Operator algebras and quantum statistical mechanics 1 2nd edition

Texts and

Monographs

in

Physics

Series Editors:
R.

Balian, Gif-sur-Yvette, France

W. Bei 9 lb6ck, Heidelberg,
t,

Germany

H. Grosse, Wien, Austria
E. H. Lieb, Princeton, NJ, USA

H.

Reshetikhin, Berkeley, CA, USA
Spohn, Minchen, Germany

W.

Thirring, Wien,

N.

Austria

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01a Bratteli
Derek W. Robinson

Operator Algebras
an d Q uantum
S tatisticaIM ec h anic s I

C*- and

W*-Algebras
Symmetry Groups
Decomposition of States
Second Edition

4
Springer

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Professor Ola Bratteli
Universitetet i Oslo

Matematisk Institutt
Moltke Moes vei 31

0316 Oslo, Norway
e-mail:
Home page:

http:Hwww. math. uio. no/-bratteli/

Professor Derek W. Robinson
Australian National

University

School of Mathematical Sciences

ACT 0200 Canberra, Australia
e-mail: Derek. Robinson@ anu. edu. au
Home page:

Library

of

http:Hwwwmaths.anu.edu.au/-derek/

Congress Cataloging-in-Publication

Data

Bratteli,Ola. Operator algebras and quantum statistical mechanics. (Texts and monographs in physics) Bibliography;v.
and W -algebras, symmetry groups, decomposition of states. 1. Operator

1, p. Includes index. Contents: v. 1. C
algebras. 2. Statistical mechanics. 3. Quantum statistics. 1. Robinson, Derek W. H. Title. III. Series.
*

*

-

QA 326.B74

1987

512'.55

86-27877

Second Edition 1987. Second

Printing

2002

ISSN 0172-5998
ISBN 3-540-17093-6 2nd Edition

Springer-Verlag

Berlin

Heidelberg

New York

This work is

subject to copyright. All rights are reserved, whether the whole or part of the material is concerned,
specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm
or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under
the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must
always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.

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a member of BertelsmannSpringer Science+Business

Media GmbH


0

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Germany

1987

Printed in
The

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of

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Preface to the Second

In this second

printing

Printing of the Second

of the second edition several minor and

matical mistake have been corrected. We
Duedahl and Reinhard Schaflitzel for

are

pointing

one

Edition

major

mathe-

indebted to Roberto Conti, Sindre
these out.

Canberra and Trondheim, 2002

Ola Bratteli
Derek W. Robinson

Preface to the Second Edition

The second edition of this book differs from the

original in three respects. First,
typographical errors. Second, we have
corrected a small number of mathematical oversights. Third, we have rewritten
several subsections in order to incorporate new or improved results. The principal
changes occur in Chapters 3 and 4.
In Chapter 3, Section 3.1.2 now contains a more comprehensive discussion
of dissipative operators and analytic elements. Additions and changes have also
been made in Sections 3.1.3, 3.1.4, and 3.1.5. Further improvements occur in
Section 3.2.4. In Chapter 4 the only substantial changes are to Sections 4.2.1 and
4.2.2. At the time of writing the first edition it was an open question whether
maximal orthogonal probability measures on the state space of a C*-algebra
were automatically maximal among all the probability measures on the space.
This question was resolved positively in 1979 and the rewritten sections now
incorporate the result.
All these changes are nevertheless revisionary in nature and do not change
the scope of the original edition. In particular, we have resisted the temptation
to describe the developments of the last seven years in the theory of derivations,
and dissipations, associated with C*-dynamical systems. The current state of this
theory is summarized in [[Bra 1]] published in Springer-Verlag's Lecture Notes in
we

have eliminated

a

large

number of

Mathematics series.

Canberra and Trondheim, 1986

Ola Bratteli
Derek W. Robinson
v

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Preface to the First Edition

In this book

describe the

elementary theory of operator algebras and
theory which are of relevance, or potentially of

relevance, to mathematical physics. Subsequently we describe various
applications to quantum statistical mechanics. At the outset of this project
we intended to cover this material in one volume but in the course of
development it was realized that this would entail the omission of various interesting
topics or details. Consequently the book was split into two volumes, the
first devoted to the general theory of operator algebras and the second to the
applications.
This splitting into theory and applications is conventional but somewhat
arbitrary. In the last 15-20 years mathematical physicists have realized the
importance of operator algebras and their states and automorphisms for
problems of field theory and statistical mechanics. But the theory of 20 years
ago was largely developed for the analysis of group representations and it
was inadequate for many physical applications. Thus after a short
honeymoon period in which the new found tools of the extant theory were
applied
to the most amenable problems a longer and more interesting period ensued
in which mathematical physicists were forced to redevelop the theory in
relevant directions. New concepts were introduced, e.g. asymptotic abelianness and KMS states, new techniques applied,
e.g. the Choquet theory of
barycentric decomposition for states, and new structural results obtained,
e.g. the existence of a continuum of nonisomorphic type-three factors. The
results of this period had a substantial impact on the subsequent development
of the theory of operator algebras and led to a continuing period of fruitful
we

parts of the advanced

Vil

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viii

Preface to the First Edition

collaboration between mathematicians and

intertwining

of the

theory

forced the formation of the
book has

a

physicists. They also led to an
applications
applications often
in
Thus
this
the
division of this
context
theory.

and

in which the

certain arbitrariness.

The two volumes of the book contain six
and two in the second. The

chapters,

four in this first volume

of the second volume

are numbered
chapters
consecutively with those of the first and the references are cumulative.
Chapter I is a brief historical introduction and it is the five subsequent
chapters that form the main body of material. We have encountered various
difficulties in our attempts to synthesize this material into one coherent book.
Firstly there are broad variations in the nature and difficulty of the different
chapters. This is partly because the subject matter lies between the mainstreams of pure mathematics and theoretical physics and partly because it is a
mixture of standard theory and research work which has not previously
appeared in book form. We have tried to introduce a uniformity and structure
and we hope the reader will find our attempts are successful. Secondly the
range of topics relevant to quantum statistical mechanics is certainly more
extensive than our coverage. For example we have completely omitted
discussion of open systems, irreversibility, and semi-groups of completely
positive maps because these topics have been treated in other recent monographs [[Dav 1]] [[Eva 1]].

This book was written between September 1976 and July t979.
Most of Chapters 1-5 were written whilst the authors were in Marseille at
the Universit6 d'Aix-Marseille 11, Luminy, and the Centre de Physique
Th6orique CNRS. During a substantial part of this period 0. Bratteli was
supported by the Norwegian Research Council for Science and Humanities
and during the complementary period by a post of Professeur Associ6 at
Luminy. Chapter 6 was partially written at the University of New South
Wales and partially in Marseille and at the University of Oslo.
Chapters 2, 3, 4 and half of Chapter 5 were typed at the Centre de Physique
Th6orique, CNRS, Marseille. Most of the remainder was typed at the
Department of Pure Mathematics, University of New South Wales. It is a
pleasure to thank Mlle. Maryse Cohen-Solal, Mme. Dolly Roche, and
Mrs. Mayda Shahinian for their work.
We have profited from discussions with many colleagues throughout the
preparation of the manuscript. We are grateful to Gavin Brown, Ed Effros,
George Elliott, Uffe Haagerup, Richard Herman, Daniel Kastler, Akitaka
Kishimoto, John Roberts, Ray Streater and Andr6 Verbeure for helpful
"

"

comments and corrections to earlier versions.

We are particularly indebted to Adam Majewski
manuscript and locating numerous errors.

Oslo and

Sydney,

1979

for

reading

the final

Ola Bratteli

Derek W. Robinson

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Contents (Volume 1)

Introduction

Notes and Remarks

C*-Algebras
2. 1.

and

16

von

Neumann

Algebras

C* -Algebras

19

2. 1. 1. Basic Definitions and Structure

19

2.2. Functional and

Spectral Analysis

2.2. 1. Resolvents,

Spectra,

and

25
Radius

25

Quotient Algebras

39

Spectral

2.2.2. Positive Elements
2.2.3.

2.3.

17

Approximate

Representations
2.3. 1.

32

Identities and

and States

42

Representations

42

2.3.2. States

48

2.3.3. Construction of

54

2.3.4.

2.3.5.

Representations
Existence of Representations
Commutative C*-Algebras

58
61
ix

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x

Contents

2.4.

von

(Volume t)

Neumann

2.4. 1.

65

Algebras

Topologies

on

2.4.2. Definition and

Y(.5)
Elementary Properties

65

of

von

Neumann

Algebras

2.4.3. Normal States and the Predual
2.4.4.

Quasi-Equivalence

of

2.5. Tomita-Takesaki Modular

79

Representations

Theory

and Standard Forms of

von

Neumann
83

Algebras

Algebras
Group
2.5.3. Integration and Analytic Elements for
Isometries on Banach Spaces

2.5. 1. a-Finite

von

84

Neumann

86

2.5.2. The Modular

2.6.

One-Parameter

Groups of
97

2.5.4. Self-Dual Cones and Standard Forms

102

Quasi-Local Algebras

118

2.6. 1. Cluster

118

2.6.2.

129

2.6.3.

Properties
Topological Properties
Algebraic Properties

133

136

2.7. Miscellaneous Results and Structure

2.7. 1.

Dynamical Systems

and Crossed Products

136
142

2.7.2. Tensor Products of
2.7.3.

Operator Algebras
Weights on Operator Algebras; Self-Dual Cones
von Neumann Algebras; Duality and Classification
Classification of C*-Algebras

Groups, Sernigroups, and
3. 1. Banach

of General
of Factors;
145

152

Notes and Remarks

Generators

157
159

Space Theory

3. 1. 1. Uniform

161

Continuity
Strong, Weak, and Weak* Continuity
3.1.3. Convergence Properties
3.1.4. Perturbation Theory
3.1.5. Approximation Theory

202

Algebraic Theory

209

3.1.2.

3.2.

71

75

3.2. 1. Positive Linear
3.2.2.

Maps and Jordan Morphisms
General Properties of Derivations

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163
t84
t93

209
233


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Functional and

Proposition 2.2.13. Let A, B,
following implications are valid:
(a)
(b)
(c)

C be elements

! - B !! 0 then

if A
if A
if A

>

0 then

>

B

11 A
All All !! A';

C*-algebra

>

0 then

W. The

0 then

>

for all C Ei W;
if W possesses an identity,
(B

PROOF.

a

of

37

B

C*AC

(d)

Spectral Analysis

A

I

(A

>

0

>

0, and A

!! B >

AT)-

+

C*BC

>

+

AT)-'.

We

adjoin an identity I to W if necessary. The spectral radius formula
gives A < 11 A 111 and hence 0 :!! B :!! 11 A 11 T. But this implies
that JIB 11 :! 11 All by a second application of the same formula.
11 A 111/2) g [ 11 A 11 /2, 11 A 11 /2] and hence u((A
11 A 111/2)2)
(b) One has a(A
[0, IIA 112 /4] by Theorem 2.2.5(d). Thus
(a)

of Theorem 2.2.5 then

-

-

-

IIIA 112

IJAJIT2
0

A

<

2

which is

equivalent

As A

(c)

-

B

c

to 0 :!!

91 +

one

4

A2

has A

B

-

D*D for

=

some

D

c-

W

by

Theorem 2.2.12.

Butthen

C*AC

by

the

same

C*BC

-

=

(DC)*(DC) c- 121+

theorem.

(d) One has
A + AT >- B + AT -> AT

and both A + AT and B + AT

(B
If, however,

Proposition

X

=

+

are

AT)-

(A

X* and X >- T

+

Finally, multiplying

+

A )1/2 (A

each side

+

u(X)

g

AT)-'(B

AT)-'

+

+ AT

)-

1/2 >

and

[1, oo>

by part (c)

I.

u(X-')

-

[0, 1] by

gives

by (B +AT

(A

invertible. Therefore

AT)(B

then

2.2.3. Thus X-' -< 1. This

(B

There

positive

1/2

:!!

1/2

(B

+

AT)1/2

and
+

<

1.

invoking part (c),

one

finds

AT)-'.

many other

interesting inequalities which may be deduced from
Proposition 2.2.13(d) by integration with suitable functions of A. For example,
if A

>

B

are

>

0

one

has

A 1/2
7r

1
7E

dA

f,

A1/2

0

"

f

0

(I

-

AT (AT +

A)

AT (AT

B)-)

dA

A1/2

B 1/2

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38

C*-Algebras

and

von

Neumann

Algebras

i.e., A 1/2 > B 1/2 > 0. By use of similar transforms one can deal with other
fractional powers and deduce that A > B ! 0 implies A' > B" > 0 for all
0 < a < 1. But this is not necessarily true for a > 1.
The

following decomposition lemma is often useful
positive elements.

and is another

applica-

identity. Every

element

tion of the structure of

Lemma 2.2.14.
A-e W has

a

Let W be a C*-algebra
decomposition of theform
A

where the Uj

are

ajUj

a2U2

with

a3U3

+

unitary elements of % and the

ai

=

+

+

a4U4
e

C

satisfy I ai I

:!!

11 A 11 /2.

PROOF. It suffices to consider the case 11AII
1. But then A
A, + iA2 with
(A + A*)/2 and A2
A,
(A
A*)12i selfadjoint, IIA111 < 1, IIA211 < 1. A
general selfadjoint element B with 11B11 < I can, however, be decomposed into two
=

=

=

unitary elements B
As

a

final

=

(U +

application

=

-

+ U _)/2

of the

by the explicit construction
of

properties

U+

=

B +

positive elements

i'll

we

-

B 2.

consider

another type of decomposition. First let us extend our definition of the
modulus. If W is a C*-algebra then A*A is positive for all A e % by Theorem
e % is then defined by I A I
1A *A. If A is selfadjoint this coincides with the previous definition. Now note that if % contains
an identity and A is invertible then A*A is invertible and its inverse is
positive.

2.2.11. The modulus of A

=

It follows that I A I is invertible and I A

A

where U

=

AIAI

Therefore U is

a

-

,I(A; A . But

"
=

one

then has

UJAJ.

=

Moreover, U* U

I

=

I and U is invertible

unitary element of W and

in

AIA
(U
C*-subalgebra
special case of the so-

fact lies in the

generated by A and A*. This decomposition of A is a
called polar decomposition. The general polar decomposition concerns
operators on a Hilbert space and represents each closed, densely defined
operator A as a product A
V(A*A) 1/2 of a partial isometry V and a
positive selfadjoint operator JAI
(A*A) 1/2 We illustrate this and other
Hilbert space properties in the following:
=

=

.

EXAMPLE 2.2.15.

Let

Y(S) )

denote the

algebra of all bounded operators on the
algebra by Example 2.1.2. If A C- Y(-5)
B*B for some B c- Y(-5)
is equivalent to A
2
and this implies that ( , A )
11B 11 > 0for all c .5. In Hilbert space theory this
last property is usually taken as the definition of positivity but it is equivalent to the
abstract definition by the following reasoning. If the values of ( , A ) are positive
then they are, in particular, real and (0, AO)
(A0, 0). Therefore the polarization
identity

Hilbert space 5; then Y(5) is
then the abstract definition of positivity

complex

a

C*-

=

=

=

3

i-k((o

( 01 Aq)
4

+

ik 9), A(0

k=O

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+

ik(p))


Functional and

demonstrates that
A

(0, Aq)

=

for all

(A0, 9)

0,

o

c-.5,

Spectral Analysis

i.e., A is

selfadjoint.

39

But if

0 then

<

JI(A

AJ)0112

_

JJA0 112

+

21AI(o, AO)

+

A2110112

A2110112
and A

-

AT is invertible.

Consequently a(A) c- [0, 11 A 11 ] and A

is

the

positive in

general

sense.

LetAc-Y(.5)andlAl=(A*A)' /2

EXAMPLE2.2.16.
on

all vectors of the form I A

VIA10
This is
0

a

consistent definition of

111 A 10 11
A 0 11

11 A 0 11 and hence

=

.

Now define

an

operator V

10 by the action

a

=

A0.

linear operator because I A 10
0 is equivalent to
0. Moreover, V is isometric because 11 V I A 10 11
=

A0

=

111 A 10 11. We may extend V to a partial isometry on .5 by setting it equal
orthogonal complement of the set I I A 10; 0 c- .51 and extending by
linearity. This yields the polar decomposition of A, i.e., A
VI A 1. This decomposition
is unique in the sense that if A
UB with B > 0 and U a partial isometry such that
0 just for o orthogonal to the range of B then U
V and B
U9

I Al. This
follows because A*A
BU*UB
B 2 and hence B is equal to the unique positive
V I A I and both U and V are equal to zero
square root I A I of A *A. But then U I A I
on the orthogonal complement of the
range of I A 1. In general, V will not be an element
of the C*-algebra 91A generated by A and A*, although we have seen that this is the
case whenever A has a bounded inverse. Nevertheless, in Section 2.4 we will see that
V is an element of the algebra obtained by adding to 91A all
strong or weak limit
points of nets of elements Of %A
=

to zero

=

the

on

=

=

=

=

=

=

=

=

-

2.2.3.

Approximate Identities
Algebras

and

Quotient

In Section 2.2.1

we gave examples of C*-algebras which failed to have an
identity element and demonstrated that it is always possible to adjoin such an

element. Nevertheless, situations often

identity is fundamental
approximate identity.
Definition 2.2.17.

identity of -1
(1)
(2)
(3)
'

<

in which the absence of

lim,,, I I E,, A

a

right
a

ideal of a

C*-algebra

symmetric (a
ot <

directed

<

7 and

c-

W then

an

net' JE, I of positive elements E,,

-

A

I1

approximate

c-

3 such that

Ep,
0 for all A

c-

3.

A set I& is said to be directed when there exists

#

an

1,

#implies E,,

of elements (x,
that

If 3 is

is defined to be

JJE,,,JJ
a <

occur

and it is therefore useful to introduce the notion of an

an order relation, (X
between certain pairs
(a :!! a), transitive ((X < # and
1,, imply a < 1), antiimply #
a) and when for each pair a, c-& there exists a y such
is a family of elements, of a general set M, which is indexed
by a

?/ which is reflexive
and

#

:!!

a

A net

=

set J//.

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40

and

C*-Algebras

The definition of

dition

(3)

is

Algebras

approximate identity of

an

left ideal is similar but

a

con-

replaced by

lim,, 11 AE,,

(3)
It

is

Neumann

von

All

-

0 for all A

=

-3.

c-

necessary to prove the existence of

Proposition

2.2-18.

Let 3 be

a

approximate

ideal

right

identities.

ofa C*-algebra W-3 possesses an

approximate identity.

PROOF.

First

adjoin

an

families of 3. The set J&

#
{B1,..., B,,J
foregoing choice of a

if necessary. Next let 41 denote the set of finite

JA,,
by inclusion, i.e., if a
# equivalent to # being a subfamily of a.
define F. c- W by

thena

=

identity to E,
can

be ordered

>

is

Ai

c-

A

and

Now for the

E,, by
E,,

As each

...,

Y AiAi*

F,,
and introduce

=

3

has

one

(E,,,Ai

-

E_ F,,

Ai)(E,,Ai

c-

=

mF,,(l

mF,,)

+

3. Furthermore

-

Ai)*

JJEJJ

(E,,
+

<

1 and

<

I)AiAi*(E,,

-

+

mF,,)

F1/2(l

+

mF,,,) -2F1/2

F,', /2(J

+

m

(

(

-

-

F,,)

+

1

1)
-

F,,(T

mF,)

-

F,,' /2

mF.)

M

1
<

-

M

Here

we

(I + mF,,)-l
proposition

have used 0 :!!

part (a) of the

same

IIE,,Ai

<

-

I and

Proposition 2.2.13(c).

Therefore

by

Ai 112<
M

and

consequently 11 E,, A
Ea,

but
the

-

-

All

Ep

-,-+

=

0 for all A

(T

+

c

nF#)-1

a > P implies mF. > nF# and hence E,,
E,, form an approximate identity.

>-

3.
-

(T

+

note that

mF,)-'

Ep by Proposition 2.2.13(d).

The existence of

an approximate identity
quotient algebras which we began
following:

discussion of

result is the

Finally,

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allows

us

to

Therefore

complete the
principal

in Section 2.1. The


Functional and

Proposition 2.2.19. Let 3 be a
follows that 3 is selfadjoint and
2.1.1 is a C*-algebra.

Spectral Analysis

41

closed two-sided ideal
the quotient

of a C* algebra %. It
algebra %/3 defined in Section

PROOF.

Let f E,,J be an approximate identity of 3. If A e 3 then
A*
11 A*E,
0. But A*E,, c- 3 and hence A* c 3 because 3 is closed. This
All
proves
that 3 is selfadjoint.
-

IIE,,A

-->

-

To

complete both the discussion of the quotient algebra given in Section 2. 1.1
proof of the proposition we must show that the norm on the quotient
algebra,
and the

IJAII

inf{IIA

=

has the C*-norm property. To prove this

we

lim

11 A

This follows

by adjoining,

IIEJ

0, and

-

III

--*

if necessary,

lim sup 11 A

-

JIT

-

first establish that
-

E,, A 11.

identity

an

lim sup 11 (T

E,:, A

IIA
The

III; Ic 31,

+

-

to

W, noting that for I

EJ (A

+

Ill.

+

lim sup 11 A

lim inf 11 A

>

The C*-norm property

is

inf{IIA

then

a

=

limll(A

=

limll(I

where I is

an

arbitrary

implies firstly

Ill; Ic-31

=

-

+

that

11,411

=

<

E,,A)(A
E,)(AA*

following

-

+

E,,A)*Il

I)(T

-

Ill,

lI'j'j*II

11A*11

<

G

EO, 1]

11,411.

element of 3. Thus

11,jI12
which

-

EJ

EAll

E,,A 112

-

-

E,, A 11

consequence of the

11,4112= limlIA

11 AA*

+

-

-

3.,

1) 11

inequality follows because a(E,,) C- EO, 1]. Therefore a(l
EJI :!! 1. But then one concludes that

IJA 11

e

11,11111,4*11,

and, secondly, that

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Ea)II

calculation:

and


2.3.

Representations

2.3. 1.

and States

Representations

partially described the abstract theory of C*general theory by examples of C*-algebras of
operators acting on a Hilbert space. Next we discuss representation theory
and develop the connection between the abstract description and the
operator examples. The two key concepts in this development are the conIn the previous sections we
algebras and illustrated the

cepts of representation and state. The states of 91 are a class of linear functionals

positive values on the positive elements of % and they are of
importance for the construction of representations. We precede
the discussion of these states by giving the precise definition of a representation and by developing some general properties of representations.
First let us define a *-morphism between two *-algebras W and 0 as a
mapping 7r; A c- W F--+ 7r(A) c- 0, defined for all A c- % and such that
which take

fundamental

(1)
(2)
(3)

7r(oA + #B)
a7r(A)
7r(AB) =7r(A)7r(B),
7r(A *) =7r(A)*
=

for all A, B

+

#7r(B),

a c- C. The name morphism is usually reserved for
only have properties (1) and (2). As all morphisms we
consider are *-morphisms we occasionally drop the * symbol.
Now each *-morphism 7r between C*-algebras % and 0 is positive because
if A > 0 then A
B*B for some B c- W by Theorem 2.2.12 and hence

mappings

c-

% and

which

=

n(A)
It is less evident that

7r

is

=

n(B*B)

=

automatically

7r(B)*n(B)

>

0.

continuous.

Proposition 2.3.1. Let % be a Banach *-algebra with identity, 0
algebra, and 7r a *-morphism of % into 0. Then n is continuous and
117T(A)II

for

all A

A

%

c-

c-

of

<

is

a

C*-

11AII

%. Moreover,

7z

a

if % is a C*-algebra
C*-subalgebra of 0.

42

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then the range

0,,

=

17T(A);


Representations
PROOF.

First

assume

A

=

A*. Then since 93 is

C*-algebra

a

and States

43

0,

one

and

7r(A)

c-

has

11 7r(A) 11

by

Theorem

2.2.5(a).

=

sup{ I A 1;

Next define P

A

c-

u(7r(A))l

n(l,) where 1, denotes the identity of W. It
a projection in 0. Hence replacing 0 by
the C*-algebra POP the projection P becomes the identity I of the new
algebra 0.
,
Moreover, n(91) s:- F8. Now it follows from the definitions of a morphism and of the
spectrum that cz(7r(A)) = uw(A). Therefore
follows from the definition of

sup I A 1;

11 n(A) 11

by Proposition

2.2.2.

Finally,

Thus

11 7r(A) 11

11 A 11 for

<

2

all A

A

c-

u,,(A)l

A 11

if A is not

with the C*-norm property and the

11 7r(A) 11

=

that P is

7r

selfadjoint one can
product inequality to

=117r(A*A)II
e

W and

7r

<

11A*A11

combine this

inequality

deduce that

11A 112.

:!!

is continuous.

The range

0,., is a *-subalgebra of 0 by definition and to deduce that it is a
C*-subalgebra we must prove that it is closed, under the assumption that W is a C*algebra.
Now introduce the kernel ker 7E of 7r by
ker
then ker

7r

7r(AB)

7r(A)7r(B)

is

7r

=

JA c- W; 7r(A)

=

01

closed two-sided *-ideal. For

example if 4 c- W and B c- ker 7r then
0. The closed0, and 7r(B*)
7r(B)7r(A)
7r(B)
ness follows from the estimate lln(A)II :!! JJAJJ. Thus we can form the
quotient
algebra W,,
W/ker 71 and %,, is a C*-algebra by Proposition 2.2.19. The elements
of 91,, are the classes 4
{A + I; I c ker 7rJ and the morphism 7r induces a
morphism ft from %,, onto !5,, by the definition A(,4)
7r(A). The kernel of A is zero
by construction and hence ir' is an isomorphism between %,, and F8, Thus we can
define a morphism A` from the *-algebra 0,, onto the C*-algebra
' t7, by

A '(A(A))
A and then applying the first statement of the proposition to i
and
A successively one obtains
=

a

=

0, 7r(BA)

=

=

=

=

=

=

=

-

=

11,411
Thus
an

11A 11

Next

A

Ili-ImAvi

<

11A(A)11

:!

11,411.

11 7r(A) 11. Consequently, if 7r(A,,) converges uniformly in 0 to
converges in 91,, to an element A and A,,
A(A) 7r(A)
is any element of the equivalence class A. Thus A,, c 07, and F8.,, is closed.
=

element

where A

11 A(A) 11
A,, then

=

we

=

A.

=

define the concept of
n of % to 0 is a

*-morphism

i.e., if the

*-isomorphism
*-isomorphism

between

=

C*-algebras.

if it is one-to-one and onto,

to 0 and each element of 93 is the image of a

range of 7r is equal
unique element of W. Thus a *-morphism 7r of the C*-algebra
algebra 0 is a *-isomorphism if, and only if, ker -g
0.

% onto

a

C*-

=

Now

we can

introduce the basic definition of representation

theory.

Definition 2.3.2.
A representation of a C*-algebra 91 is defined to be a pair
(.5, 7r), where Sn is a complex Hilbert space and 7r is a *-morphism of % into
Y(.5). The representation (.5, 7r) is said to be faithful if, and only if, 7E is a
*-isomorphism between % and n(W), i.e., if, and only if, ker 7r
{01.
=

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44

C*-Algebras
There is

and

variety

a

Neumann

von

Algebras

of rather obvious

terminology

associated with this

definition. The space .5 is called the representation space, the operators 7T(A)
are called the representatives of W and, by implicit identification of 7r and the
set of

also says that 7r is a representation of % on Sv .
preceding Definition 2.3.2 established that each representa-

representatives,

The discussion

one

(.5, 7r) of a C*-algebra % defines a faithful representation of the quotient
algebra %,,
%/ker 7r. In particular, every representation of a simple
is
faithful.
C*-algebra
Naturally, the most important representations are the
tion

=

faithful

ones

and it is useful to have criteria for faithfulness.

Proposition 2.3.3.

(.5, 7r) be a representation of the C*-algebra %.

if, and onlY if, it satisfies each of thefollowing

Let

The representation isfaithful
equivalent conditions:
ker

(1)
(2)
(3)

7r
{01 ;
117r(A) 11
11 A 11 for all A
7r(A) > Ofor all A > 0.
=

=

c-

W;

PROOF.

The equivalence of condition (1) and faithfulness is by definition. We now
(1) => (2) => (3) => (1).
(1) => (2) As ker 7r

01 we can define a morphism 7r-' from the range of 7r
into % by 7r-'(n(A))
A and then applying Proposition 2.3.1 to 9' and 7r
successively one has
prove

=

=

JJAJJ

=

JJ7r-1(7r(A))JJ

:!!

117r(A)II

:!

JJAJJ.

(2) =:> (3) If A > 0 then 11 A 11 > 0 and hence 11 7r(A) 11 > 0, or 7r(A) :A 0. But
z(A) > 0 by Proposition 2.3. t and therefore ir(A) > 0.
(3) => (1) If condition (1) is false then there is a B c- ker 7r with B :A 0 and 7U(B*B)
0. But JJB*BJJ ! 0 and as JJB*BJJ
JJBJJ' one has B*B > 0. Thus condition (3) is
=

false.

A

*-automorphiSM

into

equal
The

itself, i.e.,

-c

is

T

a

of a

C*-algebra 91 is defined to be a *-isomorphism of
*-morphism of % with range equal to % and kernel

to zero

:

foregoing argument utilizing the invertibility Of T implies the following:

Corollary 2.3.4. Each *-automorphism -r of
11 All for all A c- W.
preserving, i.e., 11 -r(A) 11

a

C*-algebra

%

is

norm

=

Now

attention to various kinds of representation and methods
decomposing representations.
First we introduce the notion of a subrepresentation. If (.5, 7r) is a representation of the C*-algebra 91 and .51 is a subspace of 5 then .51 is said to be
invariant, or stable, under 7r if 7r(A)- ',
S31 for all A c U. If 51 is a closed
subspace of .5 and P_5, the orthogonal projector with range .51 then the
invariance of .51 under 7r implies that

of

we

turn

composing

our

or

Py,1 7r(A)Py,,

=

7r(A)Pb

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Representations
for all A

Ei

and States

45

91. Hence

7r(A)Pf,,

(Pf,,7r(A*)Pf,):
P.517E(A)

for all A

e W, i.e., the projector P.5, commutes with each of the representatives
7r(A). Conversely, this commutation property implies that .51 is invariant
under 7r. Hence one deduces that .5, is invariant under 7r if, and only if,

7r(A)P.61
for all A

c-

91. Furthermore,

and if 7r, is defined

(151, 7r,)

is

a

P.517E(A)

may conclude that if

we

.5,

is invariant under

7r

by
7r,(A)

then

=

representation

7r,(A)7r,(B)

=

=

=

P,617r(A)P.51

of W, e.g.,

(Pf,17r(A))(7r(B)Pf,,)
7r,(AB).
P-5,7r(AB)Pf,
=

A

representation constructed in this manner is called a subrepresentation of
(.5, 7r).
Note that the foregoing method of passing to a subrepresentation gives a
decomposition of 7r in the following sense. If .5, is invariant under 7r then its
orthogonal complement 51' is also invariant. Setting -52
-51j- one can
define a second subrepresentation (55, 2, 7r2) by 7r2(A)
P b 27r(A)P-,2. But
.5 has a direct sum decomposition, S;,)
-51 (D 52, and each operator
7r(A) then decomposes as a direct sum n(A)
7r,(A) ED n2(A). Thus we
write 7r
051, 7rO ED 0529 n2)7rl ED 7r2 and (15, 7r)
A particularly trivial type of representation of a C*-algebra is given by
0 for all A e W. A representation might be nontrivial but
n
0, i.e., n(A)
nevertheless have a trivial part. Thus if .50 is defined by
=

=

=

=

=

=

=

=

.50

=

f ; V/

c-

.5, 7r(A)

=

0 for all A

c-

WJ

then

fV, 0 is invariant under 7r and the corresponding subrepresentation
Pf, 7rPb. is zero. With this notation a representation (.5, 7t) is said to be
nondegenerate if .50
{01. Alternatively, one says that a set T1 of bounded
operators acts nondegenerately on .5 if
7ro

=

=

{C A

=

0 for all A

E

9W}

=

{01.

An important class of nondegenerate representations is the class of cyclic
representations. To introduce these representations we first define a vector 91
in a Hilbert space -5 to be cyclic for a set of bounded operators 9M if the set

lin. span {AK2; A c- 9JI) is dense in f_-). Then we have the following:
Definition 2.3.5.

A cyclic representation of a C*-algebra % is defined to be a
triple (.5, n, 92), where (15, n) is a representation of % and 0 is a vector in
.5 which is cyclic for n, in .5.

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46

C*-Algebras

and

von

Neumann

Algebras

In the sequel, if there is no possible ambiguity we will often abbreviate the
terminology and say that Q is a cyclic vector, or Q is cyclic for 7r. There is a
more general concept than a cyclic vector which is also often useful. If R is a
closed subspace of Sv then R is called a cyclic subspace for .5 whenever the set

Y 7r(Ai)oi;

Ai

c-

W, Oi

c

R

i

is dense in

.5. The orthogonal projector P,,,
cyclic projector.

whose range is

Sk, is also called

a

It is evident from these definitions that every cyclic representation is
there is a form of converse to this statement. To describe

nondegenerate but
this

need the

converse we

Let

(.5, 7r,,),,c,

the index set I

be

can

a

general notion of a direct sum of representations.
family of representations of the C*-algebra W where

be countable

or

noncountable. The direct

sum

5,11
a C-

of the

representation

defines the direct

sum

I

in the usual manner' and

spaces .5,, is defined
representatives
7E

=

(

one

7r,,,

a e

I

by setting 7r(A) equal to the operator 7r,,(A) on the component subspace
This definition yields bounded operators 7r(A) on .5 because 11 7r,,(A) 11 :!!
IJAII, for all occ-I, by Proposition 2.3.1. It is easily checked that (25, 7r)
is a representation and it is called the direct sum of the representations

(.5a, na),,, One has the following result.
Proposition 2.3-6. Let (.5, 7r) be a nondegenerate representation of the
C*-algebra 91. It follows that 7r is the direct sum of a family of cyclic subrepresentations.
PROOF.

Let

{Q,,I,,c-j

denote

maximal

a

of

family

(7r(A)Q,,, 7r(B)Qfl)
for all A, B

it,

whenever

fl.

=

vectors in

nonzero

.5 such that

0

The existence of such

family can be deduced with
subspace formed by closing
the linear subspace J7r(A)K2_ A c- 911. This is an invariant subspace so we can introduce 7r,, by 7r,,(A)
is a cyclic
Pb,.7r(A)P5. and ilt follows that each
7r_,, Q
c-

a

:A

the aid of Zorn's lemma. Next define

!5,,

as

a

the Hilbert

=

The finite subsets F of the index set I form
consists of those families

0

(p

IiM

The scalar

product

on

.5

=

J(p,,J

+

F

-F

is

then defined

((P, M

directed set when ordered

a

of vectors such that 9_

F

=

I

Y 119"112

I-M

+

-F

by

((P., 00"',

=

I ((P-., 0.)b"

lim
F

-F

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-

,

C-

by inclusion
.5,, and

and