The minimal ideal in multiplier algebras

Abstract

Let $\mathcal{A}$ be a simple, $\sigma$-unital, non-unital, non-elementary C*-algebra and let $I_{min}$ be the intersection of all the ideals of $\mathcal{M}(\mathcal{A})$ that properly contain $\mathcal{A}$. $I_{min}$ coincides with the ideal defined by Lin (Simple C*-algebras with continuous scales and simple corona algebras. 112, (1991) Proc. Amer.Math. Soc) in terms of approximate units of $\mathcal{A}$ and $I_{min}/\mathcal{A}$ is purely infinite and simple. If $\mathcal{A}$ is separable, or if $\mathcal{A}$ has the (SP) property and its dimension semigroup $D(\mathcal{A})$ of Murray-von Neumann equivalence classes of projections of $\mathcal{A}$ is order separable, or if $\mathcal{A}$ has strict comparison of positive elements by traces, then $\mathcal{A}\ne I_{min}$. If the tracial simplex $\mathcal{T}(\mathcal{A})$ is nonempty, let $I_{con}$ be the closure of the linear span of the elements $A\in \mathcal{M}(\mathcal{A}{)}_{+}$ such that the evaluation map $\hat{A}(\tau )=\tau (A)$ is continuous. If $\mathcal{A}$ has strict comparison of positive element by traces then $I_{min}=I_{con}$. Furthermore, $I_{min}$ too has strict comparison of positive elements in the sense that if $A,B\in (I_{min}{)}_{+}$, $B\notin \mathcal{A}$ and $d_{\tau }(A)<d_{\tau }(B)$ for all $\tau \in \mathcal{T}(\mathcal{A})$ for which $d_{\tau }(B)<\infty$, then $A⪯B$. However if $\mathcal{A}$ does not have strict comparison of positive elements by traces then $I_{min}\ne I_{con}$ can occur: a counterexample is provided by Villadsen's AH algebras without slow dimension growth. If the dimension growth is flat, $I_{con}$ is the largest proper ideal of $\mathcal{M}(\mathcal{A})$.

References in corpus (2)

• The Cuntz Semigroup and Comparison of Open Projections
• Stability in the Cuntz semigroup of a commutative C*-algebra

Cited by in corpus (1)

• Purely infinite corona algebras