On tensor products of classifiable $C^*$-algebras

Wei Sun

10:00 am to 11:00 am, Feb 27th, 2014   Science Building A1510

Abstract:

Let ${\mathcal A}_1$ be the class of all unital separable simple $C^*$-algebras $A$ such that each $A\otimes U$ has tracial rank no more than one for all UHF-algebras $U$ of infinite type. It has been shown that all the amenable ${\mathcal Z}$-stable $C^*$-algebras in ${\mathcal A}_1$ which satisfy the Universal Coefficient Theorem can be classified up to isomorphisms by the Elliott invariant.

We show that $A\in {\mathcal A}_1$ if and only if $A\otimes B$ has tracial rank no more than one for some unital simple infinite dimensional AF-algebra $B.$ In fact, we show that $A\in {\mathcal A}_1$ if and only if $A\otimes B\in {\mathcal A}_1$ for some unital simple AH-algebra $B.$ Other results regarding the tensor products of $C^*$-algebras in ${\mathcal A}_1$ are also obtained.

This is a joint work with Huaxin Lin.