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Quantization in *-algebras III, A survey | ||
| AUT Journal of Mathematics and Computing | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 02 دی 1404 | ||
| نوع مقاله: Review Article | ||
| شناسه دیجیتال (DOI): 10.22060/ajmc.2025.24957.1486 | ||
| نویسندگان | ||
| Gholam Hossein Esslamzadeh* 1؛ Fatemeh Taleghani2 | ||
| 1Department of Mathematics, Shiraz University | ||
| 2Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| Quasi operator system؛ Noncommutative topology؛ Baer *-ring؛ Wold type decomposition | ||
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