Pseudotraces on Almost Unital and Finite-Dimensional Algebras

TL;DR

This paper generalizes the pseudotrace construction to almost unital and finite-dimensional algebras, establishing an isomorphism between symmetric linear functionals of such algebras and their endomorphism algebras.

Contribution

It introduces the AUF algebra class and extends pseudotrace theory to non-unital, possibly infinite-dimensional algebras with many idempotents.

Findings

Pseudotrace construction applies to AUF algebras.

An isomorphism between symmetric linear functionals of AUF algebras and their endomorphism algebras.

Non-degeneracy conditions are equivalent on both sides.

Abstract

We introduce the notion of almost unital and finite-dimensional (AUF) algebras, which are associative $\mathbb C$-algebras that may be non-unital or infinite-dimensional, but have sufficiently many idempotents. We show that the pseudotrace construction, originally introduced by Hattori and Stallings for unital finite-dimensional algebras, can be generalized to AUF algebras. Let $A$ be an AUF algebra. Suppose that $G$ is a projective generator in the category $\mathrm{Coh}_{\mathrm{L}}(A)$ of finitely generated left $A$-modules that are quotients of free left $A$-modules, and let $B = \mathrm{End}_{A,-}(G)^{\mathrm{opp}}$. We prove that the pseudotrace construction yields an isomorphism between the spaces of symmetric linear functionals $\mathrm{SLF}(A)\xrightarrow{\simeq} \mathrm{SLF}(B)$, and that the non-degeneracies on the two sides are equivalent.