I am trying to find examples of Automorphisms on C*-algebras that are not trivial in K-theory i.e. automorphisms $\alpha$ such that $K_0(\alpha)$ is not the identity map. These Automorphisms can of course not be approximately inner.
I Was trying to construct an example of such on a UHF algebra but an example on any C*-algebra would be helpful for me to understand.
On
Take $A=\mathbb C^2$, and $\phi(a,b)=(b,a)$. If I'm not wrong (my $K$-theory is ridiculously weak), you can see that $K_0(A)=\mathbb Z\times \mathbb Z$, and that $K_0(\phi)(m,n)=(n,m)$.
On
As Aweygan suggested, the antipodal map of $S^{2}$ induces the map $\begin{bmatrix}1&2\\0&-1\end{bmatrix}$ on $K_{0}C(S^{2})=K^{0}S^{2}=\mathbb{Z}\oplus\mathbb{Z}$. The first summand is generated by the class of trivial bundle $1$ and the second summand is generated by the class of the virtual bundle $1-[L]$, where $L$ is the line bundle constructed via clutching along an equatorial $S^1$ with the "identity" map $S^1\rightarrow U(1)$.
This is about as degenerate an example as one can get:
Let $A$ be your favorite $C^*$-algebra with $K_0(A)$ non-trivial, let $B=A\oplus A$, and let $\alpha:B\to B$ be given by $\alpha(x_1,x_2)=(x_2,x_1)$. Then $K_0(B)\cong K_0(A)\oplus K_0(A)$, and $K_0(\alpha)(g_1,g_2)=(g_2,g_1)$.
I'm sure a more interesting example can be taken by looking the antipodal map $\alpha:S^2\to S^2$, and considering $\alpha^*:C(S^2)\to C(S^2)$, (note that $K_0(C(S^2))=K^0(S^2)=\mathbb Z\oplus\mathbb Z$), but I haven't gone through the details. Perhaps I'll expand upon this later.