Cohomology groups of integral domains and -algebras

TL;DR

This paper develops a new homology theory to analyze linear operators in algebraic structures, characterizing integral domains and solving longstanding problems in algebra and operator theory.

Contribution

It introduces a novel homology framework for studying local multipliers and band preserving operators, providing new characterizations of integral domains and solving the Wickstead problem.

Findings

Characterized integral domains where local multipliers are multipliers

Solved the Wickstead problem for Archimedean unital f-algebras

Established a new homology approach for operator analysis

Abstract

In this paper, we introduce a new homology theory devoted to the study of linear operators such as local mutipliers and band preserving operators. The idea is to study the vanishing homology problem. This enables us to characterize integral domains in which any local multiplier is a multiplier, which gives a partial answer to a problem posed by R. V. Kadison [J. Algebra 130, No.2, (1990) 494-509]. Finally, we solve the Wickstead problem [Compositio Math, 35(3) (1977), 225--238] for the class of Archimedean unital f-algebras.