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Inverse semigroups associated to subshifts  [PDF]
Charles Starling
Mathematics , 2015,
Abstract: The dynamics of a one-sided subshift $\mathsf{X}$ can be modeled by a set of partially defined bijections. From this data we define an inverse semigroup $\mathcal{S}_{\mathsf{X}}$ and show that it has many interesting properties. We prove that the Carlsen-Matsumoto C*-algebra $\mathcal{O}_\mathsf{X}$ associated to $\mathsf{X}$ is canonically isomorphic to Exel's tight C*-algebra of $\mathcal{S}_{\mathsf{X}}$. As one consequence, we obtain that $\mathcal{O}_\mathsf{X}$ can be written as a partial crossed product of a commutative C*-algebra by a countable group.
Groupoids and inverse semigroups associated to W*-algebras  [PDF]
Anatol Odzijewicz,Aneta Sli?ewska
Mathematics , 2011,
Abstract: We investigate the Banach Lie groupoids and inverse semigroups naturally associated to W*-algebras. We also present statements describing relationship between these groupoids and the Banach Poisson geometry which follows in the canonical way from the W*-algebra structure.
Amenability and Uniqueness for Groupoids Associated with Inverse Semigroups  [PDF]
Scott M. LaLonde,David Milan
Mathematics , 2015,
Abstract: We investigate recent uniqueness theorems for reduced $C^*$-algebras of Hausdorff \'{e}tale groupoids in the context of inverse semigroups. In many cases the distinguished subalgebra is closely related to the structure of the inverse semigroup. In order to apply our results to full $C^*$-algebras, we also investigate amenability. More specifically, we obtain conditions that guarantee amenability of the universal groupoid for certain classes of inverse semigroups. These conditions also imply the existence of a conditional expectation onto a canonical subalgebra.
Inverse semigroups associated with labelled spaces and their tight spectra  [PDF]
Giuliano Boava,Gilles de Castro,Fernando de L. Mortari
Mathematics , 2015,
Abstract: The notion of a labelled space was introduced by Bates and Pask in generalizing certain classes of C*-algebras. Motivated by Exel's work on inverse semigroups and combinatorial C*-algebras, we associate each weakly left resolving labelled space with an inverse semigroup, and characterize the tight spectrum of the latter in a way that is reminiscent of the description of the boundary path space of a directed graph.
Enumeration of finite inverse semigroups  [PDF]
Martin E. Malandro
Mathematics , 2013,
Abstract: We give an efficient algorithm for the enumeration up to isomorphism of the inverse semigroups of order n, and we count the number S(n) of inverse semigroups of order n<=15. This improves considerably on the previous highest-known value S(9). We also give a related algorithm for the enumeration up to isomorphism of the finite inverse semigroups S with a given underlying semilattice of idempotents E, a given restriction of Green's D-relation on S to E, and a given list of maximal subgroups of S associated to the elements of E.
Inverse semigroup C*-algebras associated with left cancellative semigroups  [PDF]
Magnus Dahler Norling
Mathematics , 2012,
Abstract: To each discrete left cancellative semigroup $S$ one may associate a certain inverse semigroup $I_l(S)$, often called the left inverse hull of $S$. We show how the full and the reduced C*-algebras of $I_l(S)$ are related to the full and reduced semigroup C*-algebras for $S$ recently introduced by Xin Li, and give conditions ensuring that these algebras are isomorphic. Our picture provides an enhanced understanding of Li's algebras.
Heat kernels on metric graphs and a trace formula  [PDF]
Vadim Kostrykin,Jurgen Potthoff,Robert Schrader
Mathematics , 2007,
Abstract: We study heat semigroups generated by self-adjoint Laplace operators on metric graphs characterized by the property that the local scattering matrices associated with each vertex of the graph are independent from the spectral parameter. For such operators we prove a representation for the heat kernel as a sum over all walks with given initial and terminal edges. Using this representation a trace formula for heat semigroups is proven. Applications of the trace formula to inverse spectral and scattering problems are also discussed.
Graph inverse semigroups: their characterization and completion  [PDF]
David G. Jones,Mark V. Lawson
Mathematics , 2011,
Abstract: Graph inverse semigroups generalize the polycyclic inverse monoids and play an important role in the theory of C*-algebras. This paper has two main goals: first, to provide an abstract characterization of graph inverse semigroups; and second, to show how they may be completed, under suitable conditions, to form what we call the Cuntz-Krieger semigroup of the graph. This semigroup is the ample semigroup of a topological groupoid associated with the graph, and the semigroup analogue of the Leavitt path algebra of the graph.
The structure of generalized inverse semigroups  [PDF]
Ganna Kudryavtseva,Mark V. Lawson
Mathematics , 2012,
Abstract: We prove that the structure of right generalized inverse semigroups is determined by free \'etale actions of inverse semigroups. This leads to a tensor product interpretation of Yamada's classical struture theorem for generalized inverse semigroups.
Maximal subgroups of amalgams of finite inverse semigroups  [PDF]
Alessandra Cherubini,Tatiana B. Jajcayova,Emanuele Rodaro
Mathematics , 2013, DOI: 10.1007/s00233-014-9601-4
Abstract: We use the description of the Schutzenberger automata for amalgams of finite inverse semigroups given by Cherubini, Meakin, Piochi to obtain structural results for such amalgams. Schutzenberger automata, in the case of amalgams of finite inverse semigroups, are automata with special structure possessing finite subgraphs, that contain all essential information related to the whole automaton. Using this crucial fact, and the Bass-Serre theory, we show that the maximal subgroups of an amalgamated free-product are either isomorphic to certain subgroups of the original semigroups or can be described as fundamental groups of particular finite graphs of groups build from the maximal subgroups of the original semigroups.
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