Semigroup C*-algebras

Abstract

We give an overview of some recent developments in semigroup C*-algebras. Contents 1. Introduction 2. C*-algebras generated by left regular representations 3. Examples 3.1. The natural numbers 3.2. Positive cones in totally ordered groups 3.3. Monoids given by presentations 3.4. Examples from rings in general, and number theory in particular 3.5. Finitely generated abelian cancellative semigroups 4. Preliminaries 4.1. Embedding semigroups into groups 4.2. Graph products 4.3. Krull rings 5. C*-algebras attached to inverse semigroups, partial dynamical systems, and groupoids 5.1. Inverse semigroups 5.2. Partial dynamical systems 5.3. Étale groupoids 5.4. The universal groupoid of an inverse semigroup 5.5. Inverse semigroup C*-algebras as groupoid C*-algebras 5.6. C*-algebras of partial dynamical systems as C*-algebras of partial transformation groupoids 5.7. The case of inverse semigroups admitting an idempotent pure partialhomomorphism to a group 6. Amenability and nuclearity 6.1. Groups and groupoids 6.2. Amenability for semigroups 6.3. Comparing reduced C*-algebras for left cancellative semigroups and their left inverse hulls 6.4. C*-algebras generated by semigroups of projections 6.5. The independence condition 6.6. Construction of full semigroup C*-algebras 6.7. Crossed product and groupoid C*-algebra descriptions of reduced semigroup C*-algebras 6.8. Amenability of semigroups in terms of C*-algebras 1 2 XIN LI 6.9. Nuclearity of semigroup C*-algebras and the connection to amenability 68 7. Topological freeness, boundary quotients, and C*-simplicity 69 8. The Toeplitz condition 80 9. Graph products 86 9.1. Constructible right ideals 87 9.2. The independence condition 91 9.3. The Toeplitz condition 94 10. K-theory 97 11. Further developments, outlook, and open questions 99 References 101

Groupoids and $C^*$-algebras for left cancellative small categories

“…al. ([15], [5]), and took dramatic steps forward with the work of Li ( [16]; see also the survey [17] ). In particular, Li considered chiefly right LCM monoids, that is having order structure like that of the positive cone in a quasi-lattice ordered group.…”