Introduction

In Chapter 2, we mainly discussed bounded derivations on C*-algebras and special unbounded derivations which arise from a fixed physical state (and consequently associated with a W*-dynamical system). However, in quantum physics, we have to study various physical states like ground states, vacuum states and equilibrium states on a C*-algebra of quasi-local observables. In addition a partial differentiation on a mainfold is not bounded. Therefore, unbounded derivations in C*-algebras are perhaps much more important than those previously discussed.

In the previous chapter, we have seen that an everywhere-defined derivation on a C*-algebra is automatically bounded; for such derivations, the purely algebraic property implies continuity. On the other hand, for a densely defined derivation, the purely algebraic property (Leibnitz's rule) does not imply closability. Therefore it is quite difficult to study densely defined derivations from a purely algebraic point of view. -Fortunately all densely defined derivations which arise from analysis, geometry and quantum physics turn out to be closable. They are also self-adjoint with respect to the *-operation. In this chapter, we shall study closable *-derivations in C*-algebras. The study of general unbounded derivations in C*-algebras can be divided into three steps: (i) closability, (2) the domain of a closed *-derivation and (3) generators.

Closable *-derivations are also of special interest, because they include partial differentiations in manifolds. In this chapter, we shall discuss these topics.

Derivations appeared for the first time at a fairly early stage in the young field of C*-algebras, and their study continues to be one of the central branches in the field. During the past four decades, the study of derivations has made great strides. Their theory divides naturally into two major parts: bounded derivations and unbounded derivations. About thirty years ago, Kaplansky (in an excellent survey [97] on derivations) brought together two apparently unrelated results which stimulated research on continuous derivations. The first, related to quantum mechanics and due to Wielandt [190] in 1949, proved that the commutation relation ab − ba = 1 carinot be realized by bounded operators. The second, involving differentiation and due to Šilov [180] in 1947, proved that if a Banach algebra A of continuous functions on the unit interval contains all infinitely differentiable functions, then A contains all n-times differentiable functions for some n.

It is noteworthy that Kaplansky's observations three decades ago are still applicable to recent developments in the theory of unbounded derivations, although one has to replace quantum mechanics by all of quantum physics. Furthermore, the work of Šilov continues to have a strong influence on the study of unbounded derivations in commutative C*-algebras.

At an early stage, mathematicians devoted most of their efforts to the study of bounded derivations, rather than unbounded ones, even though the work of Šilov and Wielandt had already suggested the importance of unbounded derivations. This, of course, is understandable because bounded derivations are more easily handled than unbounded ones.

Introduction

In this chapter, we shall discuss a unified axiomatic treatment of quantum lattice systems and quasi-free dynamics in Fermion field theory within the framework of C*-dynamical systems. In these systems, time evolution, equilibrium states (KMS states), ground states, stability under bounded perturbations and phase transitions are important physical notions. Here we shall present an abstract treatment of these notions within the theory of C*-dynamical systems. About time evolution, we shall emphasize the approximate innerness property for the corresponding time automorphism group in a given C*-dynamical system. In fact, the approximate innerness assures the existence of a ground state, and the existence of a KMS state at each inverse temperature under the assumption of ‘the existence of a tracial state’, which is always satisfied in quantum lattice systems and in canonical anticommutation relation algebras.

We shall also discuss in detail one of the most important open problems (the Powers-Sakai conjecture) in C*-dynamical systems. In each section we shall explain relations between C*-dynamical systems and physical systems. For simplicity we shall assume that every C*-algebra has an identity (unless otherwise stated).

Approximately inner C*-dynamics

A quantum lattice system consists of a set of particles confined to a lattice and interacting at distance. There are two physical interpretations of these models. One is a lattice gas and the other is a spin system. The lattice gas views each point of the lattice as a possible site for a finite number of AT-particles, i.e. each point of the configuration space can be empty or occupied by 1, 2, …, N -particles.

In this chapter we turn to the analysis of Clifford algebras and modules, together with an analysis of the associated Dirac operators, on Riemannian manifolds more general than the open subsets of Euclidean space studied in the previous chapter. Concepts from differential geometry are needed from the outset, but in keeping with the spirit of making the material available to more classically trained analysts we have attempted to minimize the use of differential geometric machinery, possibly at the expense of clarity and elegance. In the first section, therefore, Dirac operators are introduced explicitly on a single coordinate patch of a manifold. This serves several useful purposes. It helps bring out quite simply the role that the curvature of the manifold plays in the expression for D2. For the ‘flat’ case studied earlier, (–D2) was the Euclidean Laplacian, and solutions of DF = 0 were automatically harmonic: the forerunner of the GCR property of operators. But the fundamental Bochner–Weitzenböck theorem expresses (–D2) in general as a second-order Laplacian together with a zero-order curvature operator. This idea is a basic one throughout the chapter. In the specific examples of the spinor Laplacian and the Hodge Laplacian, the curvature operator is explicitly calculated. By assuming that the coordinates form a normal coordinate system, we also express (–D2) asymptotically as a sum of an operator, which will play a fundamental role in the proof of the Atiyah–Singer index theorem, and a remainder operator.

Section 2 deals with the problem of passing from a local setting to a global setting.

Associated with any Euclidean space ℝi or Minkowski space ℝp,q is a universal Clifford algebra, denoted by and, respectively. Roughly speaking, a Clifford algebra is an associative algebra with unit into which a given Euclidean or Minkowski space may be embedded, in which the corresponding quadratic form may be expressed as the negative of a square. The real numbers ℝ, the complex numbers ℂ, and the quaternions ℝ are the simplest examples.

Our intent in this chapter is to give an elementary, coherent, and largely self-contained account of the theory of Clifford algebras. In sections 1 and 2 we present the definitions basic to all of our work. The balance of section 2 is devoted to three constructive proofs of the existence of universal Clifford algebras: two basis-free constructions using tensor algebras and exterior algebras, and a basis-dependent construction. The reader who is willing to accept the existence of Clifford algebras may wish to proceed directly to the statement of the major structural results in section 3. Sections 4, 5, and 6 explore the interconnections between Clifford algebras and orthogonal groups; the spin representation and spin groups will be studied in detail, with Spin(p, q) and Spin(p, q + 1) both being realized in using the notion of transformers. The reader who is primarily interested in the analytic applications of Clifford algebras may wish to proceed directly to the discussion of the Euclidean case in section 7. Section 8 is a discussion of spin groups as Lie groups. In section 9 we construct various realizations of Spin(p, q), p + q ≤ 6, whereby these groups are explicitly identified with classical Lie groups.