7
$\begingroup$

I am looking for functorial ways of replacing a commutative $C^*$-algebra $C$ by a simple one, say $A$ , such that the $K$-theory remains unchanged, i.e. $K_*(C) \cong K_*(A)$.

I am particularly interested in the case where $C = C_0(\mathbb{R}^{2n})$. There are stupid answers to that, like: Replace $C$ by $\mathbb{C}$ and let algebra homomorphisms $C \to C$ act trivially, but this is obviously not what I want. What I would like to have is that the automorphism group of $A$ should contain the automorphisms of $C_0(\mathbb{R}^{2n})$, i.e. the group of homeomorphisms.

Is there a simple C$^*$-algebra $A$ with $K_*(A) \cong K_*(\mathbb{C})$ such that there is an injection $Homeo(\mathbb{R}^{2n}) \to Aut(A)$?

Improve this question
$\endgroup$
1
  • 4
    $\begingroup$ Hi Ulrich, This probably fits in the "stupid answer" category but you could take $\mathbb{R}^{2n}$ many tensor copies of $M_2(\mathbb{C})$ as $A$ then the permutation group of $\mathbb{R}^{2n}$ is contained in $Aut(A)$ by just permuting the tensor factors. $\endgroup$
    – Caleb Eckhardt
    Commented Nov 21, 2013 at 5:15

0

Reset to default

You must log in to answer this question.