Direct finiteness of representable regular rings with involution: A counterexample

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 $\ast$-regular $\ast$-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 $R$.

As observed by Wehrung, the identity minus shift has no quasi-inverse in the ring of row and column finite matrices. Thus, the claimed example does not work