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.

Disclosure

“⇐⇒ J(Ai ) ⊆ [Ai , Ai ] for every i. In particular, the free products of finite-dimensional semisimple algebras are trace-RFD. These free products were recently studied via quivers in [BDGPWX24]. Acknowledgments. The author used OpenAI’s ChatGPT 5.5 to assist in developing the proof of Proposition 3.3, including Lemma 3.2. The author independently verified the resulting arguments and takes full responsibility for the mathematical content of the paper. The author was partially supp”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T19:09:51.008456Z

Structural counts