Central zero divisors in group algebras

Abstract:

A central element of the complex group algebra ${\mathbf {C}}G$ which is a zero divisor in the ${W^ * }$ group algebra $W(G)$ is also a zero divisor in ${\mathbf {C}}G$. As a corollary, if $K$ is a field of characteristic zero, $G$ is a group, $A$ is an abelian normal subgroup of $G$, and $R$ is the Ore localization of $KG$ obtained by inverting all nonzero elements of $KA$, then all matrix rings over $R$ are directly finite and $R$ has the invariant basis property.