Abelian \(C^*\)-subalgebras of \(C^*\)-algebras of real rank zero and inductive limit \(C^*\)-algebras. (English) Zbl 0869.46030
Summary: Let \(X= S^{n_1}\times S^{n_2}\times\cdots\times S^{n_k}\), or let \(X\) be an absolute retract. It is shown that if a monomorphism \(\phi: C(X)\to A\) is homotopically trivial, then \(\phi\) can be approximated pointwise by homomorphisms from \(C(X)\) into \(A\) with finite-dimensional range, provided that \(A\) belongs to a certain class of simple \(C^*\)-algebras of real rank zero. These \(C^*\)-algebras include all purely infinite simple \(C^*\)-algebras, the Bunce-Deddens algebras, and the irrational rotation algebras. It is also shown (as a consequence) that if \(A\) is a simple \(C^*\)-algebra which is the inductive limit of a sequence of \(C^*\)-algebras of the form \(C(X_k)\otimes M_{n_k}\), with each \(X_k\) a contractible compact metric space, and if \(A\) is assumed to have real rank zero and only countably many extreme traces, then \(A\) is an AF-algebra.
MSC:
| 46L05 | General theory of \(C^*\)-algebras |
| 46L55 | Noncommutative dynamical systems |
| 46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
Keywords:
monomorphism; homotopically trivial; simple \(C^*\)-algebras of real rank zero; purely infinite simple \(C^*\)-algebras; Bunce-Deddens algebras; irrational rotation algebrasReferences:
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