On regular ${}^{\ast }$-algebras of bounded linear operators: A new approach towards a theory of noncommutative Boolean algebras

Abstract

We study (von Neumann) regular ${}^{\ast }$-subalgebras of $B(H)$, which we call R${}^{\ast }$-algebras. The class of R${}^{\ast }$-algebras coincides with that of "E${}^{\ast }$-algebras that are pre-C${}^{\ast }$-algebras" in the sense of Z. Szűcs and B. Takács. We give examples, properties and questions of R${}^{\ast }$-algebras. We observe that the class of unital commutative R${}^{\ast }$-algebras has a canonical one-to-one correspondence with the class of Boolean algebras. This motivates the study of R${}^{\ast }$-algebras as that of noncommutative Boolean algebras. We explain that seemingly unrelated topics of functional analysis, like AF C${}^{\ast }$-algebras and incomplete inner product spaces, naturally arise in the investigation of R${}^{\ast }$-algebras. We obtain a number of interesting results on R${}^{\ast }$-algebras by applying various famous theorems in the literature.

39 pages

References in corpus (2)

• An elementary proof for the non-bijective version of Wigner's theorem
• Ring isomorphisms of Murray--von Neumann algebras