K-stability of continuous C(X)-algebras

TL;DR

This paper investigates the K-stability of continuous C(X)-algebras with K-stable fibers, proving that under certain topological conditions on X, the entire algebra inherits K-stability.

Contribution

It establishes that continuous C(X)-algebras with K-stable fibers are K-stable when the base space is compact, metrizable, and finite-dimensional.

Findings

K-stability of fibers implies K-stability of the algebra under specified conditions

The result applies to compact, metrizable, finite-dimensional spaces

Provides criteria for inheriting K-stability in continuous C(X)-algebras

Abstract

A C*-algebra is said to be K-stable if its nonstable K-groups are naturally isomorphic to the usual K-theory groups. We study continuous $C(X)$-algebras, each of whose fibers are K-stable. We show that such an algebra is itself K-stable under the assumption that the underlying space $X$ is compact, metrizable, and of finite covering dimension.

Equations

$$a\cdot b=a+b-ab$$

$$r(a):=1-(1-a)((1-a^{\ast })(1-a){)}^{-1/2}$$

$$k_n(A):={\pi }_{n+1}(\hat{\mathcal{U}}(A)),\text{ for }n=-1,0,\dots$$

$$a\mapsto \left(\begin{array}{cc}a& 0\\ 0& 0\end{array}\right)$$

$$A=B{\oplus }_{D}C:=\{(b,c)\in B\oplus C:\delta (b)=\gamma (c)\}$$

$$A\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\tilde{r}(f)(x):=r(f(x))$$

$$\tilde{F}(s,f)(x):=F(s,f(x))$$

$$\Lambda (c):=-\sum _{n=1}^{\infty }\frac{(ic{)}^n}{n!}$$

$$\Vert d\Vert =\Vert 1-((1-a)(1-b^{\ast }))\Vert =\Vert ((1-b)-(1-a))(1-b^{\ast })\Vert \le \Vert a-b\Vert <2$$

$$\Vert u-a\Vert =\Vert v-b\Vert =\Vert v(1-|b|)\Vert \le \Vert 1-|b|\Vert <\delta <ϵ$$

$$G(t)={\pi }_Y(b)\cdot \Lambda (t{\pi }_Y(c))$$

$$({\iota }_m{)}_{\ast }:{\pi }_0(M_{m-1}(A))\to {\pi }_0(M_m(A))$$

$${\iota }_{\ast }:{\pi }_0(\hat{\mathcal{U}}(A))\to {\pi }_0(\hat{\mathcal{U}}(M_n(A))$$

$$A\cong A(Y_{x_1})\oplus A(Y_{x_2})\oplus \dots \oplus A(Y_{x_n})$$

$$\Vert {\eta }_{Y_x}(b_x)-{\eta }_{Y_x}(\iota (e_x))\Vert <1,\Vert {\pi }_{Y_x}(e_x^{\ast }\cdot e_x)\Vert <\delta ,\text{ and}\Vert {\pi }_{Y_x}(e_x\cdot e_x^{\ast })\Vert <\delta$$

$$A\cong A(Y_{x_1})\oplus A(Y_{x_2})\oplus \dots \oplus A(Y_{x_n})$$

$$M_2(A)\cong M_2(A(Y_{x_1}))\oplus M_2(A(Y_{x_2}))\oplus \dots \oplus M_2(A(Y_{x_n}))$$

$$H(s,t)={\pi }_Y(f_2(t))\cdot \Lambda (sh(t))$$

$$F(0)=0\text{ and }F(1)=\iota (a)$$

$$L_Y(0)=0,L_Y(1)={\pi }_Y(a)$$

$$\hat{\mathcal{U}}(A)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\eta }_x\circ {\iota }_A\circ G_x={\iota }_x\circ {\pi }_x\circ G_x={\iota }_x\circ F^{\prime }={\iota }_x\circ r_x\circ {\eta }_x\circ F$$

$$\tilde{H}(0,t)={\eta }_x\circ F(t),\tilde{H}(1,t)={\eta }_x\circ {\iota }_A\circ G_x(t),\tilde{H}(s,0)=0,\tilde{H}(s,1)={\iota }_x(a(x))$$

$${\eta }_x\circ F(t)={\eta }_x\circ H_2(0,t)\text{ and }{\eta }_x\circ H_1(1,t)={\eta }_x\circ {\iota }_A\circ G_x(t)$$

$$H(0,t)={\eta }_{Y^{\prime }}\circ F(t),H(1,t)={\eta }_{Y^{\prime }}\circ {\iota }_A\circ G_x(t),\text{ and }{\eta }_x^{Y^{\prime }}\circ H(s,1)={\iota }_x(a(x))$$

$${\eta }_x^{Y^{\prime }}\circ f(s)={\iota }_x(a(x))={\eta }_x^{Y^{\prime }}\circ {\iota }_{Y^{\prime }}\circ G_{Y^{\prime }}(s)\forall s\in [0,1]$$

$$L_Y:=({\pi }_Y\circ G_x)\bullet ({\pi }_Y^{Y^{\prime }}\circ G_{Y^{\prime }})$$

$$\dim (\partial U)\le \dim (X)-1$$

Full text

Nonstable K-theory is the study of the homotopy groups of the unitary group of a C*-algebra. The study of these groups was initiated by Rieffel [13], who showed that, for an irrational rotation algebra $A$, the inclusion map from $A$ to $M_{n}(A)$ induces an isomorphism between the corresponding homotopy groups. In other words, the nonstable $K$-groups are naturally isomorphic to the usual $K$-theory groups of the algebra.

The theory was further explored by Thomsen [16], who used the notion of quasi-unitaries to profitably extend nonstable K-theory to encompass non-unital C*-algebras. Furthermore, he showed that this forms a homology theory, which allowed him to explicitly calculate these groups for certain C*-algebras. In particular, he showed that certain infinite dimensional C*-algebras (including the Cuntz algebras and simple infinite dimensional AF-algebras) satisfy the property enjoyed by the irrational rotation algebra mentioned above; a property he termed $K$-stability.

Since then, it has been proved (See Section 1.1) that a variety of interesting simple C*-algebras are $K$-stable. The goal of this paper is to enlarge this class of C*-algebras to include non-simple C*-algebras.

By the Dauns-Hoffmann theorem (See [11] or [14]), any non-simple C*-algebra may be represented as the section algebra of an upper semi-continuous C*-bundle over a compact space. If we assume that the underlying space $X$ is Hausdorff, then such an algebra carries a non-degenerate, central action of $C(X)$, and is called a $C(X)$-algebra. An interesting sub-class of $C(X)$-algebras are ones that come equipped with a natural continuity condition. These algebras, called continuous $C(X)$-algebras, are particularly tractable as one can often take phenomena that occur at each fiber and propagate them to understand local behaviour of the algebra. Using a compactness argument, one may even be able to understand global behaviour. It is this idea that we employ in this paper to prove our main theorem.

1. Preliminaries

1.1. Nonstable $K$-theory

We begin by reviewing the work of Thomsen of constructing the nonstable $K$-groups associated to a C*-algebra. For the proofs of all the facts mentioned below, the reader may refer to [16].

Let $A$ be a $C^{*}$-algebra (not necessarily unital). Define an associative composition $\cdot$ on $A$ by

[TABLE]

Henceforth, if $a,b\in A$, then $ab$ will denote the usual multiplication in the algebra, and should not be confused with $a\cdot b$, which will denote the above composition.

An element $a\in A$ is said to be quasi-invertible if there exists $b\in A$ such that $a\cdot b=b\cdot a=0$, and we write $\widehat{\mathcal{GL}}(A)$ for the set of all quasi-invertible elements in $A$. An element $u\in A$ is said to be a quasi-unitary if $u\cdot u^{\ast}=u^{\ast}\cdot u=0$, and we write $\widehat{\mathcal{U}}(A)$ for the set of all quasi-unitary elements in $A$.

If $B$ is a unital $C^{*}$-algebra, we write $GL(B)$ for the group of invertibles in $B$ and $U(B)$ for the group of unitaries in $B$. Let $A^{+}$ denote the unitization of $A$. It follows by [16, Lemma 1.2] that an element $a\in A$ is quasi-invertible if and only if $1-a\in GL(A^{+})$; and similarly, $u\in A$ is a quasi-unitary if and only if $(1-u)\in U(A^{+})$. Therefore, $\widehat{\mathcal{GL}}(A)$ is open in $A$, $\widehat{\mathcal{U}}(A)$ is closed in $A$, and they both form topological groups. Furthermore, the map $r:\widehat{\mathcal{GL}}(A)\to\widehat{\mathcal{U}}(A)$ given by

[TABLE]

is a strong deformation retract, and hence a homotopy equivalence.

For elements $u,v\in\widehat{\mathcal{U}}(A)$, we write $u\sim v$ if there is a continuous function $f:[0,1]\to\widehat{\mathcal{U}}(A)$ such that $f(0)=u$ and $f(1)=v$. We write $\widehat{\mathcal{U}}_{0}(A)$ for the set of $u\in\widehat{\mathcal{U}}(A)$ such that $u\sim 0$. The next result, which we will use repeatedly throughout the paper, follows from [16, Theorem 1.9] and [3, Theorem 4.8].

Theorem 1.1**.**

If $\varphi:A\to B$ is a surjective $\ast$-homomorphism between two C-algebras, then the induced map $\varphi:\widehat{\mathcal{U}}_{0}(A)\to\widehat{\mathcal{U}}_{0}(B)$ is a Serre fibration.*

For a C*-algebra $A$, the suspension of $A$ is defined to be $SA:=C_{0}(\mathbb{R})\otimes A$. For $n>1$, we set $S^{n}A:=S(S^{n-1}A)$. We then have

Lemma 1.2**.**

[16, Lemma 2.3]** For any C-algebra $A$, $\pi_{n}(\widehat{\mathcal{U}}(A))\cong\pi_{0}(\widehat{\mathcal{U}}(S^{n}A))$.*

Definition 1.3**.**

The nonstable $K$-groups of a C*-algebra $A$ are defined as

[TABLE]

By 1.2 and Bott periodicity, it follows [16, Proposition 2.6] that, if $A$ is a stable C*-algebra, then $k_{n}(A)\cong K_{n}(A)$, where $K_{n}(A)$ denotes the usual $K$-theory groups of $A$. Motivated by this, Thomsen defines the notion of $K$-stability of a C*-algebra.

Definition 1.4**.**

Let $A$ be a $C^{*}$-algebra and $m\geq 2$. Define $\iota_{m}:M_{m-1}(A)\to M_{m}(A)$ by

[TABLE]

We say that $A$ is $K$-stable if $(\iota_{m})_{\ast}:k_{n}(M_{m-1}(A))\to k_{n}(M_{m}(A))$ is an isomorphism for all $n=-1,0,1,2\ldots$ and all $m=2,3,4\ldots$.

Remark 1.5**.**

The following C*-algebras are known to be $K$-stable:

• •

If $\mathcal{Z}$ denotes the Jiang-Su algebra, then $A\otimes\mathcal{Z}$ is $K$-stable for any C*-algebra $A$ [7]. In particular, every separable, approximately divisible C*-algebra is $K$-stable [17].

• •

Every irrational rotation algebra is $K$-stable [13].

• •

If $\mathcal{O}_{n}$ denotes the Cuntz algebra, then $A\otimes\mathcal{O}_{n}$ is $K$-stable for any C*-algebra $A$ [16].

• •

If $A$ is an infinite dimensional simple AF-algebra, then $A\otimes B$ is $K$-stable for any C*-algebra $B$ [16] .

• •

If $A$ is a purely infinite, simple C*-algebra, and $p$ any non-zero projection of $A$, then $pAp$ is $K$-stable [18].

Note that some of the examples mentioned above may be subsumed into the first example as they are known to absorb $\mathcal{Z}$ tensorially. However, it is worth mentioning that the original proofs of $K$-stability for these algebras does not use $\mathcal{Z}$-stability. Furthermore, in the case where the algebras are $\mathcal{Z}$-stable, A may be deduced from [6, Theorem 4.6].

We conclude this section with an important observation about $K$-stable C*-algebras.

Lemma 1.6**.**

If $A$ is $K$-stable, then for any $m\geq 2,\iota_{m}(\widehat{\mathcal{U}}(M_{m-1}(A))$ is a strong deformation retract of $\widehat{\mathcal{U}}(M_{m}(A))$.

Proof.

Note that $\widehat{\mathcal{GL}}(A)$ is an absolute neighbourhood retract [12, Theorem 5], and therefore, the pair $(\widehat{\mathcal{GL}}(M_{m}(A)),\iota_{m}(\widehat{\mathcal{GL}}(M_{m-1}(A))))$ has the homotopy extension property with respect to all spaces [12, Theorem 7]. If $A$ is $K$-stable, then $\iota_{m}:\widehat{\mathcal{GL}}(M_{m-1}(A))\to\widehat{\mathcal{GL}}(M_{m}(A))$ is a weak homotopy equivalence. However, since $\widehat{\mathcal{GL}}(A)$ is an open subset of a normed linear space, $\widehat{\mathcal{GL}}(A)$ has the homotopy type of a CW-complex [10, Chapter IV, Corollary 5.5]. By Whitehead’s theorem, it follows that $\iota_{m}$ is a homotopy equivalence, so $\iota_{m}(\widehat{\mathcal{GL}}(M_{m-1}(A))$ is a strong deformation retract of $\widehat{\mathcal{GL}}(M_{m}(A))$ by [5, Theorem 0.20]. Since the retractions $r:\widehat{\mathcal{GL}}(M_{k}(A))\to\widehat{\mathcal{U}}(M_{k}(A))$ commute with the inclusion map $\iota_{k}$, we conclude that $\iota_{m}(\widehat{\mathcal{U}}(M_{m-1}(A))$ is a strong deformation retract of $\widehat{\mathcal{U}}(M_{m}(A))$. ∎

1.2. $C(X)$-algebras

Let $A$ be a C*-algebra, and $X$ a compact Hausdorff space. We say that $A$ is a $C(X)$-algebra [8, Definition 1.5] if there is a unital $\ast$-homomorphism $\theta:C(X)\to ZM(A)$, where $ZM(A)$ denotes the center of the multiplier algebra of $A$.

If $Y\subset X$ is closed, the set $C_{0}(X,Y)$ of functions in $C(X)$ that vanish on $Y$ is a closed ideal of $C(X)$. By the Cohen factorization theorem [1, Theorem 4.6.4], $C_{0}(X,Y)A$ is a closed, two-sided ideal of $A$. The quotient of $A$ by this ideal is denoted by $A(Y)$, and we write $\pi_{Y}:A\to A(Y)$ for the quotient map (also referred to as the restriction map). If $Z\subset Y$ is a closed subset of $Y$, we write $\pi^{Y}_{Z}:A(Y)\to A(Z)$ for the natural restriction map, so that $\pi_{Z}=\pi^{Y}_{Z}\circ\pi_{Y}$. If $Y=\{x\}$ is a singleton, we write $A(x)$ for $A(\{x\})$ and $\pi_{x}$ for $\pi_{\{x\}}$. The algebra $A(x)$ is called the fiber of $A$ at $x$. For $a\in A$, write $a(x)$ for $\pi_{x}(a)$. For each $a\in A$, we have a map $\Gamma_{a}:X\to{\mathbb{R}}$ given by $x\mapsto\|a(x)\|$. We say that $A$ is a continuous $C(X)$-algebra if $\Gamma_{a}$ is continuous for each $a\in A$.

If $A$ is a continuous $C(X)$-algebra, we will often have reason to consider other $C(X)$-algebras obtained from $A$. At that time, the following result of Kirchberg and Wasserman will be useful.

Theorem 1.7**.**

[9, Remark 2.6]** Let $X$ be a compact Hausdorff space, and let $A$ be a continuous $C(X)$-algebra. If $B$ is a nuclear C-algebra, then $A\otimes B$ is a continuous $C(X)$-algebra whose fiber at a point $x\in X$ is $A(x)\otimes B$.*

Finally, one fact that plays a crucial role in our investigation is that a $C(X)$-algebra may be patched together from quotients in the following way: Let $B,C,$ and $D$ be C*-algebras, and $\delta:B\to D$ and $\gamma:C\to D$ be $\ast$-homomorphisms. We define the pullback of this system to be

[TABLE]

This is describe by a diagram

[TABLE]

where $\phi(b,c)=b$ and $\psi(b,c)=c$.

Lemma 1.8**.**

[2, Lemma 2.4]** Let $X$ be a compact Hausdorff space and $Y$ and $Z$ be two closed subsets of $X$ such that $X=Y\cup Z$. If $A$ is a $C(X)$-algebra, then $A$ is isomorphic to the pullback

[TABLE]

1.3. Notational Conventions

We fix some notational conventions we will use repeatedly: If $A$ is a C*-algebra, we write $\iota_{A}:A\to M_{2}(A)$ for the natural inclusion map. When there is no ambiguity, we write $\iota$ for this map. Moreover, if $\varphi:A\to B$ is a $\ast$-homomorphism between two C*-algebras, then the induced map from $\widehat{\mathcal{U}}(A)$ to $\widehat{\mathcal{U}}(B)$ is also denoted by $\varphi$.

If $A$ is a continuous $C(X)$-algebra, then $M_{2}(A)$ is also a continuous $C(X)$-algebra with fibers $M_{2}(A(x))$ by Theorem 1.7. We will often consider both simultaneously, so we fix the following convention: If $Y\subset X$ is a closed set, we denote the restriction map by $\eta_{Y}:M_{2}(A)\to M_{2}(A(Y))$, and write $\iota_{Y}:A(Y)\to M_{2}(A(Y))$ for the natural inclusion map. If $Y=X$, we simply write $\iota$ for $\iota_{X}$. Note that $\eta_{Y}\circ\iota=\iota_{Y}\circ\pi_{Y}$. Once again, if $Y=\{x\}$, we simply write $\iota_{x}$ for $\iota_{\{x\}}$.

Finally, suppose $f$ and $g$ are two continuous paths in a topological space $Y$. If $f(1)=g(0)$, we write $f\bullet g$ for the concatenation of the two paths. If $f$ and $g$ agree at end-points, we write $f\sim_{h}g$ if there is a path homotopy between them. Furthermore, we write $\overline{f}$ for the path $\overline{f}(t):=f(1-t)$. If $Y=\widehat{\mathcal{U}}(A)$ for some C*-algebra $A$, we write $f\cdot g$ for the path $t\mapsto f(t)\cdot g(t)$, and we write $f^{\ast}$ for the path $t\mapsto f(t)^{\ast}$.

2. Main Results

Lemma 2.1**.**

Let $A$ be a $K$-stable C-algebra and $X$ a locally compact Hausdorff space, then $C_{0}(X)\otimes A$ is $K$-stable.*

Proof.

Suppose first that $X$ is compact. For simplicity of notation, we write $B$ for $C(X)\otimes A$. By 1.6, for each $m\geq 2$, there is a retraction $r:\widehat{\mathcal{U}}(M_{m}(A))\to\widehat{\mathcal{U}}(M_{m-1}(A))$ and a homotopy $F:[0,1]\times\widehat{\mathcal{U}}(M_{m}(A))\to\widehat{\mathcal{U}}(M_{m}(A))$ such that $F(0,v)=v$ and $F(1,v)=r(v)$ for all $v\in\widehat{\mathcal{U}}(M_{m}(A))$. Clearly, we may identify $\widehat{\mathcal{U}}(M_{m}(B))$ with $C(X,\widehat{\mathcal{U}}(M_{m}(A))$, where the latter denotes the space of continuous functions from $X$ to $\widehat{\mathcal{U}}(M_{m}(A))$, equipped with the uniform topology (which coincides with the compact-open topology). Therefore, we may define $\widetilde{r}:\widehat{\mathcal{U}}(M_{m}(B))\to\widehat{\mathcal{U}}(M_{m-1}(B))$ by

[TABLE]

and $\widetilde{F}:[0,1]\times\widehat{\mathcal{U}}(M_{m}(B))\to\widehat{\mathcal{U}}(M_{m}(B))$ by

[TABLE]

It is easy to see that $\widetilde{r}$ is a retraction, and that $\widetilde{F}$ is continuous, and implements a homotopy between $\iota_{M_{m-1}(B)}\circ\widetilde{r}$ and $\text{id}_{\widehat{\mathcal{U}}(M_{m}(B))}$. Thus, $B$ is $K$-stable.

If $X$ is not compact, let $X^{+}$ denote its one-point compactification. We now have a short exact sequence $0\to C_{0}(X)\otimes A\to C(X^{+})\otimes A\to A\to 0$, which induces a long exact sequence of $k$-groups by [16, Theorem 2.5]. By the first part of the argument, $C(X^{+})\otimes A$ is $K$-stable, so the result follows from the five lemma. ∎

Let $A$ be a C*-algebra, and $c\in A$ a self-adjoint element. We define

[TABLE]

Observe that, in $A^{+},\Lambda(c)=1-\exp(ic)$, so that $\Lambda(c)\sim 0$ in $\widehat{\mathcal{U}}(A)$ via the path $t\mapsto\Lambda(tc)$. The next lemma is implicit in [16, Lemma 1.7], but we spell it out since its proof is crucial to us.

Lemma 2.2**.**

Let $a,b\in\widehat{\mathcal{U}}(A)$ such that $\|a-b\|<2$, then $a\sim b$ in $\widehat{\mathcal{U}}(A)$

Proof.

Consider $A$ as an ideal in $A^{+}$. If $\|a-b\|<2$, then $d:=a\cdot b^{\ast}$ satisfies

[TABLE]

Hence, $(1-d)$ is a unitary in $A^{+}$, whose spectrum does not contain $1$. Therefore, there is a continuous function $g:S^{1}\to{\mathbb{R}}$ such that $g(1)=0$ and $\exp(ig(x))=x$ for all $x\in\sigma(1-d)$. Define $c:=g(1-d)$, then $c$ is self-adjoint and $a\cdot b^{\ast}=\Lambda(c)$. Thus $a=\Lambda(c)\cdot b\sim b$ in $\widehat{\mathcal{U}}(A)$. ∎

The next lemma is a variant of [15, Exercise 2.8] for quasi-unitaries.

Lemma 2.3**.**

Given $\epsilon>0$, there is a $\delta>0$ with the following property: If $A$ is a C-algebra and $a\in A$ such that $\|a\cdot a^{\ast}\|<\delta$ and $\|a^{\ast}\cdot a\|<\delta$, then there is a quasi-unitary $u\in\widehat{\mathcal{U}}(A)$ such that $\|a-u\|<\epsilon$.*

Proof.

Fix $0<\delta<\min\{\epsilon,1\}$, and consider $A$ as an ideal in $A^{+}$. If $b:=(1-a)$, the hypothesis implies that $\|1-bb^{\ast}\|=\|a\cdot a^{\ast}\|<\delta<1$, and $\|1-b^{\ast}b\|<1$ as well. Hence, $bb^{\ast}$ and $b^{\ast}b$ are both invertible, and therefore $b$ and $b^{\ast}$ are also invertible. Thus, $|b|=(b^{\ast}b)^{1/2}\in GL(A^{+})$, and $v:=b|b|^{-1}\in U(A^{+})$. Furthermore, $\sigma(b^{\ast}b)\subset(1-\delta,1+\delta)$, so $\sigma(|b|)\subset((1-\delta)^{1/2},(1+\delta)^{1/2})\subset(1-\delta,1+\delta)$. Hence, $\||b|-1\|<\delta$, so that $u:=1-v\in\widehat{\mathcal{U}}(A)$, and

[TABLE]

as required. ∎

Lemma 2.4**.**

Let $X$ be a compact, Hausdorff space and let $A$ be a continuous $C(X)$-algebra. Let $a\in\widehat{\mathcal{U}}(A)$ and $x\in X$ such that $a(x)\sim 0$ in $\widehat{\mathcal{U}}(A(x))$. Then there is a closed neighbourhood $Y$ of $x$ such that $\pi_{Y}(a)\sim 0$ in $\widehat{\mathcal{U}}(A(Y))$.

Proof.

Since $\pi_{x}:\widehat{\mathcal{U}}_{0}(A)\to\widehat{\mathcal{U}}_{0}(A(x))$ is a fibration, there is a path $H:[0,1]\to\widehat{\mathcal{U}}(A)$ such that $H(0)=0$ and $H(1)(x)=a(x)$. Let $b:=H(1)$, then since $A$ is a continuous $C(X)$-algebra, there is a closed neighbourhood $Y$ of $x$ such that $\|\pi_{Y}(a-b)\|<2$. It follows by 2.3 that $\pi_{Y}(a)\sim\pi_{Y}(b)$ in $\widehat{\mathcal{U}}(A(Y))$. Furthermore, the path $G:[0,1]\to\widehat{\mathcal{U}}(A(Y))$ is of the form

[TABLE]

for some self-adjoint element $c\in A$ (since every self-adjoint element in $A(Y)$ lifts to a self-adjoint element in $A$). Since $a(x)=b(x)$, the proof of 2.2 in fact ensures that we may choose $c$ such that $c(x)=0$. Therefore, $\pi_{x}\circ G(t)=a(x)$ for all $t\in[0,1]$. Concatenating the paths $\pi_{Y}\circ H$ and $G$, we obtain a path connecting [math] to $\pi_{Y}(a)$ in $\widehat{\mathcal{U}}(A(Y))$. ∎

Our proof of A is by induction on the covering dimension of the underlying space. The next theorem is the base case, and it holds even if the space is not metrizable.

Theorem 2.5**.**

Let $X$ be a compact Hausdorff space of zero covering dimension, and let $A$ be a continuous $C(X)$-algebra. If each fiber of $A$ is $K$-stable, then so is $A$.

Proof.

If $A$ is a continuous $C(X)$-algebra, then so is every suspension of $A$. Furthermore, $(S^{n}A)(x)\cong S^{n}(A(x))$ by Theorem 1.7, and $S^{n}(A(x))$ is $K$-stable by 2.1. Hence, by 1.2, it suffices to show that, for each $m\geq 2$, the map

[TABLE]

is an isomorphism. However, by Theorem 1.7, each $M_{n}(A)$ is also a continuous $C(X)$-algebra, with fibers $M_{n}(A(x))$, which is $K$-stable if $A(x)$ is $K$-stable. Therefore, suffices to show that the map

[TABLE]

is an isomorphism for each $n\geq 2$. For simplicity of notation, we fix $n=2$.

We first consider injectivity. Suppose $a\in\widehat{\mathcal{U}}(A)$ such that $\iota(a)\sim 0$ in $\widehat{\mathcal{U}}(M_{2}(A))$. Then, for any $x\in X$, $\iota_{x}(a(x))\sim 0\text{ in }\widehat{\mathcal{U}}(M_{2}(A(x))$. Since $A(x)$ is $K$-stable, $a(x)\sim 0$ in $\widehat{\mathcal{U}}(A(x))$. By 2.4, there is a closed neighbourhood $Y_{x}$ of $x$ such that $\pi_{Y_{x}}(a)\sim 0$ in $\widehat{\mathcal{U}}(A(Y_{x}))$. Since $X$ is compact and zero dimensional, we obtain disjoint open sets $\{Y_{x_{1}},Y_{x_{2}},\ldots,Y_{x_{n}}\}$ which cover $X$. Then by 1.8,

[TABLE]

via the map $b\mapsto(\pi_{Y_{x_{1}}}(b),\pi_{Y_{x_{2}}}(b),\ldots,\pi_{Y_{x_{n}}}(b))$. Since $\pi_{Y_{x_{i}}}(a)\sim 0$ for each $1\leq i\leq n$, it follows that $a\sim 0$, so $\iota_{\ast}$ is injective.

For surjectivity, choose $u\in\widehat{\mathcal{U}}(M_{2}(A))$, and we wish to construct a quasi-unitary $\omega\in\widehat{\mathcal{U}}(A)$ such that $u\sim\iota(\omega)$. To this end, fix $x\in X$. Then $u(x)\in\widehat{\mathcal{U}}(M_{2}(A(x))$. Since $A(x)$ is $K$-stable, there exists $f_{x}\in\widehat{\mathcal{U}}(A(x))$ and a path $g_{x}:[0,1]\to\widehat{\mathcal{U}}(M_{2}(A(x))$ such that $g_{x}(0)=u(x)$ and $g_{x}(1)=\iota_{x}(f_{x})$. Choose $e_{x}\in A$ such that $e_{x}(x)=f_{x}$ (Note that $e_{x}$ may not be a quasi-unitary).

Since the map $\eta_{x}:\widehat{\mathcal{U}}_{0}(M_{2}(A))\to\widehat{\mathcal{U}}_{0}(M_{2}(A(x)))$ is a fibration, $g_{x}$ lifts to a path $G_{x}:[0,1]\to\widehat{\mathcal{U}}(M_{2}(A))$ such that $G_{x}(0)=u$. Let $b_{x}:=G_{x}(1)$, and so that $b_{x}(x)=\iota_{x}(e_{x}(x))$. Choose $\delta>0$ so that conclusion of 2.3 holds for $\epsilon=1$. Since $A$ is a continuous $C(X)$-algebra, there is a closed neighbourhood $Y_{x}$ of $x$ such that

[TABLE]

By 2.3, there is a quasi-unitary $d_{x}\in\widehat{\mathcal{U}}(A(Y_{x}))$ such that $\|d_{x}-\pi_{Y_{x}}(e_{x})\|<1$, so that $\|\iota_{Y_{x}}(d_{x})-\eta_{Y_{x}}(b_{x})\|<2$. By 2.2, $\iota_{Y_{x}}(d_{x})\sim\eta_{Y_{x}}(b_{x})$ in $\widehat{\mathcal{U}}(A(Y_{x}))$. Hence, $\iota_{Y_{x}}(d_{x})\sim\eta_{Y_{x}}(u)$. As before, since $X$ is compact and zero-dimensional, we may choose disjoint, open sets $\{Y_{x_{1}},Y_{x_{2}},\ldots,Y_{x_{n}}\}$ so that

[TABLE]

via the map $a\mapsto(\pi_{Y_{x_{1}}}(a),\pi_{Y_{x_{2}}}(a),\ldots,\pi_{Y_{x_{n}}}(a))$. Similarly,

[TABLE]

via the map $b\mapsto(\eta_{Y_{x_{1}}}(b),\eta_{Y_{x_{2}}}(b),\ldots,\eta_{Y_{x_{n}}}(b))$. Therefore, there exists $\omega\in\widehat{\mathcal{U}}(A)$ such that $\pi_{Y_{x_{i}}}(\omega)=d_{x_{i}}$ for all $1\leq i\leq n$. Furthermore, for each $1\leq i\leq n,\eta_{Y_{x_{i}}}(\iota(\omega))=\iota_{Y_{x_{i}}}(d_{x_{i}})\sim\eta_{Y_{x_{i}}}(u)$ in $\widehat{\mathcal{U}}(A(Y_{x_{i}}))$, so that $\iota(\omega)\sim u$ in $\widehat{\mathcal{U}}(A)$, as required. ∎

The next two lemmas help us to extend the above argument to higher dimensional spaces.

Lemma 2.6**.**

Let $X$ be a compact Hausdorff space, and $A$ be a continuous $C(X)$-algebra. Let $f_{i}:[0,1]\to\widehat{\mathcal{U}}(A),i=1,2$ be two paths, and $x\in X$ be a point such that $\pi_{x}\circ f_{1}=\pi_{x}\circ f_{2}$.

1. (1)

There is a closed neighbourhood $Y$ of $x$ and a homotopy $H:[0,1]\times[0,1]\to\widehat{\mathcal{U}}(A(Y))$ such that $H(0)=\pi_{Y}\circ f_{1}$ and $H(1)=\pi_{Y}\circ f_{2}$. 2. (2)

If, in addition, $f_{1}(0)=f_{2}(0)$ and $f_{1}(1)=f_{2}(1)$, then the homotopy in part (1) may be chosen to be a path homotopy.

Proof.

For the first part, note that $C[0,1]\otimes A$ is itself a continuous $C(X)$-algebra by Theorem 1.7. Hence, there is a closed neighbourhood $Y$ of $x$ such that $\|\pi_{Y}\circ f_{1}-\pi_{Y}\circ f_{2}\|<2$. The result now follows from 2.2.

For the second part, an examination of 2.2 shows that the homotopy is implemented by a path $H:[0,1]\times[0,1]\to\widehat{\mathcal{U}}(A(Y))$ given by

[TABLE]

where $h(t)=g(1-\pi_{Y}(f_{1})(t)\cdot\pi_{Y}(f_{2})(t)^{\ast})$ for an appropriate branch $g:S^{1}\to{\mathbb{R}}$ of the log function such that $g(1)=0$. Since $f_{1}$ and $f_{2}$ agree at end-points, it follows that $h(0)=h(1)=0$. Hence, $H$ is a path homotopy. ∎

Lemma 2.7**.**

Let $X$ be a compact Hausdorff space, $A$ be a continuous $C(X)$-algebra, and $x\in X$ be a point such that $A(x)$ is $K$-stable. Let $a\in\widehat{\mathcal{U}}(A)$ be a quasi-unitary and $F:[0,1]\to\widehat{\mathcal{U}}(M_{2}(A))$ be a path such that

[TABLE]

Then, there is a closed neighbourhood $Y$ of $x$ and a path $L_{Y}:[0,1]\to\widehat{\mathcal{U}}(A(Y))$ such that

[TABLE]

and $\iota_{Y}\circ L_{Y}$ is path homotopic to $\eta_{Y}\circ F$ in $\widehat{\mathcal{U}}(M_{2}(A(Y)))$.

Proof.

Let $\iota_{x}:\widehat{\mathcal{U}}(A(x))\to\widehat{\mathcal{U}}(M_{2}(A(x))$ be the natural inclusion map. Since $A(x)$ is $K$-stable, 1.6 implies that there is a continuous function $r_{x}:\widehat{\mathcal{U}}(M_{2}(A(x)))\to\widehat{\mathcal{U}}(A(x))$ such that $\iota_{x}\circ r_{x}\sim\text{id}_{\widehat{\mathcal{U}}(M_{2}(A(x)))}$. Furthermore, since $r_{x}$ is a retract, the function $F^{\prime}:=r_{x}\circ\eta_{x}\circ F$ is a path in $\widehat{\mathcal{U}}(A(x))$ such that $F^{\prime}(0)=0$ and $F^{\prime}(1)=a(x)$. Consider the commutative diagram

[TABLE]

The map $\pi_{x}:\widehat{\mathcal{U}}_{0}(A)\to\widehat{\mathcal{U}}_{0}(A(x))$ is a fibration, so $F^{\prime}$ lifts to a path $G_{x}:[0,1]\to\widehat{\mathcal{U}}(A)$ such that $G_{x}(0)=0$. Set $b_{x}:=G_{x}(1)$, then $\iota_{A}\circ G_{x}:[0,1]\to\widehat{\mathcal{U}}(M_{2}(A))$ is a path such that $\iota_{A}\circ G_{x}(0)=0$ and $\iota_{A}\circ G_{x}(1)=\iota_{A}(b_{x})$. Furthermore,

[TABLE]

Since $\iota_{x}(\widehat{\mathcal{U}}(A(x))$ is a strong deformation retract of $\widehat{\mathcal{U}}(M_{2}(A(x)))$, it follows that there is a path homotopy $\widetilde{H}:[0,1]\times[0,1]\to\widehat{\mathcal{U}}(M_{2}(A(x)))$ such that, for all $s,t\in[0,1]$,

[TABLE]

The map $\eta_{x}:\widehat{\mathcal{U}}_{0}(M_{2}(A))\to\widehat{\mathcal{U}}_{0}(M_{2}(A(x))$ is a fibration, and $\eta_{x}\circ F$ has a lift, so $\widetilde{H}$ lifts to a homotopy $H_{1}:[0,1]\times[0,1]\to\widehat{\mathcal{U}}(M_{2}(A))$ such that $H_{1}(0,t)=F(t)$, and $\eta_{x}\circ H_{1}=\widetilde{H}$. Similarly, $\eta_{x}\circ\iota_{A}\circ G_{x}$ lifts to $\iota_{A}\circ G_{x}$, so $\widetilde{H}$ lifts to a homotopy $H_{2}:[0,1]\times[0,1]\to\widehat{\mathcal{U}}(M_{2}(A))$ such that $H_{2}(1,t)=\iota_{A}\circ G_{x}(t)$, and $\eta_{x}\circ H_{2}=\widetilde{H}$. Note that

[TABLE]

Therefore, by 2.4 applied to the continuous $C(X)$-algebra $C[0,1]\otimes A$, there is a closed neighbourhood $Y^{\prime}$ of $x$ and a homotopy $H:[0,1]\times[0,1]\to\widehat{\mathcal{U}}(M_{2}(A(Y^{\prime}))$ such that

[TABLE]

and a path $G_{Y^{\prime}}:[0,1]\to\widehat{\mathcal{U}}(A(Y^{\prime}))$ such that $G_{Y^{\prime}}(0)=\pi_{Y^{\prime}}(b_{x}),G_{Y^{\prime}}(1)=\pi_{Y^{\prime}}(a)$, and $\pi^{Y^{\prime}}_{x}\circ G_{Y^{\prime}}$ is a constant path $a(x)$ in $\widehat{\mathcal{U}}(A(x))$ (the last statement follows from the proof of 2.4). Furthermore, $f:[0,1]\to\widehat{\mathcal{U}}(M_{2}(A(Y^{\prime}))$ given by $f(s):=H(1-s,1)$ is a path such that $f(0)=\iota_{Y^{\prime}}\circ\pi_{Y^{\prime}}(b_{x})$ and $f(1)=\iota_{Y^{\prime}}\circ\pi_{Y^{\prime}}(a)$. Also,

[TABLE]

Thus, by 2.6 applied to $f$ and $\iota_{Y^{\prime}}\circ G_{Y^{\prime}}$, there is a closed neighbourhood $Y$ of $x$ such that $Y\subset Y^{\prime}$ and $\eta^{Y^{\prime}}_{Y}\circ\iota_{Y^{\prime}}\circ G_{Y^{\prime}}\sim_{h}\eta^{Y^{\prime}}_{Y}\circ f$ in $\widehat{\mathcal{U}}(M_{2}(A(Y))$. But $\eta_{Y}\circ F\sim_{h}(\eta_{Y}\circ\iota_{A}\circ G_{x})\bullet(\eta^{Y^{\prime}}_{Y}\circ f)$, so $\eta_{Y}\circ F\sim_{h}\iota_{Y}\circ L_{Y}$ where $L_{Y}:[0,1]\to\widehat{\mathcal{U}}(A(Y))$ is given by

[TABLE]

and $L_{Y}$ satisfies the required conditions. ∎

Remark 2.8**.**

We are now in a position to prove A, but first, we need one important fact, which allows us to use induction: If $X$ is a finite dimensional compact metric space, then covering dimension agrees with the small inductive dimension [4, Theorem 1.7.7]. Therefore, by [4, Theorem 1.1.6], $X$ has an open cover $\mathcal{B}$ such that, for each $U\in\mathcal{B}$,

[TABLE]

Now suppose $\{U_{1},U_{2},\ldots,U_{m}\}$ is an open cover of $X$ such that $\dim(\partial U_{i})\leq\dim(X)-1$ for $1\leq i\leq m$, we define sets $\{V_{i}:1\leq i\leq m\}$ inductively by

[TABLE]

and subsets $\{W_{j}:1\leq j\leq m-1\}$ by

[TABLE]

It is easy to see that $W_{j}\subset\bigcup_{i=1}^{j}\partial U_{i}$, so by [4, Theorem 1.5.3], $\dim(W_{j})\leq\dim(X)-1$ for all $1\leq j\leq m-1$

Proof of A.

Let $X$ be a compact metric space of finite covering dimension, and let $A$ be a continuous $C(X)$-algebra, each of whose fibers are $K$-stable. We wish to show that $A$ is $K$-stable. As in the proof of Theorem 2.5, it suffices to show that the map

[TABLE]

is bijective. The proof is by induction on $\dim(X)$, so by Theorem 2.5, we assume that $\dim(X)\geq 1$, and that $A(Y)$ is $K$-stable for any closed subset $Y$ of $X$ such that $\dim(Y)\leq\dim(X)-1$.

For injectivity, suppose $a\in\widehat{\mathcal{U}}(A)$ such that $\iota(a)\sim 0$ in $\widehat{\mathcal{U}}(M_{2}(A))$. Let $F:[0,1]\to\widehat{\mathcal{U}}(M_{2}(A))$ be a path such that $F(0)=\iota(a)$ and $F(1)=0$. For $x\in X$, by 2.7, there is a closed neighbourhood $Y_{x}$ of $x$ and a path $L_{Y_{x}}:[0,1]\to\widehat{\mathcal{U}}(A(Y_{x}))$ such that

[TABLE]

and $\iota_{Y_{x}}\circ L_{Y_{x}}$ is path homotopic to $\pi_{Y_{x}}\circ F$ in $\widehat{\mathcal{U}}(A(Y_{x}))$. By 2.8, we may choose $Y_{x}$ to be the closure of a basic open set $U_{x}$ such that $\dim(\partial U_{x})\leq\dim(X)-1$. Since $X$ is compact, we may choose a finite subcover $\{U_{1},U_{2},\ldots,U_{m}\}$. Now define $\{V_{1},V_{2},\ldots,V_{m}\}$ and $\{W_{1},W_{2},\ldots,W_{m-1}\}$ as in 2.8. We observe that each $V_{i}$ is a closed set such that $\pi_{V_{i}}(a)\sim 0$ in $\widehat{\mathcal{U}}(A(V_{i}))$ since $V_{i}\subset\overline{U_{i}}$ for all $1\leq i\leq m$.

Note that $W_{1}=V_{1}\cap V_{2}$, and $\dim(W_{1})\leq\dim(X)-1$. By induction hypothesis, $A(W_{1})$ is $K$-stable. Let $H_{i}:[0,1]\to\widehat{\mathcal{U}}(A(V_{i})),i=1,2$ be paths such that $H_{i}(0)=\pi_{V_{i}}(a),H_{i}(1)=0$, and $\iota_{V_{i}}\circ H_{i}\sim_{h}\eta_{V_{i}}\circ F$. Let $S:[0,1]\to\widehat{\mathcal{U}}(A(W_{1}))$ be the path

[TABLE]

Note that $S(0)=S(1)=0$, so $S$ is a loop, and

[TABLE]

Hence, $\iota_{W_{1}}\circ S$ is null-homotopic. Since the map $(\iota_{W_{1}})_{\ast}:\pi_{1}(\widehat{\mathcal{U}}(A(W_{1})))\to\pi_{1}(\widehat{\mathcal{U}}(M_{2}(A(W_{1}))))$ is injective, it follows that $S$ is null-homotopic in $\widehat{\mathcal{U}}(A(W_{1}))$. Since the map $\pi^{V_{2}}_{W_{1}}:\widehat{\mathcal{U}}_{0}(A(V_{2}))\to\widehat{\mathcal{U}}_{0}(A(W_{1}))$ is a fibration, it follows that $S$ has a lift $f:[0,1]\to\widehat{\mathcal{U}}(A(V_{2}))$ such that $f(0)=f(1)=0$ and $\pi^{V_{2}}_{W_{1}}\circ f=S$. Define $H_{2}^{\prime}:[0,1]\to\widehat{\mathcal{U}}(A(V_{2}))$ be defined by

[TABLE]

Then $H_{2}^{\prime}(0)=\pi_{V_{2}}(a),H_{2}^{\prime}(1)=0$, and

[TABLE]

By 1.8, $A(V_{1}\cup V_{2})$ is a pullback

[TABLE]

so we obtain a path $H:[0,1]\to\widehat{\mathcal{U}}(A(V_{1}\cup V_{2}))$ such that $\pi^{V_{1}\cup V_{2}}_{V_{1}}\circ H=H_{1}$ and $\pi^{V_{1}\cup V_{2}}_{V_{2}}\circ H=H_{2}^{\prime}$. In particular, $\pi^{V_{1}\cup V_{2}}_{V_{1}}\circ H(0)=\pi_{V_{1}}(a)$ and $\pi^{V_{1}\cup V_{2}}_{V_{2}}\circ H(0)=\pi_{V_{2}}(a)$ so $H(0)=\pi_{V_{1}\cup V_{2}}(a)$. Similarly, $H(1)=0$, so that $\pi_{V_{1}\cup V_{2}}(a)\sim 0$ in $\widehat{\mathcal{U}}(A(V_{1}\cup V_{2}))$.

Now observe that $W_{2}=(V_{1}\cup V_{2})\cap V_{3}$, and $\dim(W_{2})\leq\dim(X)-1$. Replacing $V_{1}$ by $V_{1}\cup V_{2}$, and $V_{2}$ by $V_{3}$ in the earlier argument, we may repeat the earlier procedure. By induction on the number of elements in the finite subcover, we see that $a\sim 0$ in $\widehat{\mathcal{U}}(A)$. This completes the proof of injectivity of $\iota_{\ast}$.

Now consider the surjectivity of $\iota_{\ast}$: Fix $u\in\widehat{\mathcal{U}}(M_{2}(A))$, and we wish to show that there is a quasi-unitary $\omega\in\widehat{\mathcal{U}}(A)$ such that $u\sim\iota(\omega)$. So fix $x\in X$. Then by $K$-stability of $A(x)$, there exists $f_{x}\in\widehat{\mathcal{U}}(A(x))$ such that $\eta_{x}(u)\sim\iota_{x}(f_{x})$. As in the proof of Theorem 2.5, there is a closed neighbourhood $Y_{x}$ of $x$ and a quasi-unitary $d_{x}\in\widehat{\mathcal{U}}(A(Y_{x}))$ such that

[TABLE]

As in the first part of the proof, we may reduce to the case where $X=V_{1}\cup V_{2}$, and there are quasi-unitaries $d_{V_{1}}\in\widehat{\mathcal{U}}(A(V_{1})),d_{V_{2}}\in\widehat{\mathcal{U}}(A(V_{2}))$ such that

[TABLE]

and if $W:=V_{1}\cap V_{2}$, then

[TABLE]

Fix paths $H_{i}:[0,1]\to\widehat{\mathcal{U}}(M_{2}(A(V_{i}))$ such that $H_{1}(0)=\iota_{V_{1}}(d_{V_{1}})$ and $H_{1}(1)=\eta_{V_{1}}(u)$, $H_{2}(0)=\eta_{V_{2}}(u)$ and $H_{2}(1)=\iota_{V_{2}}(d_{V_{2}})$ and consider the path $F:[0,1]\to\widehat{\mathcal{U}}(M_{2}(A(W))$ given by

[TABLE]

Then $F(0)=\iota_{W}\circ\pi^{V_{1}}_{W}(d_{V_{1}})$ and $F(1)=\iota_{W}\circ\pi^{V_{2}}_{W}(d_{V_{2}})$. By induction, $A(W)$ is $K$-stable, so by 1.6, there is a retraction $r_{W}:\widehat{\mathcal{U}}(M_{2}(A(W))\to\widehat{\mathcal{U}}(A(W))$. Define $H:=r_{W}\circ F$, then $H:[0,1]\to\widehat{\mathcal{U}}(A(W))$ is a path such that

[TABLE]

The map $\pi^{V_{2}}_{W}:\widehat{\mathcal{U}}_{0}(A(V_{2}))\to\widehat{\mathcal{U}}_{0}(A(W))$ is a fibration, so there is a path $H^{\prime}:[0,1]\to\widehat{\mathcal{U}}(A(V_{2}))$ such that

[TABLE]

Define $e_{V_{2}}:=H^{\prime}(0)$ so that

[TABLE]

By 1.8, $A=A(V_{1}\cup V_{2})$ is a pullback

[TABLE]

so that $\omega:=(d_{V_{1}},e_{V_{2}})$ defines a quasi-unitary in $A$. We claim that $\iota(\omega)\sim u$ in $\widehat{\mathcal{U}}(A)$. To this end, define $H_{3}:[0,1]\to\widehat{\mathcal{U}}(M_{2}(A(V_{2}))$ by

[TABLE]

then $H_{3}(0)=\eta_{V_{2}}(u)$ and $H_{3}(1)=\iota_{V_{2}}(e_{V_{2}})=\iota_{V_{2}}(\pi_{V_{2}}(\omega))=\eta_{V_{2}}(\iota_{V_{2}}(\omega))$. So if $S:[0,1]\to\widehat{\mathcal{U}}(M_{2}(A(W))$ is given by

[TABLE]

Then $S$ is a path with $S(0)=\eta_{W}(u)\cdot\eta_{W}(u)^{\ast}=0$ and $S(1)=\iota^{V_{2}}_{W}(e_{V_{2}})\cdot\eta^{V_{1}}_{W}(\iota_{V_{1}}(d_{V_{1}}))=0$. Furthermore,

[TABLE]

Hence,

[TABLE]

Hence, $S$ is null-homotopic. Once again, the map $\eta^{V_{1}}_{W}:\widehat{\mathcal{U}}_{0}(M_{2}(A(V_{1}))\to\widehat{\mathcal{U}}_{0}(M_{2}(A(W))$ is a fibration, so there is a loop $f:[0,1]\to\widehat{\mathcal{U}}(M_{2}(A(V_{1}))$ such that $f(0)=f(1)=0$ and $\eta^{V_{1}}_{W}\circ f=S$. Define $H_{4}:[0,1]\to\widehat{\mathcal{U}}(M_{2}(A(V_{1}))$ by

[TABLE]

Then $H_{4}(0)=\eta_{V_{1}}(u),H_{4}(1)=\iota_{V_{1}}(d_{V_{1}})=\eta_{V_{1}}(\iota(\omega))$. Finally, by construction

[TABLE]

Therefore, the pair $(H_{3},H_{4})$ defines a path in $\widehat{\mathcal{U}}(M_{2}(A))$ connecting $u$ to $\iota(\omega)$. This concludes the proof that $\iota_{\ast}$ is surjective. ∎

Acknowledgements

The first named author is supported by UGC Junior Research Fellowship No. 1229 and the second named author was supported by SERB Grant YSS/2015/001060.