Quotients of inverse semigroups, étale groupoids and C*-algebras (本文)

概要

The theory of operator algebras is a branch of functional analysis. This theory was initiated to formulate a mathematical framework of quantum mechanics. The theory of operator algebras itself is deeply evolving and interacting with other fields like representa- tion theory, dynamical systems, number theory and so on. Operator algebras are divided into von Neumann algebras and C*-algebras, depending on topologies. In this thesis, we study C*-algebras. Because C*-algebras are highly abstract objects, it used to be difficult to construct a C*-algebra with desired properties. Now there are many ways to construct C*-algebras from mathematical objects like groups, dynamical systems, directed graphs and so on. Many researchers have studied the relation between associated C*-algebras and their ingredients.

In this thesis, we treat C*-algebras associated to ´etale groupoids. A groupoid is a small category whose morphisms are invertible. An ´etale groupoid is a groupoid equipped with topology which has discreteness in some sense. Discrete groups and topological spaces are typical examples of ´etale groupoids. E´tale groupoids are associated to many objects like discrete group actions, directed graphs, tilings and so on. Using ´etale groupoids, we can treat many objects in a unified way.

For an ´etale groupoid G, one can associate C*-algebras C∗(G) and C∗(G), which are called the full groupoid C*-algebra and the reduced groupoid C*-algebra respectively. The study of C*-algebras associated to groupoids was initiated by Renault’s lecture note [18]. The class of groupoid C*-algebras is an important class of C*-algebras because it contains a broad class of C*-algebras and groupoid C*-algebras are somewhat treatable. Actually, many researchers have studied the relationship between ´etale groupoids G and groupoid C*-algebras C∗(G), C∗(G). For example, the simplicity of groupoid C*-algebras is studied in [3] while the intermediate subalgebras of groupoid C*-algebras are studied in [4].

As mentioned above, we can construct ´etale groupoids from many objects. In this thesis, we mainly treat ´etale groupoids associated to actions of inverse semigroups. An inverse semigroup is a special class of semigroups. Inverse semigroup actions are used to describe the local symmetry of the spaces, while group actions describe the global symmetry of the spaces. When an inverse semigroup acts on a topological space, one can associate an ´etale groupoid. An inverse semigroup acts on a certain topological space called a spectrum in a natural way. Hence, we can associate an ´etale groupoid, which is called a universal groupoid, to this action on the spectrum. The study of the universal groupoids is initiated by Paterson [14]. It is a natural task to study the relation between inverse semigroups and the universal groupoids. Because the universal groupoids are constructed only from the algebraic structure of inverse semigroups, it is expected that properties of the universal groupoids should be described in purely algebraic language.

The author of this thesis studies the relation among inverse semigroups, ´etale groupoids and C*-algebras. This research aims to give algebraic and intuitive description for infinite dimensional phenomena of C*-algebras by using inverse semigroup and ´etale groupoids. In addition, this research also aims to apply techniques in the theory of C*-algebras to the theory of inverse semigroups and ´etale groupoids. In short, the purpose of this research is to construct a framework to mutually develop the theory of inverse semigroups, ´etale groupoids and C*-algebras.

In this thesis, we study the relation among inverse semigroups, ´etale groupoids and C*-algebras from the view point of quotients. We will prove that quotients of inverse semigroups induce the quotients of ´etale groupoids. Similarly, we prove that quotients of
´etale groupoids induce the quotients of C*-algebras. Then we investigate certain quotients such as the abelianization of inverse semigroups, ´etale groupoids and C*-algebras. The main theorems in this thesis are Theorem A, B and C as described below.

This thesis is organized as follows. Chapter 1 is devoted to preliminaries. We introduce here notions which we use in this thesis.

In Chapter 2, we describe the results in [11]. Quotients of inverse semigroups and C*-algebras are fundamental notions and well-established. On the other hand, quotients of ´etale groupoids seem to be fundamental notions, but the author could not find them in literatures. Therefore, we establish the notion of quotient ´etale groupoids. One may imagine that the notion of a quotient ´etale groupoid is defined as a surjective groupoid homomorphism to another ´etale groupoid. However, such formal quotients do not induce the quotients of groupoid C*-algebras in a natural way. Therefore, in this thesis, we define the notion of quotient ´etale groupoids so that the quotients of groupoid C*-algebras are naturally induced. After we define the notion of quotient ´etale groupoids, we observe that quotients of ´etale groupoids actually induce the quotients of C*-algebras. Using these facts, we obtain the main theorem (Theorem 2.2.2.4) in this chapter. For an ´etale groupoid G, we define the abelianization Gab, which is also an ´etale groupoid. This ´etale groupoid Gab describes the abelianization of C∗(G) as follows.

Theorem A (Theorem 2.2.2.4). Let G be an ´etale groupoid with the locally compact Hausdorff unit space G(0). Then the abelianization C∗(G)ab of C∗(G) is isomorphic to C∗(Gab).
The key step in the proof of Theorem 2.2.2.4 is the calculation of one dimensional representations of C∗(G) (Theorem 2.2.1.8). At the end of this chapter, we explain the relation between these theorems and the dual of ´etale abelian group bundles.

In Chapter 3, we describe the results in [10]. We study the relation between quotients of inverse semigroups and quotients of the universal groupoids. Given an inverse semigroup S, one can associate the universal groupoid Gu(S). We observe that a quotient S ↠ S/ν of an inverse semigroup S by a congruence ν induces the invariant set Fν of Gu(S) and the normal subgroupoid Gu(ker ν)Fν ⊂ Gu(S)Fν , where Gu(S)Fν denotes the restriction of Gu(S) to Fν. Now we may consider the quotient ´etale groupoid Gu(S)Fν /Gu(ker ν)Fν and obtain one of the main theorems (Theorem 3.2.1.3).

Theorem B (Theorem 3.2.1.3). Let S be an inverse semigroup and ν be a congruence on S. Then Gu(S/ν) is isomorphic to Gu(S)Fν /Gu(ker ν)Fν .
This theorem gives us a way to compute the universal groupoids associated to quotient inverse semigroups. Indeed, we compute the universal groupoid associated to the certain quotient inverse semigroups including the abelianizations. We show that the abelianiza- tions of ´etale groupoids introduced in Chapter 2 corresponds to the abelianizations of inverse semigroups.

Theorem C (Theorem 3.2.2.3). Let S be an inverse semigroup. Then Gu(Sab) is iso- morphic to Gu(S)ab.
In the last of this chapter, we give applications and examples. We analyse Clifford inverse semigroups and compute the universal groupoids associated to the free Clifford inverse semigroups (Theorem 3.3.2.7). We also evaluate the number of fixed points in transformation groupoids associated to Boolean actions (Corollary 3.3.3.2).