# Direct finiteness of representable regular rings with involution

## TL;DR

This paper investigates conditions under which elements in certain regular rings with involution are units, aiming to establish criteria for the rings' direct finiteness, though a complete proof remains open.

## Contribution

It introduces a new condition based on finite-dimensional subspace actions for elements to be units in von Neumann *-regular rings with involution.

## Key findings

- Provides a condition for invertibility of elements in regular rings with involution
- Highlights an open problem regarding the direct finiteness of such rings
- Clarifies the gap in previous claims about ring properties

## 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.