scholarly journals Inverse Semigroup C*-Algebras Associated with Left Cancellative Semigroups

2014 ◽  
Vol 57 (2) ◽  
pp. 533-564 ◽  
Author(s):  
Magnus Dahler Norling
Keyword(s):  
Inverse Semigroup ◽  
Cancellative Semigroup ◽  
Left Inverse ◽  
C Algebras ◽  
Inverse Hull

AbstractTo each discrete left cancellative semigroup S one may associate an inverse semigroup Il(S), often called the left inverse hull of S. We show how the full and reduced C*-algebras of Il(S) are related to the full and reduced semigroup C*-algebras for S, recently introduced by Li, and give conditions ensuring that these algebras are isomorphic. Our picture provides an enhanced understanding of Li's algebras.

scholarly journals Chain conditions and semigroup graded rings

1988 ◽  
Vol 45 (3) ◽  
pp. 372-380 ◽  
Author(s):  
E. Jespers
Keyword(s):  
Inverse Semigroup ◽  
Sufficient Conditions ◽  
Cancellative Semigroup ◽  
Graded Rings ◽  
Semigroup Rings ◽  
Graded Ring ◽  
Product Group ◽  
Chain Conditions ◽  
Unique Product

AbstractThe following questions are studied: When is a semigroup graded ring left Noetherian, respectively semiprime left Goldie? Necessary sufficient conditions are proved for cancellative semigroup-graded subrings of rings weakly or strongly graded by a polycyclic-by-finite (unique product) group. For semigroup rings R[S] we also give a solution to the problem in case S is an inverse semigroup.

scholarly journals The C*-algebras of some inverse semigroups

1990 ◽  
Vol 42 (2) ◽  
pp. 335-348 ◽  
Author(s):  
Rachel Hancock ◽  
Iain Raeburn
Keyword(s):  
Inverse Semigroup ◽  
Semigroup Algebra ◽  
Inverse Semigroups ◽  
Bicyclic Semigroup ◽  
C Algebras

We discuss the structure of some inverse semigroups and the associated C* algebras. In particular, we study the bicyclic semigroup and the free monogenic inverse semigroup, following earlier work of Conway, Duncan and Paterson. We then associate to each zero-one matrix A an inverse semigroup CA, and show that the C*-algebra OA of Cuntz and Krieger is closely related to the semigroup algebra C*(CA).

scholarly journals The $K$-Theory of Some Reduced Inverse Semigroup $C^*$-Algebras

2015 ◽  
Vol 117 (2) ◽  
pp. 186 ◽  
Author(s):  
Magnus Dahler Norling
Keyword(s):  
Inverse Semigroup ◽  
Recent Result ◽  
Inverse Semigroups ◽  
Crossed Product ◽  
Crossed Products ◽  
One Dimensional ◽  
C Algebras ◽  
Morita Equivalent ◽  
K Theory

We use a recent result by Cuntz, Echterhoff and Li about the $K$-theory of certain reduced $C^*$-crossed products to describe the $K$-theory of $C^*_r(S)$ when $S$ is an inverse semigroup satisfying certain requirements. A result of Milan and Steinberg allows us to show that $C^*_r(S)$ is Morita equivalent to a crossed product of the type handled by Cuntz, Echterhoff and Li. We apply our result to graph inverse semigroups and the inverse semigroups of one-dimensional tilings.

scholarly journals Left regular bands of groups of left quotients

1991 ◽  
Vol 33 (1) ◽  
pp. 29-40 ◽  
Author(s):  
Abdulsalam El-Qallali
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Inverse Semigroups ◽  
Left Inverse ◽  
Regular Band ◽  
Left Regular ◽  
Left Regular Band ◽  
Left Regular Bands

In this paper we characterize semigroups S which have a semigroup Q of left quotients, where Q is an ℛ-unipotent semigroup which is a band of groups. Recall that an ℛ-unipotent (or left inverse) semigroup S is one in which every ℛ-class contains a unique idempotent. It is well-known that any ℛ-unipotent semigroup 5 is a regular semigroup in which the set of idempotents is a left regular band in that efe = ef for any idempotents e, fin S. ℛ-unipotent semigroups were studied by several authors, see for example [1] and [13].Bailes [1]characterized ℛ-unipotent semigroups which are bands of groups. This characterization extended the structure of inverse semigroups which are semilattices of groups. Recently, Gould studied in [7]the semigroup S which has a semigroup Q of left quotients where Q is an inverse semigroup which is a semilattice of groups.

scholarly journals Higher-Rank Graph C *-Algebras: An Inverse Semigroup and Groupoid Approach

2005 ◽  
Vol 71 (2) ◽  
pp. 159-187 ◽  
Author(s):  
Cynthia Farthing ◽  
Paul S. Muhly ◽  
Trent Yeend
Keyword(s):  
Inverse Semigroup ◽  
C Algebras ◽  
Higher Rank

scholarly journals Baer-Levi semigroups of partial transformations

2004 ◽  
Vol 69 (1) ◽  
pp. 87-106 ◽  
Author(s):  
Fernanda A. Pinto ◽  
R.P. Sullivan
Keyword(s):  
Inverse Semigroup ◽  
Cancellative Semigroup ◽  
Green’S Relations ◽  
Green's Relations ◽  
Algebraic Properties ◽  
Symmetric Inverse Semigroup ◽  
Infinite Set

Let X be an infinite set and suppose א0 ≤ q ≤ |X|. The Baer-Levi semigroup on X is the set of all injective ‘total’ transformations α: X → X such that |X\Xα| = q. It is known to be a right simple, right cancellative semigroup without idempotents, its automorphisms are “inner”, and some of its congruences are restrictions of Malcev congruences on I(X), the symmetric inverse semigroup on X. Here we consider algebraic properties of the semigroup consisting of all injective ‘partial’ transformations α of X such that |X\Xα| = q: in particular, we descried the ideals and Green's relations of it and some of its subsemigroups.

scholarly journals On inverse semigroup $C^*$-algebras and crossed products

2014 ◽  
Vol 8 (2) ◽  
pp. 485-512 ◽  
Author(s):  
David Milan ◽  
Benjamin Steinberg
Keyword(s):  
Inverse Semigroup ◽  
Crossed Products ◽  
C Algebras

scholarly journals Connes-amenability of bidual and weighted semigroup algebras

2006 ◽  
Vol 99 (2) ◽  
pp. 217 ◽  
Author(s):  
Matthew Daws
Keyword(s):  
Banach Algebra ◽  
Amenable Group ◽  
Restrictive Condition ◽  
Cancellative Semigroup ◽  
Semigroup Algebras ◽  
C Algebras ◽  
Weighted Semigroup ◽  
Connes Amenability

We investigate the notion of Connes-amenability, introduced by Runde in [10], for bidual algebras and weighted semigroup algebras. We provide some simplifications to the notion of a $\sigma WC$-virtual diagonal, as introduced in [13], especially in the case of the bidual of an Arens regular Banach algebra. We apply these results to discrete, weighted, weakly cancellative semigroup algebras, showing that these behave in the same way as ${ C}^*$-algebras with regards Connes-amenability of the bidual algebra. We also show that for each one of these cancellative semigroup algebras $l^1(S,\omega)$, we have that $l^1(S,\omega)$ is Connes-amenable (with respect to the canonical predual $c_0(S)$) if and only if $l^1(S,\omega)$ is amenable, which is in turn equivalent to $S$ being an amenable group, and the weight satisfying a certain restrictive condition. This latter point was first shown by Grønnbæk in [6], but we provide a unified proof. Finally, we consider the homological notion of injectivity, and show that here, weighted semigroup algebras do not behave like ${ C}^*$-algebras.

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