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 Murrayvon Neumann equivalence. What is known about the induced map on $K_{1}(A)$?

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

