MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $K_{1}(A)$?

share|cite|improve this question
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
Crossposted to math.SE:… – Qiaochu Yuan May 30 '13 at 0:37

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 $[u_nxu_n^*]=[u_n]+[x]-[u_n]=[x]$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.