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 $K$ be a commutative ring with unit and $S$ an inverse semigroup. We show that the semigroup algebra $KS$ can be described as a convolution algebra of functions on the universal \'etale groupoid associated to $S$ 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 $C^*$-algebra. It provides a convenient topological framework for understanding the structure of $KS$, 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 $S$ 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

Measurement of high p_T isolated prompt photons in lead-lead collisions at sqrt(s_NN)=2.76 TeV with the ATLAS detector at the LHC

Prompt photons are a powerful tool to study heavy ion collisions. Their production rates provide access to the initial state parton distribution functions and also provide a means to calibrate the expected energy of jets that are produced in the medium. The ATLAS detector measures photons with its hermetic, longitudinally segmented calorimeter, which gives excellent spatial and energy resolutions, and detailed information about the shower shape of each measured photon. This provides significant rejection against the expected background from the decays of neutral pions in jets. Rejection against jet fragmentation products is further enhanced by requiring candidate photons to be isolated. First results on the spectra of isolated prompt photons from a dataset with an integrated luminosity of approximately 0.13 nb^-1 of lead-lead collisions at sqrt(s_NN)=2.76 TeV are shown as a function of transverse momentum and centrality. The measured spectra are compared to expectations from perturbative QCD calculations.Comment: Proceedings for Hard Probes 2012, May 27 - June 1, 2012, Cagliari, Sardinia, Ital