On one-sided primitivity of Banach algebras
M. J. Crabb, J. Duncan, C. M. McGregor

TL;DR
This paper investigates the algebraic structure of certain Banach algebra completions of a semigroup algebra, demonstrating their primitivity and analyzing the properties of their modules, with implications for understanding algebraic representations.
Contribution
It introduces two specific Banach algebra completions of a semigroup algebra and proves their left-primivity and module properties, providing new insights into their algebraic structure.
Findings
Both completions are left-primitive with separating families of infinite-dimensional irreducible right modules.
The semigroup algebra itself is left-primitive but not right-primitive.
All irreducible right modules of the semigroup algebra are finite-dimensional.
Abstract
Let be the semigroup with identity, generated by and , subject to being invertible and . We study two Banach algebra completions of the semigroup algebra . Both completions are shown to be left-primitive and have separating families of irreducible infinite-dimensional right modules. As an appendix, we offer an alternative proof that is left-primitive but not right-primitive. We show further that, in contrast to the completions, every irreducible right module for is finite dimensional and hence that has a separating family of such modules.
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Taxonomy
TopicsAdvanced Topics in Algebra · Advanced Operator Algebra Research · semigroups and automata theory
