ON REGULAR *-ALGEBRAS OF BOUNDED LINEAR OPERATORS: A NEW APPROACH TOWARDS A THEORY OF NONCOMMUTATIVE BOOLEAN ALGEBRAS

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