The principal-symbol index map for an algebra of pseudodifferential operators

TL;DR

This paper describes the K-theory index map for a C*-algebra generated by zero-order pseudodifferential operators on a cylinder, linking algebraic K-theory classes to Fredholm indices of elliptic operators.

Contribution

It provides a detailed description of the principal-symbol index map for a specific algebra of pseudodifferential operators on a cylinder, including its target in K_0 of the commutator ideal.

Findings

The index map takes values in Z^2, isomorphic to K_0 of the commutator ideal.

The index map relates K_1-classes of invertible operators to Fredholm indices of elliptic operators.

The algebra includes operators with periodic symbols on a cylindrical manifold.

Abstract

A C*algebra A generated by a class of zero-order classical pseudodifferential operator on a cylinder RxB, where B is a compact riemannian manifold, containing operators with periodic symbols, is considered. A description of the K-theory index map associated to the continuous extension to A of the principal-symbol map is given. That index map takes values in K_0 of the commutator ideal E of the algebra, which is isomorphic to Z^2. It maps the K_1-class of an operator invertible modulo E to the Fredholm indices of a pair of elliptic pseudodifferentail operators on SxB, where S denotes the circle.

Equations

$$0⟶{\mathfrak{K}}_M\hookrightarrow \Psi (M)\stackrel{{\sigma }_M}{⟶}C(S^{\ast }(M))⟶0.$$

$$[a(M_x)f](t,x)=a(x)f(t,x),(t,x)\in \mathbb{R}\times B;$$

$$[b(M_t)f](t,x)=b(t)f(t,x),(t,x)\in \mathbb{R}\times B;$$

$$[e^{ik{M}_t}f](t,x)=e^{ikt}f(t,x),(t,x)\in \mathbb{R}\times B;$$

$$S^{\ast }\overline{\Omega }=\{(t,x,\tau ,\xi );(t,\tau )\in T^{\ast }[-\infty ,+\infty ],(x,\xi )\in T^{\ast }B,|\tau {|}^2+|\xi {|}^2=1\}.$$

$$\mathbf{M}=\{(t,x,\tau ,\xi ,e^{i\theta });\theta \in \mathbb{R}\text{mbox}and\theta =t\text{mbox}if|t|<\infty \}.$$

$$\sigma :\mathfrak{A}\to C(\mathbf{M})$$

$$a(x),b(t),e^{ij\theta },0,\tau ,\sum _{k=1}^n{c}^j(x){\xi }_j.$$

$$(F_{d}u{)}_j={\int }_0^{1}u(e^{2\pi i\phi })e^{-2\pi ij\phi }d\phi ,u\in L^2({\mathbb{S}}^1)\text{mbox}andj\in \mathbb{Z}$$

$$[b(M_j)u]((u_j{)}_{j\in \mathbb{Z}})=(b_j{u}_j{)}_{j\in \mathbb{Z}}\text{mbox}andb(D_{\phi })=F_d^{-1}b(M_j)F_d.$$

$${\sigma }_S:\mathfrak{S}\to C(S^1\times \{-1,+1\}),$$

$$\begin{array}[]{rcl}W:L^{2}({\mathbb{R}})&\longrightarrow&L^{2}(S^{1})\otimes\ell^{2}({\mathbb{Z}})\simeq L^{2}(S^{1};\,\ell^{2}({\mathbb{Z}}))\\ u&\longmapsto&Wu,\ \ \ \ \ \,(Wu)(e^{2\pi i\varphi})=Y_{\varphi}\tilde{u}(\varphi)\end{array}$$

$$\mathfrak{E}\stackrel{\cong }{\to }\mathfrak{S}\otimes {\mathfrak{K}}_{\mathbb{Z}}\otimes {\mathfrak{K}}_B.$$

$$\mathfrak{E}/{\mathfrak{K}}_{\Omega }\stackrel{\cong }{\to }C(S^1\times \{-1,+1\})\otimes {\mathfrak{K}}_{\mathbb{Z}}\otimes {\mathfrak{K}}_B\cong C(S^1\times \{-1,+1\};{\mathfrak{K}}_{\mathbb{Z}}\otimes {\mathfrak{K}}_B).$$

$$\gamma :\mathfrak{E}⟶C(S^1\times \{-1,+1\};\mathfrak{K}(L^2(\mathbb{Z}\times B))),$$

$$\gamma :\mathfrak{A}⟶C(S^1\times \{-1,+1\};\mathfrak{B}(L^2(\mathbb{Z}\times B))).$$

$$\Gamma :\mathfrak{A}⟶C(S^1\times \{-1,+1\};\mathfrak{B}(L^2(S^1\times B)))$$

$${\Gamma }_A(e^{2\pi i\phi },\pm 1)=F_d^{-1}{\Gamma }_A(e^{2\pi i\phi },\pm 1)F_d,e^{2\pi i\phi }\in S^1.$$

$$[e^{i\phi M_{\theta }}u](e^{i\theta })=e^{i\phi \theta }u(e^{i\theta }),-\pi <\theta \le \pi ,u\in L^2(S^1).$$

$$(D_{\theta }u)(e^{i\theta })=-i{\tilde{u}}^{\prime }(\theta ),\tilde{u}(\theta )=u(e^{i\theta }),\theta \in \mathbb{R},u\in C^{\infty }(S^1).$$

$$a(M_x),b(\pm \infty )I,e^{ij{M}_{\theta }},e^{-i\phi M_{\theta }}[1+(D_{\theta }-\phi {)}^2-{\Delta }_B{]}^{-1/2}{e}^{i\phi M_{\theta }},$$

$$e^{-i\phi M_{\theta }}(D_{\theta }-\phi )[1+(D_{\theta }-\phi {)}^2-{\Delta }_B{]}^{-1/2}{e}^{i\phi M_{\theta }},$$

$$e^{-i\phi M_{\theta }}{D}_x[1+(D_{\theta }-\phi {)}^2-{\Delta }_B{]}^{-1/2}{e}^{i\phi M_{\theta }},$$

$$\sigma \oplus \Gamma :M_k(\mathfrak{A})⟶M_k(C(\mathbf{M}))\oplus M_k(C(S^1\times \{-1,+1\};\Psi (S^1\times B))).$$

$$0⟶\mathfrak{E}\hookrightarrow \mathfrak{A}\stackrel{\sigma }{⟶}C(\mathbf{M})⟶0,$$

$$0⟶{\mathfrak{K}}_{S^1}\hookrightarrow \mathfrak{S}\stackrel{{\sigma }_S}{⟶}C(S^1\times \{-1,+1\})⟶0.$$

$$\begin{array}[]{ccccc}\mathbb{Z}\cong K_{0}({\mathfrak{K}}_{S^{1}})&\!{\longrightarrow}&\!K_{0}({\mathfrak{S}})&\!{\longrightarrow}&\!K_{0}(C(S^{1}\times\{-1,+1\}))\\ \\ {\delta}_{1}\ \uparrow&\!{}&\!{}&\!{}&\!\downarrow\ {\delta}_{0}\\ \\ K_{1}(C(S^{1}\times\{-1,+1\}))&\!{\longleftarrow}&\!K_{1}({\mathfrak{S}})&\!{\longleftarrow}&\!K_{1}({\mathfrak{K}}_{S^{1}})=0\end{array}$$

$$K_0(\mathfrak{E})\cong K_0(\mathfrak{E}/{\mathfrak{K}}_{\Omega })\cong K_1(\mathfrak{A}/\mathfrak{E})\cong {\mathbb{Z}}^2\text{mbox}and{K}_1(\mathfrak{E})\cong \mathbb{Z}.$$

$$0⟶{\mathfrak{K}}_{\Omega }\hookrightarrow \mathfrak{E}\stackrel{\pi }{⟶}\frac{\mathfrak{E}}{{\mathfrak{K}}_{\Omega }}⟶0$$

$$\begin{array}[]{ccccc}\mathbb{Z}&\!{\longrightarrow}&\!{\mathbb{Z}}^{2}&\!{\longrightarrow}&\!{\mathbb{Z}}^{2}\\ \\ {\delta}_{1}\ \uparrow&\!{}&\!{}&\!{}&\!\downarrow\ {\delta}_{0}\\ \\ {\mathbb{Z}}^{2}&\!{\longleftarrow}&\!{\mathbb{Z}}&\!{\longleftarrow}&\!0\end{array}$$

Full text

The problem of computing the K-theory index map $\delta_{1}(K_{1}({\mathfrak{A}}/{\mathfrak{E}}))\to K_{0}({\mathfrak{E}})\simeq{\mathbb{Z}}^{2}$ on an element of $K_{1}({\mathfrak{A}}/{\mathfrak{E}})$ is reduced to the problem of computing the Fredholm indices of two elliptic operators on the compact manifold $S^{1}\times B$.

For $B=S^{1}$, Hess went further and proved in her thesis that $K_{0}({\mathfrak{A}})\cong{\mathbb{Z}}^{5}$ and $K_{1}({\mathfrak{A}})\cong{\mathbb{Z}}^{4}$.

Introduction

More than fifty years after the Atiyah-Singer index theorem, people are still looking for generalizations of their theorem for classes of non-compact manifolds, for manifolds with boundary, corners or edges, or for manifolds with singularities in a more general sense. There are, in fact, a great variety of different classes of problems, and of different approaches or tecnhiques that could be used.

One of the approaches is to consider Calgebras generated by zero-order pseudodiferential operators which contain an order-reducing device. That means that there are linear isomorphisms between Sobolev spaces so that certain classes of differential operators multiplied on the right by them fall into the Calgebra. In the case of a compact manifold, the kernel of the continuous extension to the C*-algebra of the principal symbol map coincides with the ideal of compact operators, and the Fredholm index can be regarded as the K-theory index map associated to the exact sequence (1) below. For a noncompact manifold, one might argue that the relevant index is no longer the Fredholm index, but rather the K-theory index map taking values in $K_{0}$ of the kernel of the principal symbol.

Monthubert and Nistor followed that track in [24, 25], using algebras of pseudodifferential operators on groupoids which relate to (regular) algebras of pseudodifferential operators on a noncompact manifold. In their case, the relevant compactification of the original manifold is itself a manifold with corners. In this paper we consider an algebra of pseudodifferential operators on a cylinder ${\mathbb{R}}\times B$, where $B$ is compact, which includes operators with periodic symbols. In our case the relevant compactification of the manifold is not even locally path connected.

The $K$-groups of our algebra, for the special cases of $B$ equals a point and of $B$ equals $S^{1}$, have been computed in [23] and [13], respectively.

The algebra considered here is an example of a comparison algebra, in the sense of Cordes, who pioneered [4, 5, 6, 9, 10] the use of Banach-algebra techniques in the study of the spectral theory of partial differential operators and singular integral operators on noncompact manifolds. Those algebras were defined in [8] as C*-algebras of bounded operators on $L^{2}(M)$ for a noncompact manifold $M$ equipped with a smooth positive measure and a positive second-order elliptic differential operator $H$. Typically, $M$ is equipped with a riemannian metric and its induced measure, and $H$ is equal to the identity minus the Laplacian. One defines $\Lambda=H^{-1/2}$, where $H$ also denotes the Friedrichs extension of the original operator defined on $C_{c}^{\infty}(M)$, which of course coincides with the unique self-adjoint extension in case the operator is essentially self-adjoint. Given a class of smooth functions $\mathcal{A}^{\sharp}$ and a class of first-order differential operators $\mathcal{D}^{\sharp}$ on $M$ satisfying certain conditions, a comparison algebra is defined as the C*-subalgebra of the algebra of bounded operators on $L^{2}(M)$ generated by all operators of multiplication by $a\in\mathcal{A}^{\sharp}$ and by all $D\Lambda$, $D\in\mathcal{A}^{\sharp}$. In his monograph [8], Cordes develops general techniques to describe the ideal structure of a comparison algebra and how to find differential operators within reach of the algebra, meaning differential operators which, composed with an isomorphism from $L^{2}$ to a suitable Sobolev space, belongs to the algebra.

In case the manifold is compact, there is only one possible choice of classes: $\mathcal{A}^{\sharp}=C^{\infty}(M)$ and $\mathcal{D}^{\sharp}$ equals to the set of all first order differential operators with smooth coefficients. In this case, the resulting comparison algebra coincides [21] with the Calgebra generated by all zero-order classical pseudodifferential operators on $M$, which we will denote by $\Psi(M)$. The principal symbol of operators, a priori a -homomorphism defined on the subalgebra of classical pseudodifferential operators and taking values in smooth functions on the co-sphere bundle $S^{*}M$, extends to a surjective C-algebra homomorphism $\sigma_{M}:\Psi(M)\to C(S^{*}M)$ with kernel equal to the ideal of compact operators ${\mathfrak{K}}_{M}$. In other words, we have the short exact sequence of C-algebras

[TABLE]

In Cordes’ approach, the map $\sigma_{M}$ arises as the Gelfand map for the commutative C*algebra $\Psi(M)/{\mathfrak{K}}_{M}$. Classically, (1) follows from the equality between the norm, modulo compact operators, of a singular integral operator on a compact manifold and the supremum norm of its symbol, proven by Gohberg [15] and Seeley [29]. Proofs of that result in the language of pseudodifferential operators appeared in [16, 18].

1. Preliminaries

In this section, we define the C*-algebra ${\mathfrak{A}}$ and review results about its structure that will be needed in the next and final section.

Let $B$ denote a compact Riemannian manifold of dimension $n$ without boundary. Consider the cylinder $\Omega=\mathbb{R}\times B$ and denote by $\Delta_{\Omega}=\Delta_{\mathbb{R}}+\Delta_{B}$ its Laplacian (we use the classical analysis convention sign for the Laplacian, $-\Delta_{\Omega}\geq 0$) Define ${\mathfrak{A}}$ as the C*-subalgebra of ${{\mathfrak{B}}}(L^{2}(\Omega))$ (we denote by ${\mathfrak{B}}({\mathcal{H}})$ the C*-algebra of all bounded operators on a Hilbert space ${\mathcal{H}}$) generated by:

• (i)

all $A_{1}=a(M_{x})$, operators of multiplication by $a\in C^{\infty}({B})$,

[TABLE]

• (ii)

all $A_{2}=b(M_{t})$, operators of multiplication by $b\in C([-\infty,+\infty])$ (this denotes the set of all continuous functions on $\mathbb{R}$ which have limits at $-\infty$ and $+\infty$),

[TABLE]

• (iii)

all operators $A_{3}=e^{ikM_{t}}$, $k\in{\mathbb{Z}}$, defined by

[TABLE]

• (iv)

$A_{4}=\Lambda:=(1-\Delta_{\Omega})^{-1/2};$

• (v)

$A_{5}=\frac{1}{i}\frac{\partial}{\partial t}\Lambda;$

• (vi)

$D_{x}\Lambda$, where $D_{x}=-i\sum_{k=1}^{n}c^{k}(x)\partial_{x_{k}}$ is a smooth vector field on $B$.

Since $\Omega$ is a complete manifold, its Laplacian, defined on $C_{c}^{\infty}(\Omega)$, has a unique extension (also denoted by $\Delta_{\Omega}$) as an unbounded self-adjoint operator on $L^{2}(\Omega)$ ([14], [8, Theorem IV.1.8]) and one can then define $\Lambda$ by functional calculus . The operators in (v) and (vi), a priori defined on the dense subset $\Lambda^{-1}(C_{c}^{\infty}(\Omega))$, have continuous extensions to $L^{2}(\Omega)$ [8, Section V.1].

Moreover, if $L$ denotes an $M$-th order differential operator on $\Omega$ whose (smooth) coefficients approach $2\pi$-periodic functions as $t$ tends to $-\infty$ and as $t$ tends to $+\infty$, the operator $L\Lambda^{M}$ can be extended to a bounded operator on $L^{2}(\Omega)$, which belongs to ${\mathfrak{A}}$ . These $L\Lambda^{M}$ are zero-order classical pseudodifferential operators (meaning that their local symbols have asymptotic expansions in homogeneous components).

Applying Cordes’ comparison algebra techniques and using results from [7, 11], the statements about the structure of ${\mathfrak{A}}$ contained in the rest of this Section were proven in [20] (see also [19]).

Let ${\mathfrak{E}}$ denote the commutator ideal of ${\mathfrak{A}}$, i.e, the smallest closed ideal containing all commutators $AB-BA$, $A$ and $B$ in ${\mathfrak{A}}$. Let $\overline{\Omega}$ denote the compactfication $[-\infty,+\infty]\times B$ of $\Omega$, and let $S^{*}\!M$ denote the cosphere bundle of any manifold $M$. We may then write

[TABLE]

Theorem 1**.**

The spectrum of the commutative C-algebra ${\mathfrak{A}}/{\mathfrak{E}}$ is homeomorphic to the following subset of $(S^{*}\overline{\Omega})\times S^{1}:$*

[TABLE]

Denoting by

[TABLE]

the composition of the Gelfand isomorphism ${\mathfrak{A}}/{\mathfrak{E}}\simeq C(\mathbf{M})$ with the projection ${\mathfrak{A}}\to{\mathfrak{A}}/{\mathfrak{E}}$, for each of the generators of ${\mathfrak{A}}$ listed above, $\sigma(A_{k})(t,x,\tau,\xi,e^{i\theta})$, $k=1,\cdots,6$, is equal to, respectively,

[TABLE]

Since $M$ is a compactification of $S^{*}\Omega$, we may regard $C(\mathbf{M})$ as a subspace of $C(S^{*}\Omega)$. Theorem 1 can then be rephrased by saying that the usual principal symbol of classical pseudodifferential operators, defined on a dense subalgebra of ${\mathfrak{A}}$, has a continuous extension $\sigma$ to ${\mathfrak{A}}$ whose image is equal to $C(\mathbf{M})$ and whose kernel is ${\mathfrak{E}}$.

Let $F_{d}:L^{2}(S^{1})\rightarrow{\ell}^{2}(\mathbb{Z})$ denote the discrete Fourier transform

[TABLE]

($S^{1}$ is equipped with the Lebesgue measure pushforwarded by $[0,1)\ni\varphi\mapsto e^{2\pi i\varphi}\in S^{1}$; hence $F_{d}$ is unitary). For each sequence $(b_{j})_{j\in{\mathbb{Z}}}$ with limits as $j\to-\infty$ and as $j\to+\infty$, we define bounded operators $b(M_{j})$ on $\ell^{2}({\mathbb{Z}})$ and $b(D_{\varphi})$ on $L^{2}(S^{1})$ by

[TABLE]

Given $a\in C(S^{1})$, let $a(M_{\varphi})\in{\mathfrak{B}}(L^{2}(S^{1}))$ denote the operator of multiplication by $a$. Finally let ${\mathfrak{S}}$ denote the C*-subalgebra of ${\mathcal{L}}(L^{2}({S}^{1}))$ generated by the operators of the form $a(M_{\varphi})$ and $b(D_{\varphi})$, for $a\in C(S^{1})$ and $b=(b_{j})_{j\in{\mathbb{Z}}}$ a sequence as above.

In Cordes’ language, ${\mathfrak{S}}$ is the only comparison algebra over $S^{1}$. The principal symbol, defined on the dense -subalgebra of all classical zero-order pseudos, extends to a surjective C-algebra homomorphism (see the Introduction)

[TABLE]

whose kernel is equal to the set ${\mathfrak{K}}_{S^{1}}$ of all compact operators on $L^{2}(S^{1})$. On our set of generators, one has $[\sigma(a(M_{\varphi}))](e^{2\pi i\varphi},\pm 1)=a(e^{2\pi i\varphi})$, for $a\in C(S^{1})$, and $[\sigma(b(D_{\varphi}))](e^{2\pi i\varphi},\pm 1)=b(\pm\infty)$, for $b=(b_{j})_{j\in{\mathbb{Z}}}$ a sequence having limits as $j\to+\infty$ and as $j\to-\infty$. A detailed exposition of some of these facts can be found in [1].

For each $\varphi\in\mathbb{R}$, let $U_{\varphi}$ be the operator on $L^{2}(S^{1})$ given by $U_{\varphi}f(z)=z^{-\varphi}f(z)$, $z\in S^{1}$ (we take the principal branch of $z^{-\varphi}=e^{-\varphi\log z}$, $z\neq 0$). Let $Y_{\varphi}:=F_{d}U_{\varphi}F_{d}^{-1}$. This defines a smooth family $\varphi\mapsto Y_{\varphi}$ of unitary operators on $\ell^{2}(\mathbb{Z})$ such that for all $k\in\mathbb{Z},~{}(Y_{k}u)_{j}=u_{j+k},$ and $Y_{\varphi}Y_{\omega}=Y_{\varphi+\omega}$. Given $u\in L^{2}({\mathbb{R}})$, the sequence $\tilde{u}(\varphi)=(u(\varphi-j)_{j\in{\mathbb{Z}}})$ belongs to $\ell^{2}({\mathbb{Z}})$ for almost every $\varphi\in[0,1)$ and the map

[TABLE]

is a Hilbert space isomorphism.

We have used $\otimes$ to denote Hilbert-space tensor product. Given $A\in\mathfrak{B}(\mathcal{H})$ and $B\in\mathfrak{B}(\mathcal{K})$, we also denote by $A\otimes B$ the bounded operator on $\mathcal{H}\otimes\mathcal{K}$ defined by $(A\otimes B)(u\otimes v)=Au\otimes Bv$. And also, given C*-algebras $\mathfrak{A}_{H}\subset\mathfrak{B}(\mathcal{H})$ and $\mathfrak{A}_{K}\subset\mathfrak{B}(\mathcal{K})$, we denote by $\mathfrak{A}_{H}\otimes\mathfrak{A}_{K}\subset\mathfrak{B}(\mathcal{H}\otimes\mathcal{K})$ the closed linear span of all $A\otimes B$, $A\in\mathfrak{A}_{H}$ and $B\in\mathfrak{A}_{K}$.

We will denote by ${\mathfrak{K}}_{X}$ denote the sets of all compact operators on $L^{2}(X)$, for $X=\Omega$, $S^{1}$, ${\mathbb{Z}}$, $B$, ${\mathbb{Z}}\times B$, $S^{1}\times B$. We will also write $W$ and $F$ meaning $W\otimes I_{B}$ and $F\otimes I_{B}$, where $I_{B}$ denotes the identity map on $L^{2}(B)$. Similarly we will also denote by $\sigma_{S}$ the map $\sigma_{S}\otimes I_{{\mathfrak{K}}_{{\mathbb{Z}}}}\otimes I_{{\mathfrak{K}}_{B}}$ defined on ${\mathfrak{S}}\otimes{\mathfrak{K}}_{\mathbb{Z}}\otimes{\mathfrak{K}}_{B}$.

The -isomorphism $\mathfrak{B}(L^{2}(\Omega))\ni A\ \longmapsto\ WF^{-1}AFW^{-1}\in\mathfrak{B}(L^{2}(S^{1})\otimes\ell^{2}({\mathbb{Z}})\otimes L^{2}(B))$ restricted to ${\mathfrak{E}}$ defines a C-isomorphism

[TABLE]

The kernel of the surjective map $\sigma_{S}:{\mathfrak{S}}\otimes{\mathfrak{K}}_{\mathbb{Z}}\otimes{\mathfrak{K}}_{B}\to C(S^{1}\times\{-1,+1\})\otimes{\mathfrak{K}}_{\mathbb{Z}}\otimes{\mathfrak{K}}_{B}$ is equal to ${\mathfrak{K}}_{S^{1}}\otimes{\mathfrak{K}}_{\mathbb{Z}}\otimes{\mathfrak{K}}_{B}\cong{\mathfrak{K}}_{\Omega}$ and hence $\sigma_{S}$ induces a C*-algebra isomorphism

[TABLE]

The composition of this isomorphism with the canonical projection ${\mathfrak{E}}\to{\mathfrak{E}}/{\mathfrak{K}}_{\Omega}$ defines a C*-algebra homomorphism

[TABLE]

which can be extended, via the left regular representation, to a map from ${\mathfrak{A}}$ to the algebra of bounded operators on the Banach space $C(S^{1}\times\{-1,+1\};\,{\mathfrak{K}}(L^{2}({\mathbb{Z}}\times B))$. The image of this Banach-algebra homomorphism is contained in $C(S^{1}\times\{-1,+1\};\,{\mathfrak{B}}(L^{2}({\mathbb{Z}}\times B))$ (an $f\in C(S^{1}\times\{-1,+1\};{\mathfrak{B}}(L^{2}({\mathbb{Z}}\times B))$ is here regarded as a multiplication operator on $C(S^{1}\times\{-1,+1\};{\mathfrak{K}}(L^{2}({\mathbb{Z}}\times B))$ ) and this defines a C*-algebra homomorphism, which we also denote by $\gamma$,

[TABLE]

We define yet another *-homomorphism

[TABLE]

[TABLE]

The image of $\Gamma$ is contained in $C(S^{1}\times\{-1,+1\};\,\Psi(S^{1}\times B))$, with $\Psi(S^{1}\times B)$ as defined at the end of the Introduction. It follows almost by definition that the image of ${\mathfrak{E}}$ under $\Gamma$ equals $C(S^{1}\times\{-1,+1\};\,{\mathfrak{K}}_{S^{1}\times B})$.

In order to give a more explicit description of $\Gamma$, we need two more definitions. For each $\varphi\in{\mathbb{R}}$, let us denote by $e^{i\varphi M_{\theta}}$ the multiplication operator (by a possibly discontinuous function)

[TABLE]

And let us denote by $D_{\theta}$ the densely defined operator on $L^{2}(S^{1})$

[TABLE]

For each of the generators of ${\mathfrak{A}}$ defined on page (iii), the value of $\Gamma_{A_{k}}(e^{2\pi i\varphi},\pm 1)$, $1\leq k\leq 6$, equals

[TABLE]

[TABLE]

[TABLE]

respectively (the first three classes of generators are mapped to constant functions).

For any positive integer $k$, the homomorphisms $\sigma$ and $\Gamma$ defined in (2) and (9) canonically induce homomorphisms, which we also denote by $\sigma$ and $\Gamma$, on the algebra $M_{k}({\mathfrak{A}})$ of $k$-by-$k$ matrices with entries in ${\mathfrak{A}}$. Regarding $M_{k}({\mathfrak{A}})$ as a C*-subalgebra of $\mathfrak{B}(L^{2}(\Omega)^{k})$, it turns out that an $A\in M_{k}({\mathfrak{A}})$ is a Fredholm operator if and only if $\sigma(A)$ and $\Gamma(A)$ are invertible [20, Theorem 3.2]. Moreover, the quotient of $M_{k}({\mathfrak{A}})$ by the ideal of compact operators on $L^{2}(\Omega)^{k}$ is isomorphic to the image of the total symbol

[TABLE]

2. The principal-symbol exact sequence

In this section we analyse the $K$-theory index map associated to the exact sequence of C*-algebras

[TABLE]

induced by the continuous extension to ${\mathfrak{A}}$ of the usual principal symbol of pseudodifferential operators (see Theorem 1).

We first consider the exact sequence of C*-algebras associated to the map $\sigma_{S}$ defined in (4),

[TABLE]

This sequence tells us that an $A\in{\mathfrak{S}}$ is a Fredholm operator if and only if $\sigma_{S}(A)$ never vanishes. A well-known index formula for zero-order pseudodifferential operators on the circle, which essentially goes back to Noether [26] (see also the example right after Theorem 19.2.4 in [17]), implies that the index of a Fredholm operator $A\in{\mathfrak{S}}$ is equal to the difference of the winding numbers of the restrictions of $\sigma_{S}(A)$ to the two copies of $S^{1}$ in $S^{1}\times\{-1,+1\}$. A detailed proof of this fact without explicit mention of pseudodifferential or singular integral operators can be found in [1, Seção 10]. It then follows that there is a Fredholm operator of index 1 in ${\mathfrak{S}}$ and, therefore, the index map $\delta_{1}$ in the standard exact sequence of $K$-groups associated to (13)

[TABLE]

is surjective. We then get $K_{0}({\mathfrak{S}})\cong{\mathbb{Z}}^{2}$ and $K_{1}({\mathfrak{S}})\cong{\mathbb{Z}}$.

Proposition 1**.**

The canonical projection $\pi:{\mathfrak{E}}\to{\mathfrak{E}}/{\mathfrak{K}}_{\Omega}$ induces a $K_{0}$-isomorphism.

**Proof: **The previous considerations and the isomorphisms (5) and (6) imply that

[TABLE]

The standard exact sequence of $K$-groups associated to

[TABLE]

then becomes

[TABLE]

The index map $\delta_{1}$ is non-zero (otherwise there would be an exact sequence $0\to{\mathbb{Z}}\to{\mathbb{Z}}^{2}\to{\mathbb{Z}}^{2}\to 0$). If it were not surjective, the image of the upper left homomorphism would be a finite non-trivial group, which could not be a subgroup of ${\mathbb{Z}}^{2}$. The exactness of the sequence then provides the desired result. $\Box$

The following rephrasing of the fact that $\delta_{1}$ in (14) is surjective is perhaps interesting in itself.

Corollary 1**.**

There exists $k>0$ and a Fredholm operator of index $1$ in $M_{k}({\mathfrak{E}})\subset{\mathfrak{B}}(L^{2}(\Omega)^{k})$.

We will need to use Banach algebra $K$-theory [3], in which $K_{0}$ consists of formal differences of classes of idempotents, while the elements of $K_{1}$ are classes of invertibles. The following lemma was used before in [13, 23, 27], see [23, Lemma 1] for a proof.

Lemma 1**.**

Let $\mathcal{A}$ be a unital Banach algebra, $\mathcal{J}$ a closed ideal of $\mathcal{A}$, and let $\delta_{1}:K_{1}(\mathcal{A}/\mathcal{J})\rightarrow K_{0}(\mathcal{J})$ denote the index map associated to the exact sequence $0\rightarrow\mathcal{J}\rightarrow\mathcal{A}\rightarrow\mathcal{A}/\mathcal{J}\rightarrow 0$. If $u\in M_{k}(\mathcal{A}/\mathcal{J})$ is invertible, $\pi(a)=u$ and $\pi(b)=u^{-1},$ then

[TABLE]

Next we consider the exact sequence obtained from (12) quotienting ${\mathfrak{E}}$ and ${\mathfrak{A}}$ by the compact ideal,

[TABLE]

The following Proposition describes the connecting map $\delta_{1}:K_{1}({\mathfrak{A}}/{\mathfrak{E}})\to K_{0}({\mathfrak{E}}/{\mathfrak{K}}_{\Omega})$ in terms of the index of Fredholm operators on $L^{2}(S^{1}\times B)$. We recall that each element of $K_{1}({\mathfrak{A}}/{\mathfrak{E}})$ is of the form $[[A]_{{\mathfrak{E}}}]_{1}$, where $A\in M_{k}({\mathfrak{A}})$ is invertible modulo $M_{k}({\mathfrak{E}})$ for some positive integer $k$ and $[A]_{{\mathfrak{E}}}$ denotes its class in the quotient $M_{k}({\mathfrak{A}})/M_{k}({\mathfrak{E}})$.

Proposition 2**.**

Let $A\in M_{k}({\mathfrak{A}})$ be such that $[A]_{{\mathfrak{E}}}$ is invertible in $M_{k}({\mathfrak{A}}/{\mathfrak{E}})$ for some positive integer $k$. Then ${\Gamma}_{A}(z,-1)$ and ${\Gamma}_{A}(z,+1)$ are Fredholm operators on $L^{2}(S^{1}\times B)^{k}$ for every $z\in S^{1}$ and

[TABLE]

where $E$ is a rank one projection on $L^{2}(S^{1}\times B)$ and $\mathsf{ind}$ denotes the Fredholm index $($​It will become clear in the proof how the left side of $(\ref{ph})$ can be regarded as an element of $K_{1}({\mathfrak{E}}/{\mathfrak{K}}_{\Omega}))$.

**Proof: **Recall that the image of ${\mathfrak{E}}$ under $\Gamma$ equals $C(S^{1}\times\{-1,+1\};\,{\mathfrak{K}}_{S^{1}\times B})$.

Let $B\in M_{k}({\mathfrak{A}})$ be such that $I-AB$ and $I-BA$ belong to $M_{k}({\mathfrak{E}})$. Then, $I-\Gamma_{A}\Gamma_{B}$ and $I-\Gamma_{B}\Gamma_{A}$ belong to $C(S^{1}\times\{-1,+1\};\,M_{k}({\mathfrak{K}}_{S^{1}\times B}))$. This means that, at each $(z,\pm 1)$ in $S^{1}\times\{-1,+1\}$, $\Gamma_{A}(z,\pm 1)$ is invertible modulo compacts, and hence it is a Fredholm operator.

Let $a=[A]_{{\mathfrak{K}}}$ and $b=[B]_{{\mathfrak{K}}}$. By Lemma 1,

[TABLE]

(we are regarding ${\mathfrak{E}}/{\mathfrak{K}}_{\Omega}$ as an ideal of ${\mathfrak{A}}/{\mathfrak{K}}_{\Omega}$). Using the isomorphism (6) and the definitions of $\gamma$ and $\Gamma$ that follow (6), we see that the map

[TABLE]

is an isomorphism. Using this isomorphism as an identification, we may replace $a$ and $b$ by $\Gamma_{A}$ and $\Gamma_{B}$ in (17) and get $\delta_{1}([[A]_{{\mathfrak{E}}}]_{1})=$

[TABLE]

[TABLE]

Let us consider now the map $p:C(S^{1};\,{\mathfrak{K}}_{S^{1}\times B})\to{\mathfrak{K}}_{S^{1}\times B}$, $p(f)=f(1)$. The map $q:{\mathfrak{K}}_{S^{1}\times B}\to C(S^{1};\,{\mathfrak{K}}_{S^{1}\times B})$ that regards a compact operator as a constant function is a right inverse for $p$. Moreover, the kernel of $p$ is the suspension of ${\mathfrak{K}}$, and then its $K_{0}$-group is 0. All that imply that $p$ induces a $K_{0}$-isomorphism. Using this isomorphism as an identification, we may write $\delta_{1}([[A]_{{\mathfrak{E}}}]_{1})$ as the element of $K_{0}({\mathfrak{K}}_{S^{1}\times B})\oplus K_{0}({\mathfrak{K}}_{S^{1}\times B})$

[TABLE]

[TABLE]

where we have denoted $\Gamma_{A}(1,\pm 1)$ by $A^{\pm}$ and analogously for $\Gamma_{B}(1,\pm 1)$.

If we apply Lemma 1 to $\mathcal{A}=\mathcal{B}(L^{2}(S^{1}\times B))$, $\mathcal{J}={\mathfrak{K}}_{S^{1}\times B}$ and $u=[A^{\pm}]_{{\mathfrak{K}}}$, recalling that the K-theory index map is a generalization of the Fredholm index [28], we conclude that this element of $K_{0}({\mathfrak{K}}_{S^{1}\times B})\oplus K_{0}({\mathfrak{K}}_{S^{1}\times B})$ that we have just written above is equal to the left side of (16). $\Box$

Theorem 2 below follows immediately from Proposition 1 and Proposition 2. To simplify notation, in its statment we regard as identifications the canonical group isomorphisms

[TABLE]

Theorem 2**.**

Let $\delta_{1}^{/}:K_{1}({\mathfrak{A}}/{\mathfrak{K}})\to K_{0}({\mathfrak{E}})$ denote the index map associated to the exact sequence $0\to{\mathfrak{E}}\to{\mathfrak{A}}\to{\mathfrak{A}}/{\mathfrak{E}}\to 0$, let $\Gamma_{*}:K_{0}({\mathfrak{E}})\to{\mathbb{Z}}\oplus{\mathbb{Z}}$ denote the isomorphism induced by the surjective map $\Gamma:{\mathfrak{E}}\to C(S^{1}\times\{-1,+1\};{\mathfrak{K}}_{S^{1}\times B})$, and denote $\delta_{1}=\Gamma_{*}\circ\delta_{1}^{/}$. Given $A\in M_{k}({\mathfrak{A}})$ such that $[A]_{{\mathfrak{E}}}$ is invertible in $M_{k}({\mathfrak{A}}/{\mathfrak{E}})$ for some positive integer $k$, then $A^{-}={\Gamma}_{A}(z,-1)$ and $A^{+}={\Gamma}_{A}(z,+1)$ are Fredholm operators on $L^{2}(S^{1}\times B)^{k}$ and

[TABLE]

where $\mathsf{ind}$ denotes the Fredholm index.

The final goal should be to describe the index map associated to $(\ref{1})$ in purely topological terms. That can now be easily achieved if we invoke the Atiyah-Singer Theorem.

In the following we adopt topological K-theory definitions and notation of [3, Chapter I]. We denote, as usual, by $T^{*}M$, $B^{*}M$ and $S^{*}M$, respectively, the tangent bundle, the bundle of unit closed balls in the tangent bundle, and the co-sphere bundle of any manifold $M$.

Theorem 3**.**

Let $\delta_{1}$ denote the index map associated to $(\ref{1})$, and let $f\in M_{k}(C(\mathbf{M}))$ be invertible for some positive $k$. The restrictions $f_{-}$ and $f_{+}$ of $f$ to the points of $\mathbf{M}$ with, respectively, $t=-\infty$ and $t=+\infty$ are invertible elements of $M_{k}(C(S^{*}(S^{1}\times B)))$. Let

[TABLE]

denote the classes of the triples $(E^{k},E^{k},f_{\pm})$, where $E^{k}$ denotes the trivial bundle $(S^{1}\times B)\times{\mathbb{C}}^{k}$. Then

[TABLE]

where $ind_{t}:K(T^{*}(S^{1}\times B))\to{\mathbb{Z}}$ denotes the topological index of Atiyah and Singer [2].

**Proof: *Let $A\in M_{k}({\mathfrak{A}})$ be such that $\sigma(A)=f$. Define $A^{\pm}=\Gamma_{A}(1,\pm 1)$. We claim that $A^{-}$ and $A^{+}$ belong to $M_{k}(\Psi(S^{1}\times B))$ and $\sigma_{S^{1}\times B}(A^{\pm})=f_{\pm}$. By definition, $f_{\pm}=\sigma(A)|_{t=\pm\infty}$. The maps ${\mathfrak{A}}\ni A\mapsto\sigma(A)|_{t=\pm\infty}$, ${\mathfrak{A}}\ni A\mapsto\Gamma_{A}(1,\pm 1))|_{t=\pm\infty}$ and $\Psi(M)\ni T\mapsto\sigma_{S^{1}\times B}(T)$ are C-algebra homomorphisms. So, it is enough to verify our claim on a set of generators of ${\mathfrak{A}}$. For each of the generators $A_{1},A_{2},\cdots,A_{6}$ defined on page (iii), the claim follows immediately from (10).

Our theorem would already follow from the Atiyah-Singer Theorem [2] if $A^{+}$ and $A^{-}$ were classical pseudodiferential operators. For the general case, we need to show that the topological index of the class $[(E^{k},E^{k},\sigma(T))]\in K(T^{*}(S^{1}\times B))$ is equal to the Fredholm index of any Fredholm operator $T\in\Psi(S^{1}\times B)$ (the Atiyah-Singer Theorem gives that for those $T$ that belong to the dense subalgebra of all classical pseudodifferential operators).

In the rest of this proof, we will replace $S^{1}\times B$ by an arbitrary compact manifold $M$.

Let $\delta^{tbs}$ denote index mapping for the exact sequence of C*algebras

[TABLE]

and let $\iota:K_{0}(C_{0}(T^{*}M))\mapsto K(T^{*}M)$ be the canonical isomorphism between these two groups. We then have

[TABLE]

For a proof of (19) in a slightly more general setting, see [22, Proposition 15], for example.

Now let $T\in M_{k}(\Psi(M))$ be a Fredholm operator. Its Fredholm index $\mathsf{ind}(T)$ coincides with $\delta^{M}([\sigma_{M}(T)]_{1})$, where $\delta^{M}$ denotes the index map of (1). Because every $f\in C^{\infty}(S^{*}M)$ is the symbol of a zero-order classical pseudodifferential operator, there exists such a pseudo $T_{0}$ with invertible symbol so close to $T$ that $[\sigma(T)]_{1}=[\sigma(T_{0})]_{1}$. Then

[TABLE]

[TABLE]

The Atiyah-Singer Theorem was used in the third of the above equalities. Twice we used (19). $\Box$

Example

Some statements made here about operators in ${\mathfrak{A}}$ are proven in [20], using results from [8].

Let $L:C^{\infty}(\Omega)^{k}\to C^{\infty}(\Omega)^{k}$ be a differential operator of the form

[TABLE]

where each $a_{j,\alpha}\in M_{k}(C^{\infty}(\Omega))$ has support contained in $U\times{\mathbb{R}}$, $U$ the domain of a chart in $B$. Moreover, let us assume that $a_{j,\alpha}$ is semi-periodic in the following sense. There exist $2\pi$-periodic (in $t$) $a_{j,\alpha}^{\pm}\in M_{k}(C^{\infty}(\Omega))$ and, for a choice of real-valued $\chi^{\pm}\in C^{\infty}({\mathbb{R}})$ such that $\chi^{+}(t)+\chi^{-}(t)=1$ for all $t$ and $\chi_{\pm}(t)=0$ for $\mp t>1$, there exists $a^{0}_{j,\alpha}\in M_{k}(C^{\infty}(\Omega))$ vanishing at infinity such that

[TABLE]

The operator $A=L\Lambda^{N}$​, a priori defined on the dense subspace $[\Lambda^{-N}(C_{c}^{\infty}(\Omega))]^{k}$ of $L^{2}(\Omega)^{k}$, extends to a bounded operator $A\in M_{k}({\mathfrak{A}})$ ([8, Sections VII-3 and IX-3], [20, Theorem 3.7]). Each of the generators $A_{1},\cdots,A_{5}$ defined on page (iii) is a particular example of such an $A$, with $k=1$ and $N$ equal to 0 or 1. And because $B$ has a finite atlas, each $A_{6}$ is a finite sum of such $A$’s.

The symbol $\sigma_{A}$ coincides with the continuous extension to the compactification $\mathbf{M}$ of $S^{*}\Omega$ of the principal symbol of $L$,

[TABLE]

It follows that $A$ is invertible modulo ${\mathfrak{E}}$ if and only if $L$ is uniformily elliptic in the sense that

[TABLE]

We also have

[TABLE]

($\theta$ denotes the argument of a point in $S^{1}$). These are zero-order elliptic classical pseudodifferential operators on $S^{1}\times B$ whose principal symbols are

[TABLE]

Having smooth symbols, their indices (and then $\delta_{1}([[A]_{\mathfrak{E}}]_{1})$) can be more explicitly calculated as the integral of certain diferential forms on $S^{*}(S^{1}\times B)$.

With the aid of a computer algebra system, Patrícia Hess [13] was able to follow this path and, using Fedosov’s formula [12, formula (1)], find the index of a 2-by-2 pseudodifferential operator on $S^{1}\times S^{1}$. That result was used in her proof that, for the case $B=S^{1}$, $K_{0}({\mathfrak{A}})\cong{\mathbb{Z}}^{5}$ and $K_{1}({\mathfrak{A}})\cong{\mathbb{Z}}^{4}$.