Let $A$ be a separable unital quasidiagonal $C^*$-algebra. What can be said about the $K$-theory of $A$, for example some properties? Especially, are there some criterions to decide whether or not $K_*(A)$ has torsion? I appreciate any reference request in this direction. Thank you.
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This is not necessarily an answer, but it was too long for a comment:
Note that for any separable unital $C^*$-algebra $A$ its suspension $SA := C_0(\mathbb{R}) \otimes A$ is quasidiagonal. This can be found as Corollary 7.3.7 in the (excellent) book "$C^*$-algebras and Finite-dimensional Approximations" by Brown and Ozawa. Here is the link.
Using $K_i(SA) \cong K_{i+1}(A)$ and Bott periodicity it follows that quasidiagonality does not impose any restrictions on the K-theory groups, if you allow non-unital $C^*$-algebras (like $SA$).
answered Jun 7, 2017 at 14:31
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2$\begingroup$ ... and if you then take the unitization of $SA$, which is again quasidiagonal you see that even in the unital case there are no restrictions on having torsion in the K-theory. $\endgroup$ Commented Jun 7, 2017 at 14:40