Cornel Pasnicu: $\mathcal{D}$-stable $C^*$-algebras, the ideal property and real rank zero p.177-192

Abstract:

Let $\mathcal{D}$ be a strongly self-absorbing, $K_1$-injective $C^*$-algebra (e.g., the Jiang-Su algebra $\mathcal{Z}$ and $\mathcal{O}_{\infty}$). We characterize, in particular, when $A \otimes \mathcal{D}$ has the ideal property, where $A$ is a separable, purely infinite $C^*$-algebra. Answering a natural question, we prove that there is a separable, nuclear $C^*$-algebra $B$ such that RR($B$) = RR( $B \otimes \mathcal{Z}$) = sr($B$) = $\mbox{sr}(B \otimes \mathcal{Z})$ = 1 and Prim($B$) has two elements (in particular, Prim($B$) has a basis consisting of compact-open sets) but $B \otimes \mathcal{Z}$ does not have the ideal property. We also study some (permanence) properties of large classes of separable, $\mathcal{D}$-stable $C^*$-algebras with the ideal property. For "many" separable $C^*$-algebras $C$ we characterize when RR( $C \otimes \mathcal{Z}) = 0$.

Key Words: $C^*$-algebra, minimal tensor product of $C^*$-algebras, ideal property, strongly self-absorbing, the Jiang-Su algebra, weakly purely infinite, purely infinite, strongly purely infinite, exact $C^*$-algebra, nuclear $C^*$-algebra, primitive ideal spectrum, real rank, stable rank, $K$-theory groups, $K_1$-injective, $\mathcal{D}$-stable, $C(X)$-algebra, $K_0$-liftable..

2000 Mathematics Subject Classification: Primary: 46L05,
Secondary: 46L06.

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