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.
Citation
Michiya Mori. "On regular $^*$-algebras of bounded linear operators: A new approach towards a theory of noncommutative Boolean algebras." Tohoku Math. J. (2) 75 (3) 423 - 463, 2023. https://doi.org/10.2748/tmj.20220316
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