1,837 research outputs found
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
Noncommmutative theorems: Gelfand Duality, Spectral, Invariant Subspace, and Pontryagin Duality
We extend the Gelfand-Naimark duality of commutative C*-algebras, "A
COMMUTATIVE C*-ALGEBRA -- A LOCALLY COMPACT HAUSDORFF SPACE" to "A
C*-ALGEBRA--A QUOTIENT OF A LOCALLY COMPACT HAUSDORFF SPACE". Thus, a
C*-algebra is isomorphic to the convolution algebra of continuous regular Borel
measures on the topological equivalence relation given by the above mentioned
quotient. In commutative case this reduces to Gelfand-Naimark theorem.
Applications: 1) A simultaneous extension, to arbitrary Hilbert space
operators, of Jordan Canonical Form and Spectral Theorem of normal operators 2)
A functional calculus for arbitrary operators. 3) Affirmative solution of
Invariant Subspace Problem. 4) Extension of Pontryagin duality to nonabelian
groups, and inevitably to groups whose underlying topological space is
noncommutative.Comment: 10 page
New developments in the theory of Groebner bases and applications to formal verification
We present foundational work on standard bases over rings and on Boolean
Groebner bases in the framework of Boolean functions. The research was
motivated by our collaboration with electrical engineers and computer
scientists on problems arising from formal verification of digital circuits. In
fact, algebraic modelling of formal verification problems is developed on the
word-level as well as on the bit-level. The word-level model leads to Groebner
basis in the polynomial ring over Z/2n while the bit-level model leads to
Boolean Groebner bases. In addition to the theoretical foundations of both
approaches, the algorithms have been implemented. Using these implementations
we show that special data structures and the exploitation of symmetries make
Groebner bases competitive to state-of-the-art tools from formal verification
but having the advantage of being systematic and more flexible.Comment: 44 pages, 8 figures, submitted to the Special Issue of the Journal of
Pure and Applied Algebr
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