Chapter 1. Introduction and Overview

1

INTRODUCTION AND OVERVIEW

CHAPTER 1.

For lack o f

This book is about algebras of operators.

1.

a better name, we shall refer to these algebras a s a L g e b r a s o f

This is because most people think about

unbounded operators.

* C -algebras

and

von Neumann

"operator algebras"

*

not be

C -algebras, but

*

algebras

appears, and

when

the

shorter

our algebras will

they are utilized

in the study of

C -algebras as we shall show in Part I11 of the book.

used

especially

geometry which

in

that

part

of

noncommutative

is concerned with

structures on simple

title

typically

They are

differential

the study of differentiable

*

C -algebras.

(This subdiscipline has not

really become a theory yet because only a few classes of simple

*

C -algebras are understood up to now.)

The algebras which we consider arise frequently in the study

of representations of Lie groups (both finite-dimensional, and infinite-dimensional ones).

The theory is best developed for

finite-dimensional groups, but

are

playing

an

increasingly

recent work on loop groups. In some cases

structures for

group

actions

important

the problem

3c

the infinite-dimensional groups

the

assumed that the given charts," and we

in

of understanding

C -algebras boils down

on

role

*

C -algebra

* C -algebra

the

exciting

differentiable

to finding smooth Lie

It will

in question.

be

has an atlas of "coordinate

shall see in examples how

to

identify

this.

Then the smoothness is a relative notion and refers to a given,

more o r less canonical, group action defined naturally in terms of the generators for the given

*

C -algebra.

The more classical and f a m i l i a r side of the subject is the

study

space

of

%).

unitary

representations

of

Lie

groups

(on

Hilbert

Such a representation has an enveloping algebra of

unbounded operators on

3,

The

is a

and i t is obtained by differentia-

tion of the given unitary representation along the Lie algebra. resulting

algebra

*-algebra,

i.e., comes

with

a

natural involutory anti-automorphism o r , equivalently, is given by a Hermitian representation.

J o rgensen

2

The enveloping algebra of a unitary representation is useful

because the elements in i t are operators which play a central role in Quantum Mechanics.

Examples of such algebras include

the Weyl algebra of partial differential operators with poly-

nomial coefficients.

magnetic

We shall also show how Hamiltonians for

fields can be

unitary

representations

completely of

spectrally

certain

analyzed using

nilpotent

Lie

Finally we shall show how the representation theory of can be used for constructing models space.

for operators

The converse problem is to decide i f a given

unbounded unitary

operators

can

representation.

be

integrated

This

is

groups. SL2(!R)

in Hilbert

*-algebra

(exponentiated)

the

more

interesting

to

of

a

and

difficult part, and i t too has its roots in Quantum Mechanics since

the

*-algebra

is

typically

given

from

the*ysical

context, and i t is generated by raising and lowering operators (the

notion

theory).

Verma-module

is

now

popular

"reconstruct" the unitary representation.

If

in

representation

then

this procedure

The problem is then to find the Hilbert space, and to

the problem

is not

"unitarizable"

fails, and one tries instead to get a representation of the Lie

group by bounded operators in some Banach space. techniques which

Nelson’s

analytic

are used vector

in solving

method, m

bation methods.

the

C -vector

based on complete positivity,

Arveson-Powers

include method

methods, and pertur-

The latter two types of methods were developed

recently by R . T. Moore and the author in [ J M ] . Recently,

The types of

these problems

heat

equation

methods

have

been

developed

by

0. Bratteli, F.M. Goodman, D.W. Robinson and the author [ B G J R ] , and

they were shown to apply to the most general instance of

the integrability problem in the setting of Banach spaces.

In this book we shall include a systematic treatment of the

methods which have been developed more recently, and we shall apply them to some classical problems in Quantum Mechanics, and to some recent problems in

theory.

*

C -algebras, and general operator