Rational \(K\)-stability of continuous \(C(X)\)-algebras. (English) Zbl 1533.46051
As mentioned in the Introduction, the following two theorems are obtained as the main result and its application.
Main Theorem. Let \(X\) be a compact metric space of finite covering dimension. Let \(A\) be a continuous \(C(X)\)-algebra. Suppose that each fiber of \(A\) is rationally \(K\)-stable. Then so is \(A\).
Recall that a \(C^*\)-algebra \(A\) is said to be a \(C(X)\)-algebra for \(X\) a compact Hausdorff space if there is a unital homomorphism from the \(C^*\)-algebra \(C(X)\) of complex-valued continuous functions on \(X\) to the center of the multiplier \(C^*\)-algebra of \(A\) (non-unital or not). A continuous \(C(X)\)-algebra is a \(C(X)\)-algebra \(A\) with a norm continuous map on \(X\) to the fibers of \(A\). Also, a \(C^*\)-algebra \(A\) is said to be rationally \(K\)-stable if the rational, non-stable \(K\)-theory as homotopy theory groups of unitary matrix groups over \(A\) are naturally isomorphic to the rational \(K\)-theory groups of \(A\) (tensored with the field of rational numbers).
Application Theorem. Suppose that an action of a compact Lie group \(G\) on a separable \(C^*\)-algebra \(A\) has finite Rokhlin dimension with commuting towers. If \(A\) is rationally \(K\)-stable or \(K\)-stable, then so is the crossed product \(C^*\)-algebra of \(A\) by the action of \(G\).
Note that Rokhlin dimension being \(d\) means that there exist \(d+1\) contractive completely positive maps from \(C(G)\) to \(A\) satisfying several approximate conditions with respect to products, commutators, action compositions, and partition of unity, for elements of finite subsets of \(A\) and \(C(G)\).
Main Theorem. Let \(X\) be a compact metric space of finite covering dimension. Let \(A\) be a continuous \(C(X)\)-algebra. Suppose that each fiber of \(A\) is rationally \(K\)-stable. Then so is \(A\).
Recall that a \(C^*\)-algebra \(A\) is said to be a \(C(X)\)-algebra for \(X\) a compact Hausdorff space if there is a unital homomorphism from the \(C^*\)-algebra \(C(X)\) of complex-valued continuous functions on \(X\) to the center of the multiplier \(C^*\)-algebra of \(A\) (non-unital or not). A continuous \(C(X)\)-algebra is a \(C(X)\)-algebra \(A\) with a norm continuous map on \(X\) to the fibers of \(A\). Also, a \(C^*\)-algebra \(A\) is said to be rationally \(K\)-stable if the rational, non-stable \(K\)-theory as homotopy theory groups of unitary matrix groups over \(A\) are naturally isomorphic to the rational \(K\)-theory groups of \(A\) (tensored with the field of rational numbers).
Application Theorem. Suppose that an action of a compact Lie group \(G\) on a separable \(C^*\)-algebra \(A\) has finite Rokhlin dimension with commuting towers. If \(A\) is rationally \(K\)-stable or \(K\)-stable, then so is the crossed product \(C^*\)-algebra of \(A\) by the action of \(G\).
Note that Rokhlin dimension being \(d\) means that there exist \(d+1\) contractive completely positive maps from \(C(G)\) to \(A\) satisfying several approximate conditions with respect to products, commutators, action compositions, and partition of unity, for elements of finite subsets of \(A\) and \(C(G)\).
Reviewer: Takahiro Sudo (Nishihara)
MSC:
| 46L85 | Noncommutative topology |
| 46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
| 46L55 | Noncommutative dynamical systems |
| 46L05 | General theory of \(C^*\)-algebras |
| 46L40 | Automorphisms of selfadjoint operator algebras |
Keywords:
nonstable \(K\)-theory; \(K\)-theory; homotopy; \(C^*\)-algebra; unitary; continuous field; compact space; crossed product; Rokhlin; \(K\)-stableReferences:
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