605 research outputs found
Simplicial cohomology of band semigroup algebras
We establish simplicial triviality of the convolution algebra ,
where is a band semigroup. This generalizes results of the first author
[Glasgow Math. J. 2005, Houston J. Math. 2010]. To do so, we show that the
cyclic cohomology of this algebra vanishes in all odd degrees, and is
isomorphic in even degrees to the space of continuous traces on .
Crucial to our approach is the use of the structure semilattice of , and the
associated grading of , together with an inductive normalization procedure
in cyclic cohomology; the latter technique appears to be new, and its
underlying strategy may be applicable to other convolution algebras of
interest.Comment: v1: AMS-LaTeX, 24 pages, 1 figure. v2: some typos corrected; a few
minor adjustments made for clarity; references updated. Accepted June 2011 by
Proc. Royal Soc. Edinburgh Sect.
Transference Principles for Semigroups and a Theorem of Peller
A general approach to transference principles for discrete and continuous
operator (semi)groups is described. This allows to recover the classical
transference results of Calder\'on, Coifman and Weiss and of Berkson, Gillespie
and Muhly and the more recent one of the author. The method is applied to
derive a new transference principle for (discrete and continuous) operator
semigroups that need not be groups. As an application, functional calculus
estimates for bounded operators with at most polynomially growing powers are
derived, culminating in a new proof of classical results by Peller from 1982.
The method allows a generalization of his results away from Hilbert spaces to
\Ell{p}-spaces and --- involving the concept of -boundedness --- to
general Banach spaces. Analogous results for strongly-continuous one-parameter
(semi)groups are presented as well. Finally, an application is given to
singular integrals for one-parameter semigroups
Semigroup graded algebras and codimension growth of graded polynomial identities
We show that if is any of four semigroups of two elements that are not
groups, there exists a finite dimensional associative -graded algebra over a
field of characteristic such that the codimensions of its graded polynomial
identities have a non-integer exponent of growth. In particular, we provide an
example of a finite dimensional graded-simple semigroup graded algebra over an
algebraically closed field of characteristic with a non-integer graded
PI-exponent, which is strictly less than the dimension of the algebra. However,
if is a left or right zero band and the -graded algebra is unital, or
is a cancellative semigroup, then the -graded algebra satisfies the
graded analog of Amitsur's conjecture, i.e. there exists an integer graded
PI-exponent. Moreover, in the first case it turns out that the ordinary and the
graded PI-exponents coincide. In addition, we consider related problems on the
structure of semigroup graded algebras.Comment: 21 pages; Minor misprints are correcte
Non-commutative Stone duality: inverse semigroups, topological groupoids and C*-algebras
We study a non-commutative generalization of Stone duality that connects a
class of inverse semigroups, called Boolean inverse -semigroups, with a
class of topological groupoids, called Hausdorff Boolean groupoids. Much of the
paper is given over to showing that Boolean inverse -semigroups arise
as completions of inverse semigroups we call pre-Boolean. An inverse
-semigroup is pre-Boolean if and only if every tight filter is an
ultrafilter, where the definition of a tight filter is obtained by combining
work of both Exel and Lenz. A simple necessary condition for a semigroup to be
pre-Boolean is derived and a variety of examples of inverse semigroups are
shown to satisfy it. Thus the polycyclic inverse monoids, and certain Rees
matrix semigroups over the polycyclics, are pre-Boolean and it is proved that
the groups of units of their completions are precisely the Thompson-Higman
groups . The inverse semigroups arising from suitable directed graphs
are also pre-Boolean and the topological groupoids arising from these graph
inverse semigroups under our non-commutative Stone duality are the groupoids
that arise from the Cuntz-Krieger -algebras.Comment: The presentation has been sharpened up and some minor errors
correcte
Spectral expansions of non-self-adjoint generalized Laguerre semigroups
We provide the spectral expansion in a weighted Hilbert space of a
substantial class of invariant non-self-adjoint and non-local Markov operators
which appear in limit theorems for positive-valued Markov processes. We show
that this class is in bijection with a subset of negative definite functions
and we name it the class of generalized Laguerre semigroups. Our approach,
which goes beyond the framework of perturbation theory, is based on an in-depth
and original analysis of an intertwining relation that we establish between
this class and a self-adjoint Markov semigroup, whose spectral expansion is
expressed in terms of the classical Laguerre polynomials. As a by-product, we
derive smoothness properties for the solution to the associated Cauchy problem
as well as for the heat kernel. Our methodology also reveals a variety of
possible decays, including the hypocoercivity type phenomena, for the speed of
convergence to equilibrium for this class and enables us to provide an
interpretation of these in terms of the rate of growth of the weighted Hilbert
space norms of the spectral projections. Depending on the analytic properties
of the aforementioned negative definite functions, we are led to implement
several strategies, which require new developments in a variety of contexts, to
derive precise upper bounds for these norms.Comment: 162page
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