Cornel Pasnicu: The weak ideal property in tensor products, 119-125

Abstract:

Let $A$ and $B$ be C*-algebras such that $A$ or $B$ is exact. We describe the largest ideal in $A \otimes B$ which has the weak ideal property. For many C*-algebras $A$ and $B$ as above we characterize when the largest ideal in $A \otimes B$ which has the weak ideal property is the tensor product of the largest ideals in $A$ and $B$ which have the weak ideal property (this is not always true if $A$ or $B$ is exact). Assume that the C*-algebras $A$ and $B$ have the weak ideal property (and one of them is exact). We characterize (in an interesting particular case and also in general) when $A \otimes B$ has the weak ideal property (these two characterizations are totally different in nature).

Key Words: Weak ideal property, tensor product C*-algebra, largest ideal which has the weak ideal property, primitive spectrum, ideal property.

2010 Mathematics Subject Classification: Primary 46L06; Secondary 46L05.