It may well be a trivial question but I was wondering if there is any relation between $K$-groups and ultrapowers of $C^\*$-algebras. For instance, if $A$ is a $C^\*$-algebra does $K\_0(A^U)$ depend on the choice of a free ultrafilter $U$? What if $A$ is a von Neumann algebra with a trace and $A^U$ is the tracial ultrapower? For instance, what is $K\_0(R^U)$, where $R$ is the hyperfinite $II\_1$ factor?



K-theory of ultrapowers
If $R$ is type II finite AW\* or W\* factor, then $K\_0(R^U) $ is naturally order isomorphic to the reals, as $R^U$ is again a type II finite AW\* or W\* factor. More generally, if $A$ is a C\* algebra with stable range 1, then $l^{\infty}(A)$ (the algebra of bounded sequences of elements of $A$) has the interesting property that all of its simple images whose kernels contain $c\_0(A)$ (the ideal consisting of sequences that converge to zero) are at least finite AW\*-factors. 
A reference for the latter is 


D Handelman [me], *Homomorphisms from C${^\*}$-algebras to AW$^\*$-algebras,* Michigan Math. J, 1981, 229--241. Here is the [link to the paper.](http://wwwmath.uni-muenster.de/u/cuntz/HandelmanAW.pdf)


