3,737 research outputs found
Simplicity, primitivity and semiprimitivity of etale groupoid algebras with applications to inverse semigroup algebras
This paper studies simplicity, primitivity and semiprimitivity of algebras
associated to \'etale groupoids. Applications to inverse semigroup algebras are
presented. The results also recover the semiprimitivity of Leavitt path
algebras and can be used to recover the known primitivity criterion for Leavitt
path algebras.Comment: Updated after referee report and corrected misprint
A Groupoid Approach to Discrete Inverse Semigroup Algebras
Let be a commutative ring with unit and an inverse semigroup. We show
that the semigroup algebra can be described as a convolution algebra of
functions on the universal \'etale groupoid associated to by Paterson. This
result is a simultaneous generalization of the author's earlier work on finite
inverse semigroups and Paterson's theorem for the universal -algebra. It
provides a convenient topological framework for understanding the structure of
, including the center and when it has a unit. In this theory, the role of
Gelfand duality is replaced by Stone duality.
Using this approach we are able to construct the finite dimensional
irreducible representations of an inverse semigroup over an arbitrary field as
induced representations from associated groups, generalizing the well-studied
case of an inverse semigroup with finitely many idempotents. More generally, we
describe the irreducible representations of an inverse semigroup that can
be induced from associated groups as precisely those satisfying a certain
"finiteness condition". This "finiteness condition" is satisfied, for instance,
by all representations of an inverse semigroup whose image contains a primitive
idempotent
The averaging trick and the Cerny conjecture
The results of several papers concerning the \v{C}ern\'y conjecture are
deduced as consequences of a simple idea that I call the averaging trick. This
idea is implicitly used in the literature, but no attempt was made to formalize
the proof scheme axiomatically. Instead, authors axiomatized classes of
automata to which it applies
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