over $C\left(\mathbb{S}_{H}^{2n-1}\right)$. We now define

[TABLE]

and the projections

[TABLE]

over $C\left(\mathbb{S}_{H}^{2n-1}\right)$. Note that $\mathfrak{P}_{m}^{\prime}\left(C\left(\mathbb{S}_{H}^{2n-1}\right)\right)=\left\{\boxplus^{m}\tilde{I}\right\}$ for $m>0$, where $\tilde{I}$ denotes the identity element of $C\left(\mathbb{S}_{H}^{2n-1}\right)$.

Also note that $Q_{0,\operatorname{id},l}=0$ for all $l$, where $\operatorname{id}\equiv\operatorname{id}_{\left\{1,2,...,n\right\}}$ is the only $\left(0,n\right)$-shuffle. The monoid homomorphism

[TABLE]

with

[TABLE]

“truncated” from $\rho$ induces a well-defined monoid homomorphism

[TABLE]

in the sense that $\rho=\rho_{\partial}\circ\partial_{n}$. Indeed for $\left(j,\Theta\right)\in\Omega_{0}$, i.e. with $j>0$, the quotient map

[TABLE]

factors through $\partial_{n}$ since the unit space $\overline{\mathbb{Z}_{\geq}^{n}}\backslash\mathbb{Z}_{\geq}^{n}$ of $\mathfrak{G}_{n}$ contains $X_{\Theta}$, and hence the map $\rho_{\left(j,\Theta\right)}$ factors through $\partial_{n}$.

We call a $\boxplus$-sum of $Q_{j,\Theta,l}$ indexed by $\prec$-unrelated $\left(j,\Theta\right)\in\Omega_{0}$ (i.e. $1\leq j\leq n$) and $l\equiv l_{\left(j,\Theta\right)}>0$ depending on $\left(j,\Theta\right)$ to be a reduced $\boxplus$-sum of standard projections over $C\left(\mathbb{S}_{H}^{2n-1}\right)$. (The degenerate empty $\boxplus$-sum [math] is taken as a reduced $\boxplus$-sum.) Two such reduced $\boxplus$-sums are called different when they have different sets of (mutually $\prec$-unrelated) indices $\left(j,\Theta\right)\in\Omega_{0}$ or have different weight functions $l$ of $\left(j,\Theta\right)$. Each $Q_{j,\Theta,l}$ with $j,l>0$ is a reduced $\boxplus$-sum of standard projections over $C\left(\mathbb{S}_{H}^{2n-1}\right)$.

Proposition 5. Different reduced $\boxplus$-sums of standard projections over $C\left(\mathbb{S}_{H}^{2n-1}\right)$ are mutually inequivalent projections over $C\left(\mathbb{S}_{H}^{2n-1}\right)$, and they form a graded submonoid

[TABLE]

of the monoid $\mathfrak{P}\left(C\left(\mathbb{S}_{H}^{2n-1}\right)\right)$, with its monoid structure explicitly determined by $Q_{j,\Theta,l}\boxplus Q_{j^{\prime},\Theta^{\prime},l^{\prime}}\sim Q_{j,\Theta,l}$ for $\left(j^{\prime},\Theta^{\prime}\right)\prec\left(j,\Theta\right)$ with $j,j^{\prime},l,l^{\prime}>0$. Furthermore the monoid homomorphism

[TABLE]

is injective.

Proof. The submonoid $\mathfrak{P}^{\prime}\left(C\left(\mathbb{S}_{H}^{2n-1}\right)\right)=\partial_{n}\left(\mathfrak{P}^{\prime}\left(C\left(\mathcal{T}^{\otimes n}\right)\right)\right)$ consists of reduced $\boxplus$-sums of $Q_{j,\Theta,l}=\partial_{n}\left(\Theta\left(P_{j,l}\right)\right)$ with $j>0$, since $Q_{0,\operatorname{id},l}=0$.

Let $\mathfrak{M}$ be the subset of $\mathfrak{P}^{\prime}\left(C\left(\mathcal{T}^{\otimes n}\right)\right)$ consisting of all reduced $\boxplus$-sums $P$ of $\Theta\left(P_{j,l}\right)$ with $j>0$. Then $\partial_{n}|_{\mathfrak{M}}:\mathfrak{M}\rightarrow\mathfrak{P}^{\prime}\left(C\left(\mathbb{S}_{H}^{2n-1}\right)\right)$ is still surjective, and $\rho_{0}|_{\mathfrak{M}}$ still factors through $\rho_{\partial}$, i.e. $\rho_{0}|_{\mathfrak{M}}=\rho_{\partial}\circ\partial_{n}|_{\mathfrak{M}}$. These imply that $\rho_{\partial}$ is injective if $\rho_{0}|_{\mathfrak{M}}$ is injective.

For any reduced $\boxplus$-sum $P\in\mathfrak{M}$ of $\Theta\left(P_{j,l}\right)$ with $j>0$, the $\left(j,\Theta\right)$-component of $\rho\left(P\right)$ is the same as that of $\rho_{0}\left(P\right)$ for all $\left(j,\Theta\right)\in\Omega_{0}$, while the only other component, namely, the $\left(0,\operatorname{id}\right)$-component of $\rho\left(P\right)$ is $\infty$ since $\rho_{\left(0,\operatorname{id}\right)}\left(\Theta\left(P_{j,l}\right)\right)=\infty$ for any $j>0$. Thus we get $\rho\left(P\right)=\left(\infty,\rho_{0}\left(P\right)\right)$ for all $P\in\mathfrak{M}$. Hence the injectivity of $\rho|_{\mathfrak{M}}$ implies the injectivity of $\rho_{0}|_{\mathfrak{M}}$ on $\mathfrak{M}$, and hence the injectivity of $\rho_{\partial}$.

Since two different reduced $\boxplus$-sums $Q,Q^{\prime}$ over $C\left(\mathbb{S}_{H}^{2n-1}\right)$ are of the form $\partial_{n}\left(P\right),\partial_{n}\left(P^{\prime}\right)$ respectively for two different reduced $\boxplus$-sums $P,P^{\prime}\in\mathfrak{M}$ over $C\left(\mathcal{T}^{\otimes n}\right)$ which are inequivalent over $C\left(\mathcal{T}^{\otimes n}\right)$ and hence $\rho_{0}\left(P\right)\neq\rho_{0}\left(P^{\prime}\right)$, we get $\rho_{\partial}\left(Q\right)\neq\rho_{\partial}\left(Q^{\prime}\right)$ showing that $Q,Q^{\prime}$ are different equivalence classes in $\mathfrak{P}^{\prime}\left(C\left(\mathbb{S}_{H}^{2n-1}\right)\right)$.

The property that $\Theta\left(P_{j,l}\right)\boxplus\Theta^{\prime}\left(P_{j^{\prime},l^{\prime}}\right)\sim\Theta\left(P_{j,l}\right)$ over $\mathcal{T}^{\otimes n}$ for $\left(j^{\prime},\Theta^{\prime}\right)\prec\left(j,\Theta\right)$ is clearly preserved under the quotient map $\partial_{n}$, i.e. $Q_{j,\Theta,l}\boxplus Q_{j^{\prime},\Theta^{\prime},l^{\prime}}\sim Q_{j,\Theta,l}$ over $C\left(\mathbb{S}_{H}^{2n-1}\right)$.

$\square$

Theorem 3. For $n>1$ and $m\in\mathbb{N}$, if $\mathfrak{P}_{m}\left(\mathcal{T}^{\otimes n-1}\right)=\mathfrak{P}_{m}^{\prime}\left(\mathcal{T}^{\otimes n-1}\right)$ and $GL_{m}\left(\mathcal{T}^{\otimes n-1}\right)$ is connected, then $\mathfrak{P}_{m}^{\prime}\left(C\left(\mathbb{S}_{H}^{2n-1}\right)\right)=\mathfrak{P}_{m}\left(C\left(\mathbb{S}_{H}^{2n-1}\right)\right)$.

Proof. Many arguments used to prove a similar theorem for $\mathcal{T}^{\otimes n}$ instead of $C\left(\mathbb{S}_{H}^{2n-1}\right)$ can be used again here with minor modifications. In this proof, $I$ and $\tilde{I}$ denote respectively the identity element of $\mathcal{T}^{\otimes n-1}$ and $C\left(\mathbb{S}_{H}^{2n-1}\right)$.

Let $P\in\mathfrak{P}_{m}\left(C\left(\mathbb{S}_{H}^{2n-1}\right)\right)$. The idempotent $\lambda_{n}\left(P\right)$ over $\mathcal{T}^{\otimes n-1}\otimes C\left(\mathbb{T}\right)$ satisfies that for any $z\in\mathbb{T}$,

[TABLE]

which is of rank $m$ pointwise, and hence

[TABLE]

i.e. $\lambda_{n}\left(P\right)\left(z\right)\sim\boxplus^{m}I$ over $\mathcal{T}^{\otimes n-1}$. As before, for some large $k$, there is an idempotent-valued continuous loop $\Gamma:\mathbb{T}\rightarrow M_{k}\left(\mathcal{T}^{\otimes n-1}\right)$ starting and ending at $\boxplus^{m}I$ with $\Gamma\left(e^{i\theta}\right)=\left(\boxplus^{m}I\right)\boxplus\left(\boxplus^{k-m}0\right)$, say, for all $\theta\in\left[3\pi/2,2\pi\right]$, and homotopic to the loop $\lambda_{n}\left(P\right)$ via idempotents. Consequently, there is a continuous path of invertibles $U_{t}\in GL_{k}\left(\mathcal{T}^{\otimes n-1}\otimes C\left(\mathbb{T}\right)\right)$ with $U_{0}=I_{k}$ such that $U_{1}\lambda_{n}\left(P\right)U_{1}^{-1}=\Gamma$, which can be lifted along $\lambda_{n}$ to a continuous path of invertible $V_{t}\in GL_{k}\left(C\left(\mathbb{S}_{H}^{2n-1}\right)\right)$ with $V_{0}=I_{k}$ such that $\lambda_{n}\left(V_{1}PV_{1}^{-1}\right)=\Gamma$.

Replacing $P$ by the equivalent idempotent $V_{1}PV_{1}^{-1}$, we may now assume directly that the idempotent $\lambda_{n}\left(P\right)$ over $\mathcal{T}^{\otimes n-1}\otimes C\left(\mathbb{T}\right)$ is a continuous loop of idempotents in $M_{k}\left(\mathcal{T}^{\otimes n-1}\right)$ such that $\lambda_{n}\left(P\right)\left(e^{i\theta}\right)=\left(\boxplus^{m}I\right)\boxplus\left(\boxplus^{k-m}0\right)$ for all $\theta\in\left[3\pi/2,2\pi\right]$. As before, by the connectedness assumption on $GL_{m}\left(\mathcal{T}^{\otimes n-1}\right)$, after suitably increasing the size $k$, we can find a well-defined continuous loop

[TABLE]

i.e. $W\in GL_{k}\left(\mathcal{T}^{\otimes n-1}\otimes C\left(\mathbb{T}\right)\right)$, satisfying

[TABLE]

So the idempotent $\lambda_{n}\left(P\right)$ over $\mathcal{T}^{\otimes n-1}\otimes C\left(\mathbb{T}\right)$ is equivalent to the idempotent $\boxplus^{m}I$.

Replacing $P$ by the equivalent idempotent $\tilde{W}\left(P\boxplus\left(\boxplus^{k}0\right)\right)\tilde{W}^{-1}$ for any fixed lifting $\tilde{W}\in GL_{2k}^{0}\left(C\left(\mathbb{S}_{H}^{2n-1}\right)\right)$ of $W\boxplus W^{-1}\in GL_{2k}^{0}\left(\mathcal{T}^{\otimes n-1}\otimes C\left(\mathbb{T}\right)\right)$ along $\lambda_{n}$, we may now assume that

[TABLE]

and proceed to show that $P\sim\boxplus^{m}\tilde{I}$ over $C\left(\mathbb{S}_{H}^{2n-1}\right)$, where we use $\tilde{I}$ to denote the identity element in $C\left(\mathbb{S}_{H}^{2n-1}\right)$ so as to distinguish it from the identity element $I$ of $\mathcal{T}^{\otimes n-1}$.

With $P-\left(\left(\boxplus^{m}\tilde{I}\right)\boxplus\left(\boxplus^{2k-m}0\right)\right)\in M_{2k}\left(C\left(\mathbb{S}_{H}^{2n-3}\right)\otimes\mathcal{K}\right)$ and $M_{\infty}\left(\mathbb{C}\right)$ dense in $\mathcal{K}$, we may replace $P$ by a suitable equivalent idempotent and assume that

[TABLE]

for some $K\in M_{2k}\left(C\left(\mathbb{S}_{H}^{2n-3}\right)\otimes M_{N}\left(\mathbb{C}\right)\right)$ and some $N\in\mathbb{N}$.

As before, by rearranging entries via conjugation, we get

[TABLE]

for some

[TABLE]

which must be an idempotent over $C\left(\mathbb{S}_{H}^{2n-3}\right)$. More precisely, we can lift $P$ to

[TABLE]

for some $\hat{K}\in M_{2k}\left(\mathcal{T}^{\otimes n-1}\otimes M_{N}\left(\mathbb{C}\right)\right)$ and conjugate it by the unitary $U_{k,N}$ over $\mathcal{T}^{\otimes n}$ to get the form $\left(\left(\boxplus^{m}I_{\mathcal{T}^{\otimes n}}\right)\boxplus\left(\boxplus^{2k-m}0\right)\right)\boxplus\hat{R}$ with $\hat{R}\in M_{2kN}\left(\mathcal{T}^{\otimes n-1}\right)$ as we did for the case of $\mathcal{T}^{\otimes n}$. Then the above $R$ is $\partial_{n}\left(\hat{R}\right)$. Note that even though $\hat{P}$ and $\hat{R}$ are not necessarily idempotents, $R$ is since it is the idempotent $P$ conjugated by the unitary $\partial_{n}\left(U_{k,N}\right)$ over $C\left(\mathbb{S}_{H}^{2n-1}\right)$.

Since $K_{0}\left(C\left(\mathbb{S}_{H}^{2n-3}\right)\right)=\mathbb{Z}$ [12], $R\boxplus\left(\boxplus^{r}\hat{I}\right)\sim\left(\boxplus^{r+\left[R\right]}\hat{I}\right)$ for a sufficiently large $r\in\mathbb{N}$ where $\left[R\right]\in\mathbb{Z}$ denotes the class of $R$ in $K_{0}\left(C\left(\mathbb{S}_{H}^{2n-3}\right)\right)$ and $\hat{I}$ is the identity element of $C\left(\mathbb{S}_{H}^{2n-3}\right)$. So there is an invertible $U\in GL_{d}\left(C\left(\mathbb{S}_{H}^{2n-3}\right)\right)$ for some large $d\geq\max\left\{2kN+r,r+\left[R\right]\right\}$ such that

[TABLE]

As before, with $m>0$, by rearranging entries via conjugation, we can get

[TABLE]

where the idempotent

[TABLE]

when conjugated by the invertible $U\equiv U\boxplus\left(\tilde{I}-\hat{I}\otimes P_{d}\right)\in\left(C\left(\mathbb{S}_{H}^{2n-3}\right)\otimes\mathcal{K}\right)^{+}$ becomes

[TABLE]

So we get

[TABLE]

the latter of which as before is equivalent to $\tilde{I}\boxplus\left(\boxplus^{m-1}\tilde{I}\right)\boxplus\left(\boxplus^{2k+1-m}0\right)$ by a further conjugation by $U_{d-r-\left[R\right]}^{-1}$. Thus $P\sim\left(\boxplus^{m}\tilde{I}\right)\boxplus\left(\boxplus^{2k+1-m}0\right)\equiv\boxplus^{m}\tilde{I}$.

$\square$

Corollary 4. $\mathfrak{P}_{m}\left(C\left(\mathbb{S}_{H}^{2n-1}\right)\right)=\mathfrak{P}_{m}^{\prime}\left(C\left(\mathbb{S}_{H}^{2n-1}\right)\right)\equiv\left\{\boxplus^{m}\tilde{I}\right\}$ for all $m\geq\left\lfloor\frac{n-1}{2}\right\rfloor+3$ and any $n\in\mathbb{N}$, where $\tilde{I}$ is the identity element of $C\left(\mathbb{S}_{H}^{2n-1}\right)$.

Proof. The case of $n=1$ is well known. For $n>1$, since $\mathfrak{P}_{m}^{\prime}\left(\mathcal{T}^{\otimes n-1}\right)=\mathfrak{P}_{m}\left(\mathcal{T}^{\otimes n-1}\right)$ for all $m\geq\left\lfloor\frac{n-2}{2}\right\rfloor+3$ and $GL_{m}\left(\mathcal{T}^{\otimes n-1}\right)$ is connected for all $m\geq\left\lfloor\frac{n-1}{2}\right\rfloor+3$, the above theorem implies that $\mathfrak{P}_{m}^{\prime}\left(C\left(\mathbb{S}_{H}^{2n-1}\right)\right)=\mathfrak{P}_{m}\left(C\left(\mathbb{S}_{H}^{2n-1}\right)\right)$ for all $m\geq\left\lfloor\frac{n-1}{2}\right\rfloor+3$.

$\square$

It is not clear whether there are (low-rank) idempotents over $C\left(\mathbb{S}_{H}^{2n-1}\right)$ of non-standard (equivalence) type and whether the cancellation law holds for them.

7 Projective modules over $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)$

In this section we study the problem of classification of finitely generated projective modules over the multipullback quantum complex projective space $\mathbb{P}^{n-1}\left(\mathcal{T}\right)$ that was introduced and studied by Hajac, Kaygun, Zieliński in [9].

In [12], $K_{0}\left(C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)\right)=\mathbb{Z}^{n}$ and $K_{1}\left(C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)\right)=0$ are computed, and $\mathbb{P}^{n-1}\left(\mathcal{T}\right)$ is shown to be a quantum quotient space of $\mathbb{S}_{H}^{2n-1}$. More precisely, the C*-algebra $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)$ is isomorphic to the invariant C*-subalgebra $\left(C\left(\mathbb{S}_{H}^{2n-1}\right)\right)^{U\left(1\right)}$ of $C\left(\mathbb{S}_{H}^{2n-1}\right)$ under the canonical diagonal $U\left(1\right)$-action on $C\left(\mathbb{S}_{H}^{2n-1}\right)\cong\mathcal{T}^{\otimes n}/\mathcal{K}^{\otimes n}$, which in the groupoid context can be implemented by the multiplication operator

[TABLE]

for $\zeta\in U\left(1\right)\equiv\mathbb{T}$ where

[TABLE]

is a groupoid character. Then $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)$ is realized as the groupoid C*-algebra $C^{\ast}\left(\left(\mathfrak{G}_{n}\right)_{0}\right)$ of the subgroupoid $\left(\mathfrak{G}_{n}\right)_{0}$ of $\mathfrak{G}_{n}$, where

[TABLE]

for $k\in\mathbb{Z}$. Furthermore, $C^{\ast}\left(\mathfrak{G}_{n}\right)$ becomes a (completion of the) graded algebra $\oplus_{k\in\mathbb{Z}}\overline{C_{c}\left(\left(\mathfrak{G}_{n}\right)_{k}\right)}$ with the component $\overline{C_{c}\left(\left(\mathfrak{G}_{n}\right)_{k}\right)}$ being the quantum line bundle $C\left(\mathbb{S}_{H}^{2n-1}\right)_{k}$ [12] of degree $k$ over the quantum space $\mathbb{P}^{n-1}\left(\mathcal{T}\right)$.

It is easy to see that the standard projections $Q_{j,\Theta,l}\equiv\partial_{n}\left(\Theta\left(P_{j,l}\right)\right)$ over $C\left(\mathbb{S}_{H}^{2n-1}\right)$ with $j,l>0$ found in the previous section lie in $M_{\infty}\left(C^{\ast}\left(\left(\mathfrak{G}_{n}\right)_{0}\right)\right)$ since $P_{j,l}=\boxplus^{l}\left(\left(\otimes^{j}I\right)\otimes\left(\otimes^{n-j}P_{1}\right)\right)$ is in $C^{\ast}\left(\left(\mathfrak{T}_{n}\right)_{0}\right)$, and hence are also projections over $C^{\ast}\left(\left(\mathfrak{G}_{n}\right)_{0}\right)\equiv C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)$. Furthermore with $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)\subset C\left(\mathbb{S}_{H}^{2n-1}\right)$, inequivalent $\boxplus$-sums of standard projections over $C\left(\mathbb{S}_{H}^{2n-1}\right)$ must be inequivalent over $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)$ as well.

Proposition 6. Different reduced $\boxplus$-sums of standard projections $Q_{j,\Theta,l}$ over $C\left(\mathbb{S}_{H}^{2n-1}\right)$ with $j,l>0$ when viewed as projections over $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)$ are mutually inequivalent over $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)$, and they form a graded submonoid

[TABLE]

of the monoid $\mathfrak{P}\left(C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)\right)$. Furthermore the monoid homomorphism

[TABLE]

inherited from $\rho_{\partial}$ is injective.

However, for $\left(j^{\prime},\Theta^{\prime}\right)\prec\left(j,\Theta\right)$ with $j,j^{\prime},l,l^{\prime}>0$, it is no longer true in general that $Q_{j,\Theta,l}\boxplus Q_{j^{\prime},\Theta^{\prime},l^{\prime}}\sim Q_{j,\Theta,l}$ over $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)$, even though $Q_{j,\Theta,l}\boxplus Q_{j^{\prime},\Theta^{\prime},l^{\prime}}\sim Q_{j,\Theta,l}$ over $C\left(\mathbb{S}_{H}^{2n-1}\right)$ since the invertible matrix over $C\left(\mathbb{S}_{H}^{2n-1}\right)$ intertwining $Q_{j,\Theta,l}\boxplus Q_{j^{\prime},\Theta^{\prime},l^{\prime}}$ and $Q_{j,\Theta,l}$ may not be replaced by one over the subalgebra $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)$ of $C\left(\mathbb{S}_{H}^{2n-1}\right)$.

In the following, we show that the standard projections $Q_{j,\operatorname{id},1}$ with $j>0$ provide a set of representatives of $K_{0}$-classes that freely generate the abelian $K_{0}$-group of $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)$.

The subgroupoid $\mathfrak{H}_{j}:=\mathfrak{G}_{j}\times\left(\mathbb{Z}^{n-j}\ltimes\mathbb{Z}_{\geq}^{n-j}\right)$ of $\mathfrak{G}_{n}$ for $1\leq j\leq n$ is the groupoid $\mathfrak{G}_{n}$ restricted to the open invariant subset $\left(\overline{\mathbb{Z}_{\geq}^{j}}\times\mathbb{Z}_{\geq}^{n-j}\right)\backslash\mathbb{Z}_{\geq}^{n}$ and inherits the grading of $\mathfrak{G}_{n}$. The grade-[math] part $\left(\mathfrak{H}_{j}\right)_{0}$ of $\mathfrak{H}_{j}$ is the groupoid $\left(\mathfrak{G}_{n}\right)_{0}$ restricted to $\left(\overline{\mathbb{Z}_{\geq}^{j}}\times\mathbb{Z}_{\geq}^{n-j}\right)\backslash\mathbb{Z}_{\geq}^{n}$, and from the increasing chain of $\left(\mathfrak{H}_{j}\right)_{0}$, we get an increasing composition sequence of closed ideals of $C^{\ast}\left(\left(\mathfrak{G}_{n}\right)_{0}\right)$ as

[TABLE]

such that with $\left(\overline{\mathbb{Z}_{\geq}^{j}}\times\mathbb{Z}_{\geq}^{n-j}\right)\backslash\left(\overline{\mathbb{Z}_{\geq}^{j-1}}\times\mathbb{Z}_{\geq}^{n-j+1}\right)=\overline{\mathbb{Z}_{\geq}^{j-1}}\times\left\{\infty\right\}\times\mathbb{Z}_{\geq}^{n-j}$,

[TABLE]

because the groupoid $\left(\left.\mathfrak{G}_{n}\right|_{\overline{\mathbb{Z}_{\geq}^{j-1}}\times\left\{\infty\right\}\times\mathbb{Z}_{\geq}^{n-j}}\right)_{0}$ is isomorphic to the groupoid $\left.\mathfrak{T}_{n-1}\right|_{\overline{\mathbb{Z}_{\geq}^{j-1}}\times\mathbb{Z}_{\geq}^{n-j}}$ via the groupoid isomorphism

[TABLE]

where

[TABLE]

with $\sum_{i=1}^{j-1}m_{i}+k+\sum_{i=1}^{n-j}l_{i}=0$ and hence $k=-\sum m-\sum l$ determined by $m,l$.

Since $K_{1}\left(\mathcal{T}^{\otimes j-1}\otimes\mathcal{K}\left(\mathbb{Z}_{\geq}^{n-j}\right)\right)=0$ and $K_{0}\left(\mathcal{T}^{\otimes j-1}\otimes\mathcal{K}\left(\mathbb{Z}_{\geq}^{n-j}\right)\right)=\mathbb{Z}$, it is easy to conclude from the cyclic six-term exact sequence of $K$-groups for the pair $C^{\ast}\left(\left(\mathfrak{H}_{j-1}\right)_{0}\right)\vartriangleleft C^{\ast}\left(\left(\mathfrak{H}_{j}\right)_{0}\right)$ that the following sequence is exact and splits

[TABLE]

where the projection $\left(\otimes^{j-1}I\right)\otimes\left(\otimes^{n-j}P_{1}\right)$ is a generator of $K_{0}\left(\mathcal{T}^{\otimes j-1}\otimes\mathcal{K}\left(\mathbb{Z}_{\geq}^{n-j}\right)\right)$. Note that this $\left(\otimes^{j-1}I\right)\otimes\left(\otimes^{n-j}P_{1}\right)$ lifts to the projection element $\chi_{A_{j}}\in C_{c}\left(\left(\mathfrak{H}_{j}\right)_{0}\right)\subset C^{\ast}\left(\left(\mathfrak{H}_{j}\right)_{0}\right)$ given by the characteristic function of the set

[TABLE]

Furthermore $\chi_{A_{j}}=Q_{j,\operatorname{id},1}$ in $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)\subset C\left(\mathbb{S}_{H}^{2n-1}\right)$. So we get

[TABLE]

with $K_{0}\left(C^{\ast}\left(\left(\mathfrak{H}_{j-1}\right)_{0}\right)\right)$ canonically embedded in $K_{0}\left(C^{\ast}\left(\left(\mathfrak{H}_{j}\right)_{0}\right)\right)$.

Putting together these results for all $j$, we get

[TABLE]

and hence $Q_{j,\operatorname{id},1}$ freely generate the abelian group $K_{0}\left(C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)\right)$. Note that $Q_{j,\operatorname{id},l}=\boxplus^{l}Q_{j,\operatorname{id},1}$ and $\left[Q_{j,\operatorname{id},l}\right]=l\left[Q_{j,\operatorname{id},1}\right]$ in $K_{0}\left(C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)\right)$ for any $l\in\mathbb{N}$.

We now summarize the above discussion.

Theorem 4. The projections $Q_{j,\Theta,l}$ over $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)$ with $l\in\mathbb{N}$ and $\Theta$ a $\left(j,n-j\right)$-shuffle for $0<j\leq n$ are mutually inequivalent, and the projections $Q_{j,\operatorname{id},1}$ with $0<j\leq n$ freely generate the abelian group $K_{0}\left(C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)\right)$, such that if $\left[p\right]=\sum_{j=1}^{n}m_{j}\left[Q_{j,\operatorname{id},1}\right]$ for a projection $p$ over $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)$, then the coefficient $m_{n}$ of $\left[Q_{n,\operatorname{id},1}\right]$ is the rank of $p$.

Proof. We only need to note that the rank of $Q_{n,\operatorname{id},1}$ is $1$ and the rank of any other $Q_{j,\operatorname{id},1}$ is [math]. $\square$

Remark. Since any permutation $\Theta$ on $\left\{1,2,...,n\right\}$ canonically induces a $U\left(1\right)$-equivariant (outer) C*-algebra automorphism of $\mathcal{T}^{\otimes n}$ permuting its tensor factors and preserving its ideal $\mathcal{K}^{\otimes n}$, the above expression of free generators $\left[\partial_{n}\left(\otimes^{j}I\otimes\otimes^{n-j}P_{1}\right)\right]$ with $0<j\leq n$ of $K_{0}\left(C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)\right)$ can be changed by a permutation to yield some other free generators. For example, both $\left\{\left[\partial_{3}\left(1\otimes P_{1}\otimes P_{1}\right)\right],\left[\partial_{3}\left(1\otimes 1\otimes P_{1}\right)\right],\left[\partial_{3}\left(1\otimes 1\otimes 1\right)\right]\right\}$ and

$\quad\left\{\left[\partial_{3}\left(P_{1}\otimes P_{1}\otimes 1\right)\right],\left[\partial_{3}\left(P_{1}\otimes 1\otimes 1\right)\right],\left[\partial_{3}\left(1\otimes 1\otimes 1\right)\right]\right\}$ are sets of free generators of $K_{0}\left(C\left(\mathbb{P}^{2}\left(\mathcal{T}\right)\right)\right)$.

The above theorem shows that for $\left(j^{\prime},\operatorname{id}\right)\prec\left(j,\operatorname{id}\right)$ in $\Omega_{0}$, i.e. $0<j^{\prime}<j$, it is not true that $Q_{j,\operatorname{id},1}\boxplus Q_{j^{\prime},\operatorname{id},1}\sim Q_{j,\operatorname{id},1}$ over $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)$ because

[TABLE]

in $K_{0}\left(C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)\right)$.

Next we consider the positive cone of $K_{0}\left(C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)\right)$. In the following, we use $\hat{I}$ and $\tilde{I}$ to denote the identity elements of $\mathcal{T}^{\otimes n-1}$ and $\mathcal{T}^{\otimes n}$ respectively.

First, it is easy to see that for $k>0$, the projection $\hat{I}\otimes P_{k}$ is a sum of $k$ mutually orthogonal projections $\hat{I}\otimes e_{jj}$, each equivalent to $\hat{I}\otimes P_{1}$ over $\left(\mathcal{T}^{\otimes n-1}\otimes\mathcal{K}\right)^{+}\subset\mathcal{T}^{\otimes n}$, and hence the projection $\partial_{n}\left(\hat{I}\otimes P_{k}\right)$ is a sum of $k$ mutually orthogonal projections $\partial_{n}\left(\hat{I}\otimes e_{jj}\right)$, each equivalent to $\partial_{n}\left(\hat{I}\otimes P_{1}\right)$ over $C\left(\mathbb{S}_{H}^{2n-1}\right)$. So

[TABLE]

and $\partial_{n}\left(\hat{I}\otimes P_{k}\right)\sim\boxplus^{k}Q_{n-1,\operatorname{id},1}\equiv Q_{n-1,\operatorname{id},k}$ over $C\left(\mathbb{S}_{H}^{2n-1}\right)$. Similarly, by rearranging entries via conjugation by shifts, the projection $\hat{I}\otimes P_{-k}$ is equivalent to $\tilde{I}$ over $\mathcal{T}^{\otimes n}$, and hence $\partial_{n}\left(\hat{I}\otimes P_{-k}\right)\sim\partial_{n}\left(\tilde{I}\right)$ over $C\left(\mathbb{S}_{H}^{2n-1}\right)$. However such equivalences no longer hold over the algebra $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)\subset C\left(\mathbb{S}_{H}^{2n-1}\right)$. For example, $\partial_{n}\left(\hat{I}\otimes P_{-k}\right)\boxplus\partial_{n}\left(\hat{I}\otimes P_{k}\right)\sim\partial_{n}\left(\tilde{I}\right)$ over $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)$ since $\partial_{n}\left(\hat{I}\otimes P_{-k}\right)$ and $\partial_{n}\left(\hat{I}\otimes P_{k}\right)$ are orthogonal projections in $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)$ which add up to $\tilde{I}$. So

[TABLE]

showing that

[TABLE]

and $\partial_{n}\left(\hat{I}\otimes P_{-1}\right)$ is not even stably equivalent over $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)$ to any $\boxplus$-sum of the $K_{0}$-generating projections $Q_{j,\operatorname{id},1}$ with $0<j\leq n$.

From now on, we include all projections of the form $\partial_{n}\left(\left(\otimes^{j-1}I\right)\otimes P_{k}\otimes\left(\otimes^{n-j}P_{1}\right)\right)$ with $k\in\mathbb{Z}$ as elementary projections over $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)$, where it is understood that for $k=0$, we take $P_{k}:=P_{-0}\equiv I$ instead of $P_{0}\equiv 0$.

Theorem 5. The positive cone of $K_{0}\left(C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)\right)\cong\mathbb{Z}^{n}\equiv\bigoplus_{j=1}^{n}\mathbb{Z}\left[Q_{j,\operatorname{id},1}\right]$ contains

[TABLE]

which is the part of the cone generated/spanned by the equivalence classes of the elementary projections $\partial_{n}\left(\left(\otimes^{j-1}I\right)\otimes P_{k}\otimes\left(\otimes^{n-j}P_{1}\right)\right)$ with $k\in\mathbb{Z}$ and $1\leq j\leq n$, where for $k=0$, we take $P_{k}:=P_{-0}\equiv I$.

Proof. In [29], it has been established that in the case of $n=2$, the positive cone of

[TABLE]

consists of $\left(k,m\right)\in\mathbb{Z}^{2}$ with either $k\geq 0$ or the rank $m>0$, such that $\left[\partial_{2}\left(I\otimes P_{k}\right)\right]=k\left[\partial_{2}\left(I\otimes P_{1}\right)\right]=\left(k,0\right)$ and

[TABLE]

in $K_{0}\left(C\left(\mathbb{P}^{1}\left(\mathcal{T}\right)\right)\right)$ for all $k>0$.

By induction on $n$, we can show that the positive cone of

[TABLE]

contains the set $\left(\mathbb{Z}_{\geq}^{n-1}\times\left\{0\right\}\right)\cup\left(\mathbb{Z}^{n-1}\times\mathbb{N}\right)$ consisting of $\left(k_{1},..,k_{n-1},m\right)\in\mathbb{Z}^{n}$ with either $k_{j}\geq 0$ for all $j$ or the rank $m>0$.

Indeed, under the canonical embedding

[TABLE]

due to the degree-preserving groupoid embedding of $\left.\left(\mathbb{Z}^{n-1}\ltimes\overline{\mathbb{Z}}^{n-1}\right)\right|_{\overline{\mathbb{Z}_{\geq}}^{n-1}}$ in $\left.\left(\mathbb{Z}^{n}\ltimes\overline{\mathbb{Z}}^{n}\right)\right|_{\overline{\mathbb{Z}_{\geq}}^{n}}$ as $\left.\left(\left(\mathbb{Z}^{n-1}\times\left\{0\right\}\right)\ltimes\left(\overline{\mathbb{Z}}^{n-1}\times\left\{0\right\}\right)\right)\right|_{\overline{\mathbb{Z}_{\geq}}^{n-1}\times\left\{0\right\}}$, a projection $p$ (for example, $\partial_{n-1}\left(P_{k_{1}}\otimes\cdots\otimes P_{k_{n-1}}\right)$) over $C\left(\mathbb{P}^{n-2}\left(\mathcal{T}\right)\right)$ becomes the projection $p\otimes P_{1}$ (for example, $\partial_{n}\left(P_{k_{1}}\otimes\cdots\otimes P_{k_{n-1}}\otimes P_{1}\right)$) over $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)$. Furthermore if $p\sim q$ over $C\left(\mathbb{P}^{n-2}\left(\mathcal{T}\right)\right)$, say, $upu^{-1}=q$ for some $u\in GL_{\infty}\left(C\left(\mathbb{P}^{n-2}\left(\mathcal{T}\right)\right)\right)$ then the equivalence $p\otimes P_{1}\sim q\otimes P_{1}$ over $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)$ can be explicitly constructed as

[TABLE]

with $\left(u\otimes P_{1}\right)+\partial_{n}\left(I\otimes P_{-1}\right)\in GL_{\infty}\left(C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)\right)$. Now consider the well-defined group homomorphism

[TABLE]

mapping the positive cone of $K_{0}\left(C\left(\mathbb{P}^{n-2}\left(\mathcal{T}\right)\right)\right)$ into that of $K_{0}\left(C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)\right)$. Since under $\iota$, the projection $Q_{j,\operatorname{id},1}$ over $C\left(\mathbb{P}^{n-2}\left(\mathcal{T}\right)\right)$ for $0<j\leq n-1$ is sent to the projection $Q_{j,\operatorname{id},1}$ over $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)$, by induction hypothesis, we get that the positive cone of $K_{0}\left(C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)\right)\cong\mathbb{Z}^{n}$ contains $\left(\mathbb{Z}_{\geq}^{n-2}\times\left\{0\right\}\times\left\{0\right\}\right)\cup\left(\mathbb{Z}^{n-2}\times\mathbb{N}\times\left\{0\right\}\right)$, and hence $\left(\mathbb{Z}_{\geq}^{n-1}\times\left\{0\right\}\right)\cup\left(\mathbb{Z}^{n-2}\times\mathbb{N}\times\mathbb{Z}_{\geq}\right)$.

On the other hand, for $k>0$,

[TABLE]

where $I^{\prime}$ denotes the identity element of $\mathcal{T}^{\otimes n-2}$ and $e_{ij}$ with $i,j\in\mathbb{Z}_{\geq}$ represents a matrix unit projection, because $\hat{I}\otimes P_{-k}$ is the sum of orthogonal projections $\left(\hat{I}\otimes P_{-\left(k+1\right)}\right)$ and $\left(\hat{I}\otimes e_{kk}\right)$, and $\left(\hat{I}\otimes e_{kk}\right)\boxplus 0$ when conjugated by

[TABLE]

becomes $0\boxplus\left(I^{\prime}\otimes P_{-k}\otimes P_{1}\right)$. Since $\partial_{n}\left(u_{k}\right)$ of total degree [math] is in $M_{2}\left(C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)\right)$, we get

[TABLE]

and hence

[TABLE]

because $\left[\partial_{n-1}\left(I^{\prime}\otimes P_{-k}\right)\right]\in\mathbb{Z}^{n-2}\times\left\{1\right\}$ for the rank-one projection $I^{\prime}\otimes P_{-k}$ over $\mathcal{T}^{\otimes n-1}$. With

[TABLE]

we get inductively

[TABLE]

for all $k>0$. Thus the positive cone of $K_{0}\left(C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)\right)\cong\mathbb{Z}^{n}$ contains $\left(\mathbb{Z}_{\geq}^{n-1}\times\left\{0\right\}\right)\cup\left(\mathbb{Z}^{n-1}\times\left\{1\right\}\right)$ and hence $\left(\mathbb{Z}_{\geq}^{n-1}\times\left\{0\right\}\right)\cup\left(\mathbb{Z}^{n-1}\times\mathbb{N}\right)$. On the other hand, the positive cone of $K_{0}\left(C\left(\mathbb{P}^{n-2}\left(\mathcal{T}\right)\right)\right)$ is mapped into the positive cone of $K_{0}\left(C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)\right)$ by the homomorphism $\cdot\times\left\{0\right\}\equiv K_{0}\left(\iota\right)$. So it is easy to get inductively the conclusion. $\square$

We note that for the case of $n=2$, the finitely generated projective modules over $C\left(\mathbb{P}^{1}\left(\mathcal{T}\right)\right)$ are completely classified with the positive cone of $K_{0}\left(C\left(\mathbb{P}^{1}\left(\mathcal{T}\right)\right)\right)$ explicitly identified in [29].

8 Quantum line bundles

In this section, we identify the quantum line bundles $L_{k}:=C\left(\mathbb{S}_{H}^{2n-1}\right)_{k}$ of degree $k$ over $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)$ with a concrete (equivalence class of) projection described in terms of the elementary projections defined in the previous section. We continue to use $\hat{I}$ and $\tilde{I}$ to denote the identity elements of $\mathcal{T}^{\otimes n-1}$ and $\mathcal{T}^{\otimes n}$ respectively, and we start to use $0^{\left(l\right)}$ to denote the zero of $\mathbb{Z}^{l}$.

To distinguish between ordinary function product and convolution product, we denote the groupoid C*-algebraic (convolution) multiplication of elements in $C_{c}\left(\mathcal{G}\right)\subset C^{\ast}\left(\mathcal{G}\right)$ by $\ast$, while omitting $\ast$ when the elements are presented as operators or when they are multiplied together pointwise as functions on $\mathcal{G}$. We also view $C_{c}\left(\mathfrak{G}_{n}\right)$ or $C_{c}\left(\left(\mathfrak{G}_{n}\right)_{k}\right)$ (also abbreviated as $C_{c}\left(\mathfrak{G}_{n}\right)_{k}$) as left $C_{c}\left(\mathfrak{G}_{n}\right)_{0}$-modules with $C_{c}\left(\mathfrak{G}_{n}\right)$ carrying the convolution algebra structure as a subalgebra of the groupoid C*-algebra $C^{\ast}\left(\mathfrak{G}_{n}\right)$. Similarly, for a closed subset $X$ of the unit space of $\mathfrak{G}_{n}$, the inverse image $\mathfrak{G}_{n}\upharpoonright_{X}$ of $X$ under the source map of $\mathfrak{G}_{n}$ or its grade-$k$ component $\left(\mathfrak{G}_{n}\upharpoonright_{X}\right)_{k}$ gives rise to a left $C_{c}\left(\mathfrak{G}_{n}\right)_{0}$-module $C_{c}\left(\mathfrak{G}_{n}\upharpoonright_{X}\right)$ or $C_{c}\left(\mathfrak{G}_{n}\upharpoonright_{X}\right)_{k}$.

We define a partial isometry in $C\left(\mathbb{S}_{H}^{2n-1}\right)\equiv C^{\ast}\left(\mathfrak{G}_{n}\right)$ for each $k\in\mathbb{Z}$ as the characteristic function $\chi_{B_{k}}$ of the compact open set

[TABLE]

It is easy to verify that $\chi_{B_{k}}\in C_{c}\left(\mathfrak{G}_{n}\right)_{k}$ and $\left(\chi_{B_{k}}\right)^{\ast}\in C_{c}\left(\mathfrak{G}_{n}\right)_{-k}$ such that

[TABLE]

and

[TABLE]

For $k\geq 0$, we have $C_{c}\left(\mathfrak{G}_{n}\right)_{k}\ast\left(\chi_{B_{k}}\right)^{\ast}\subset C_{c}\left(\mathfrak{G}_{n}\right)_{0}$ and

[TABLE]

which implies that the convolution operator $\cdot\ast\chi_{B_{k}}$ maps $C_{c}\left(\mathfrak{G}_{n}\right)_{0}$ onto $C_{c}\left(\mathfrak{G}_{n}\right)_{k}$. Since $\chi_{B_{k}}\ast\left(\chi_{B_{k}}\right)^{\ast}=\chi_{\left\{0^{\left(n\right)}\right\}\times\left(\left(\overline{\mathbb{Z}_{\geq}^{n-1}}\times\overline{\mathbb{Z}_{\geq k}}\right)\backslash\mathbb{Z}_{\geq}^{n}\right)}$, we get $\cdot\ast\chi_{B_{k}}$ mapping $C_{c}\left(\mathfrak{G}_{n}\right)_{0}\ast\chi_{\left\{0^{\left(n\right)}\right\}\times\left(\left(\overline{\mathbb{Z}_{\geq}^{n-1}}\times\overline{\mathbb{Z}_{\geq k}}\right)\backslash\mathbb{Z}_{\geq}^{n}\right)}$ bijectively onto $C_{c}\left(\mathfrak{G}_{n}\right)_{k}$ with $\cdot\ast\left(\chi_{B_{k}}\right)^{\ast}$ as the inverse. Furthermore $\cdot\ast\chi_{B_{k}}$ is a left $C_{c}\left(\mathfrak{G}_{n}\right)_{0}$-module homomorphism. With $\chi_{B_{k}}$ being a partial isometry, $\cdot\ast\chi_{B_{k}}$ and $\cdot\ast\left(\chi_{B_{k}}\right)^{\ast}$ extend continuously to provide an isomorphism between the $C^{\ast}\left(\mathfrak{G}_{n}\right)_{0}$-modules

[TABLE]

and $C^{\ast}\left(\mathfrak{G}_{n}\right)_{k}\equiv\overline{C_{c}\left(\mathfrak{G}_{n}\right)_{k}}$. So the quantum line bundle $C^{\ast}\left(\mathfrak{G}_{n}\right)_{k}$ is identified with the projection $\partial_{n}\left(\hat{I}\otimes P_{-k}\right)$.

For $k<0$, we consider the direct sum decomposition as left $C_{c}\left(\mathfrak{G}_{n}\right)_{0}$-modules

[TABLE]

From

[TABLE]

and

[TABLE]

we see that $\cdot\ast\chi_{B_{\left|k\right|}}$ is a left $C_{c}\left(\mathfrak{G}_{n}\right)_{0}$-module isomorphism between $C_{c}\left(\mathfrak{G}_{n}\right)_{0}$ and $C_{c}\left(\mathfrak{G}_{n}\right)_{k}\ast\chi_{\left\{0^{\left(n\right)}\right\}\times\left(\left(\overline{\mathbb{Z}_{\geq}^{n-1}}\times\overline{\mathbb{Z}_{\geq\left|k\right|}}\right)\backslash\mathbb{Z}_{\geq}^{n}\right)}$ with $\cdot\ast\left(\chi_{B_{k}}\right)^{\ast}$ as its inverse.

On the other hand, in the $C_{c}\left(\mathfrak{G}_{n}\right)_{0}$-module decomposition

[TABLE]

each $C_{c}\left(\mathfrak{G}_{n}\upharpoonright_{\left(\overline{\mathbb{Z}_{\geq}^{n-1}}\backslash\mathbb{Z}_{\geq}^{n-1}\right)\times\left\{j\right\}}\right)_{k}$ is isomorphic to the $C_{c}\left(\mathfrak{G}_{n}\right)_{0}$-module $C_{c}\left(\mathfrak{G}_{n}\upharpoonright_{\left(\overline{\mathbb{Z}_{\geq}^{n-1}}\backslash\mathbb{Z}_{\geq}^{n-1}\right)\times\left\{0\right\}}\right)_{k+j}$ with $k+j<0$ via the homeomorphism

[TABLE]

where the implicit condition $l+j\geq 0$ is equivalent to $l\geq-j$. So we focus on analyzing $C_{c}\left(\mathfrak{G}_{n}\right)_{0}$-modules of the form

[TABLE]

with $r\geq 0$. Note that the $C^{\ast}\left(\mathfrak{G}_{n}\right)_{0}$-module

[TABLE]

is identified with the projection $\partial_{n}\left(\hat{I}\otimes P_{1}\right)\equiv Q_{n-1,\operatorname{id},1}$.

For $r>0$, similar to the argument used above, it can be checked that the compact open subset

[TABLE]

defines a partial isometry $\chi_{B_{-r}^{\prime}}\in C_{c}\left(\mathfrak{G}_{n}\right)_{-r}$ with $\left(\chi_{B_{-r}^{\prime}}\right)^{\ast}\in C_{c}\left(\mathfrak{G}_{n}\right)_{r}$ such that

[TABLE]

and

[TABLE]

In the decomposition

[TABLE]

[TABLE]

[TABLE]

the second summand is isomorphic, via the right convolution $\cdot\ast\left(\chi_{B_{-r}^{\prime}}\right)^{\ast}$ by the partial isometry $\left(\chi_{B_{-r}^{\prime}}\right)^{\ast}$, to the $C_{c}\left(\mathfrak{G}_{n}\right)_{0}$-module

[TABLE]

Now we introduce the notation of a $C_{c}\left(\mathfrak{G}_{n}\right)_{0}$-module

[TABLE]

for $r\geq 0$ and $1\leq l\leq n-1$. We note that the $C_{c}\left(\mathfrak{G}_{n}\right)_{0}$-module

[TABLE]

is isomorphic to

[TABLE]

[TABLE]

via the homeomorphism

[TABLE]

Applying the same kind of arguments as shown above, we get the isomorphism of $C_{c}\left(\mathfrak{G}_{n}\right)_{0}$-modules

[TABLE]

[TABLE]

[TABLE]

[TABLE]

for $2\leq l\leq n-1$. This provides a recursive formula to reduce the index $l$ of the module $A_{r,l}$.

For $n>2$, we define a combinatorial number $\nu_{n}\left(m,l\right)$ recursively by

[TABLE]

and $\nu_{n}\left(m,1\right):=1$, for $m\geq 0$ and $2\leq l\leq n-1$, to be used in the following theorem.

Thanks to Thomas Timmermann, as he pointed out to the author, $\nu_{n}\left(m,l\right)$ can be identified with a familiar combinatorial number, namely,

[TABLE]

for all $m\geq 0$ and $l\geq 1$. Indeed, if either $l=1$ or $m=0$ (e.g. when $m+l\leq 2$), we get easily from the definition, $\nu_{n}\left(m,l\right)=1=C_{m}^{m+l-1}$. On the other hand, for $l\geq 2$ and $m\geq 1$, since

[TABLE]

the identification can be proved by an induction on $m+l\geq 3$ as shown in

[TABLE]

which is valid due to either the induction hypothesis for $m+l-1$ (in the case of $m+l-1>2$) or the already established identification (for the case of $m+l-1=2$).

Theorem 6. For $n>2$, the quantum line bundle $L_{k}\equiv C\left(\mathbb{S}_{H}^{2n-1}\right)_{k}$ of degree $k\in\mathbb{Z}$ over $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)$ is isomorphic to the finitely generated projective left module over $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)$ determined by the projection $\partial_{n}\left(\otimes^{n-1}I\otimes P_{-k}\right)$ if $k\geq 0$, and the projection

[TABLE]

if $k<0$.

Proof. Only the case of $k<0$ remains to be proved as follows.

For $k<0$, starting with the established isomorphism

[TABLE]

[TABLE]

we apply repeatedly the recursive formula

[TABLE]

reducing $l$ for $A_{r,l}$ with $2\leq l\leq n$ until $l$ reaches $2$ with

[TABLE]

in order to convert all terms to $C_{c}\left(\mathfrak{G}_{n}\right)_{0}$-modules of the form $C_{c}\left(\mathfrak{G}_{n}\right)_{0}\partial_{n}\left(I^{\otimes j}\otimes P_{1}^{\otimes n-j}\right)$ for some $0<j\leq n$.

In fact, we check inductively on $1\leq j\leq n-2$ that

[TABLE]

The case of $j=1$ is our starting point already proved. Now assuming that it holds for $j$, we get by the above recursive formula

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

which verifies (*) for $j+1$.

For $j=n-2$, (*) says

[TABLE]

[TABLE]

[TABLE]

[TABLE]

After completion, we get the $C^{\ast}\left(\left(\mathfrak{G}_{n}\right)_{0}\right)$-module $L_{k}$ isomorphic to

[TABLE]

which corresponds to the projection

[TABLE]

$\square$

Little is known about the cancellation problem and hence the classification problem for finitely generated projective modules over $C\left(\mathbb{P}^{n-1}\left(\mathcal{T}\right)\right)$. We expect that these problems will be far more complicated than those for over $C\left(\mathbb{S}_{H}^{2n-1}\right)$ and $C\left(\mathcal{T}^{\otimes n}\right)$.

The recent work of Farsi, Hajac, Maszczyk, and Zieliński [7] identifies one of three free generators of $K_{0}\left(C\left(\mathbb{P}^{2}\left(\mathcal{T}\right)\right)\right)$ as $\left[L_{1}\right]+\left[L_{-1}\right]-2\left[I\right]$ (in addition to $\left[L_{1}\right]-\left[I\right]$ and $\left[I\right]$) constructed from a Milnor module and then expresses it in terms of elementary projections, showing a perfect consistency with our result.