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Abstract
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We describe when a general complex algebraic
-algebra is
pre--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
, where
is a finite-dimensional
Baer
-algebra and
is a commutative
algebraic Baer
-algebra.
The summand
is
-isomorphic
to a finite direct sum of full complex matrix algebras of size at least
. The commutative
summand
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
is finite exactly when the complex group algebra
is an algebraic
Baer
-algebra.
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Keywords
algebraic algebra, von Neumann regular algebra, Baer
*-algebra
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Mathematical Subject Classification
Primary: 16W10
Secondary: 22D15, 46L99
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Milestones
Received: 13 August 2021
Revised: 8 November 2021
Accepted: 21 December 2021
Published: 6 April 2022
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