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Direct finiteness of representable regular rings with involution

Christian Herrmann

TL;DR

This paper addresses the open problem of direct finiteness for representable $*$-regular rings. It recalls a criterion (*) that would ensure invertibility of an element $r$ by its action on finite-dimensional subspaces, and proves that this sufficiency condition indeed guarantees that $r$ is a unit when the ring is represented on a pre-hermitian space $V_F$. The work also discusses the limitations of the criterion, showing it is not necessary via a counterexample, and emphasizes that the overarching direct finiteness question remains unresolved for $*$-regular rings and their representable subclasses. Overall, the paper clarifies gaps in earlier claims and delineates the boundary between invertibility criteria and direct finiteness in this algebraic setting.

Abstract

For von Neumann *-regular rings R of endomorphisms (the involution given by taking adjoints) of inner product spaces we provide a condition on r in R (in terms of action of r on finite dimensional subspaces) for r being a unit. It remains open whether this result can be used to prove direct finiteness of R. This was claimed in earlier versions but no proof was given.

Direct finiteness of representable regular rings with involution

TL;DR

This paper addresses the open problem of direct finiteness for representable -regular rings. It recalls a criterion (*) that would ensure invertibility of an element by its action on finite-dimensional subspaces, and proves that this sufficiency condition indeed guarantees that is a unit when the ring is represented on a pre-hermitian space . The work also discusses the limitations of the criterion, showing it is not necessary via a counterexample, and emphasizes that the overarching direct finiteness question remains unresolved for -regular rings and their representable subclasses. Overall, the paper clarifies gaps in earlier claims and delineates the boundary between invertibility criteria and direct finiteness in this algebraic setting.

Abstract

For von Neumann *-regular rings R of endomorphisms (the involution given by taking adjoints) of inner product spaces we provide a condition on r in R (in terms of action of r on finite dimensional subspaces) for r being a unit. It remains open whether this result can be used to prove direct finiteness of R. This was claimed in earlier versions but no proof was given.

Paper Structure

This paper contains 4 sections, 2 theorems, 3 equations.

Table of Contents

  1. Erratum
  2. Introduction
  3. Preliminaries
  4. A sufficient condition for invertibility

Key Result

Lemma 3

Every regular $\ast$-subring $R$ of $\mathop{\mathrm{{\sf End}^\ast}}\nolimits(V_F)$ extends to a regular $\ast$-subring $\hat{R}$ of $\mathop{\mathrm{{\sf End}^\ast}}\nolimits(V_F)$ containing $\hat{J}(V_F)$ and such that $J(V_F)$ is an ideal of $\hat{R}$.

Theorems & Definitions (4)

  • Lemma 3
  • proof
  • Proposition 4
  • proof