Trace radicals and cocenters of free products

Abstract

We call a unital associative algebra $A$ trace residually finite-dimensional if its elements are separated by finite-dimensional representations and its cocenter $A/[A,A]$ is separated by the corresponding trace functionals. For RFD algebras $A$ and $B$, we prove that the trace radical of $A*B$, the common kernel of all finite-dimensional trace functionals, is the direct sum of the trace radicals of $A$ and $B$, which implies that $A*B$ is trace RFD if and only if both $A$ and $B$ are trace RFD. The proof combines an explicit cocenter decomposition with a construction of finite-dimensional representations whose traces detect nonzero classes of words of length greater than one.