Vol. 316, No. 2, 2022

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The structure of algebraic Baer $^*$-algebras

Zsolt Szűcs and Balázs Takács

Vol. 316 (2022), No. 2, 431–452
Abstract

We describe when a general complex algebraic -algebra is pre-C-normed, and investigate its structure when the -algebra is a Baer -ring in the presence of algebraicity. Our main result is that every complex algebraic Baer -algebra can be decomposed as a direct sum M B, where M is a finite-dimensional Baer -algebra and B is a commutative algebraic Baer -algebra. The summand M is -isomorphic to a finite direct sum of full complex matrix algebras of size at least 2 × 2. The commutative summand B is -isomorphic to the linear span of the characteristic functions of the clopen sets in a Stonean topological space.

As an application we show that a group G is finite exactly when the complex group algebra [G] is an algebraic Baer -algebra.

Keywords
algebraic algebra, von Neumann regular algebra, Baer *-algebra
Mathematical Subject Classification
Primary: 16W10
Secondary: 22D15, 46L99
Milestones
Received: 13 August 2021
Revised: 8 November 2021
Accepted: 21 December 2021
Published: 6 April 2022
Authors
Zsolt Szűcs
Department of Differential Equations
Budapest University of Technology and Economics
Budapest
Hungary
Balázs Takács
Department of Analysis
Budapest University of Technology and Economics
Budapest
Hungary