Direct finiteness of representable regular rings with involution: A counterexample

TL;DR

This paper constructs a specific $*$-regular $*$-ring of endomorphisms on a Hilbert space demonstrating that direct finiteness does not always hold in such algebraic structures, providing a counterexample to previous assumptions.

Contribution

It introduces a novel counterexample of a $*$-regular $*$-ring where direct finiteness fails, expanding understanding of algebraic properties in operator rings.

Findings

Counterexample of a $*$-regular $*$-ring with non-finite behavior

Demonstrates failure of direct finiteness in certain algebraic structures

Uses shift operators on Hilbert space to construct the example

Abstract

Bruns and Roddy constructed a $3$-generated modular ortholattice $L$ which cannot be embedded into any complete modular ortholattice. Motivated by their approach, we use shift operators to construct a $*$-regular $*$-ring $R$ of endomorphisms of an inner product space (which can be chosen as the Hilbert space $\ell^2$) such that direct finiteness fails for $R$.