Quantization in *-algebras III, A survey

Articles in Press

Document Type : Review Article

Authors

1 Department of Mathematics, Shiraz University, Shiraz, Iran

2 Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran

Abstract

Our main goal here is to show that many of essential results in quantized functional analysis rely on the algebraic structure of the unital ring $B(H)$ of bounded operators on a Hilbert space $H$. The wide spectrum of structures on this ring is the main motivation for investigating the role of algebraic structure of $B(H)$ in different major results in this field. Our strategy for dealing with this general problem is finding the right category, containing operator algebras, in which a specific result remains true. The authors and their collaborators, have approached this problem from three directions, a survey of which is presented here. In the first approach, major theorems of quantized functional analysis such as Arveson's extension theorem, Ruan's theorem and Choi-Effros characterization of operator systems were proved in the much larger category of unital $*$-algebras. Moreover we unify all generalizations of the notion of operator systems. The second approach is devoted to investigating existence of projections properties in the category of $*$-algebras and constructing some noncommutative topology results. In particular some characterizations of Rickart $*$-algebras and some other types of $*$-algebras in terms of topological properties, were proved. In the third approach we work in the category of Baer $*$-rings ,that is, $*$-rings which only possess the lattice structure of projections of $B(H)$ but not necessarily the other structures. In this part major decomposition theorems of Wold, Nagy-Foias-Langer and Halmos-Wallen were proved in the purely algebraic setting.

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AUT Journal of Mathematics and Computing

Articles in Press, Accepted Manuscript
Available Online from 23 December 2025

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