Tohoku Mathematical Journal
2023
September
SECOND SERIES VOL. 75, NO. 3

Tohoku Math. J.
75 (2023), 423-463

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

Author Michiya Mori

(Received December 2, 2021, revised February 28, 2022)
Abstract. 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. Szűcs and B. Takács. 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.

Mathematics Subject Classification. Primary 06E75; Secondary 03G12, 06C20, 16E50, 47C15, 47L40, 81P10.
Key words and phrases. Nonclosed self-adjoint operator algebra, von Neumann regular ring, Boolean algebra, AF C$^*$-algebra, projection lattice, complemented modular lattice, inner product space.

To the top of this page

Back to the Contents