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Ulrich Pennig
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Replacing commutative C*-algebras by simple ones

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 $Homeo(\mathbb{R}^{2n}) \to Aut(A)$?

Ulrich Pennig
  • 7.6k
  • 1
  • 27
  • 65