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It is known that any inner automorphism of a unital $C^{\ast}$-algebra $A$ induces the identity map on $K_{0}(A)$ because unitary equivalence implies Murray-von Neumann equivalence. What is known about the induced map on $K_{1}(A)$?

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For example, if the inner automorphism is homotopic to identity, then the induced map on $K_{1}(A)$ is the identity map. But, this is too strong. Does there exist any weaker condition? What is known when $A=C(\mathbb{T},M_{n}(\mathbb{C}))$? –  David May 29 '13 at 23:18
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Crossposted to math.SE: math.stackexchange.com/questions/406088/… –  Qiaochu Yuan May 30 '13 at 0:37

1 Answer 1

Inner automorphisms also induce the identity map on $K_1(A)$.

Let $u\in A$ be the unitary inducing the inner automorphism $\alpha$. Let $x$ be a unitary $n\times n$-matrix over $A$. Then $\alpha(x)$ is equal to $u_nxu_n^*$ where $u_n$ is the diagonal matrix with all diagonal entries $u$. Hence the class in $K_1(A)$ of $\alpha(u)$ is $[u_nxu_n^*]=[u_n]+[x]-[u_n]=[x]$.

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