On higher real and stable ranks for 𝐶𝐶𝑅 𝐶*-algebras

Brown, Lawrence G.

doi:10.1090/tran/6616

On higher real and stable ranks for $CCR$ $C^*-$algebras
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Trans. Amer. Math. Soc. 368 (2016), 7461-7475 Request permission

Abstract:

We calculate the real rank and stable rank of $CCR$ algebras which either have only finite dimensional irreducible representations or have finite topological dimension. We show that either rank of $A$ is determined in a good way by the ranks of an ideal $I$ and the quotient $A/I$ in four cases: when $A$ is $CCR$; when $I$ has only finite dimensional irreducible representations; when $I$ is separable, of generalized continuous trace and finite topological dimension, and all irreducible representations of $I$ are infinite dimensional; or when $I$ is separable, stable, has an approximate identity consisting of projections, and has the corona factorization property. We also present a counterexample on higher ranks of $M(A)$, $A$ subhomogeneous, and a theorem of P. Green on generalized continuous trace algebras.

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