Advanced Theory

680 Pages·1983·11.06 MB·ii-xi, 399-1074\680

by  Richard V. Kadison and John R. Ringrose (Eds.)

This is a volume in
PURE AND APPLIED MATHEMATICS
A Series of Monographs and Textbooks
Editors: SAMUEELIL ENBERAGND HYMANB ASS
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FUNDAMENTALS OF THE THEORY
OF OPERATOR ALGEBRAS
VOLUME I1
Advanced Theory
Richard V. Kadison John R. Ringrose
Department of Mathematics School of Mathematics
University of Pennsylvania University of Newcastle
Philadelphia, Pennsylvania Newcastle upon Tyne, England
1986
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Library of Congress Cataloging in Publication Data
(Revised for vol. 2)
Kadison, Richard V., Date
Fundamentals of the theory of operator algebras.
(Pure and applied mathematics ;1 00)
Includes bibliographies and indexes.
Contents: v. 1. Elementary theory - v. 2.
Advanced theory.
1. Operator algebras. I. Ringrose, John R.
11. Title. 111. Series: Pure and applied mathematics
(Academic Press) ; 100.
QA3.PX VO~.1 00 510 s (512'.55] 82-1376
[ QA3 26 1
ISBN 0-12-393302-1 (v. 2 : alk. paper)
~6 x7 XK xy VK7h54?!1
PREFACE
Most of the comments in the preface appearing at the beginning of
Volume I are fully applicable to this second volume. This is particularly so for
the statement of our primary goal: to teach the subject rather than be
encyclopaedic. Some of those comments refer to possible styles of reading and
using Volume I. The reader who has studied the first volume following the
plan that avoids all the material on unbounded operators can continue in this
volume, deferring Lemma 6.1.10, Theorem 6.1.1 1, and Theorem 7.2.1’ with its
associated discussion to a later reading. This program will take the reader to
Section 9.2, where Tomita’s modular theory is developed. At that point, an
important individual decision should be made: Is it time to retrieve the
unbounded operator theory or shall the first reading proceed without it? The
reader can continue without that material through all sections of Chapters 9
(other than Section 9.2), 10, 11, and 12 (ignoring Subsection 11.2, Tensor
products of unbounded operators, which provides an alternative approach to
the commutant formula for tensor products of von Neumann algebras).
However, avoiding Section 9.2 makes a large segment of the post-1970
literature of von Neumann algebras unavailable. Depending on the purposes
of the study of these volumes, that might not be a workable restriction. Very
little of Chapter 13 is accessible without the results of Section 9.2, but Chapter
14 can be read completely.
Another shortened path through this volume can be arranged by omitting
some of the alternative approaches to results obtained in one way. For
example, the first subsection of Section 9.2 may be read and the last two
omitted on the first reading. The last subsection of Section 11.2 may also be
omitted. It is not recommended that Section 7.3 be omitted on the first
reading although it does deal primarily with an alternative approach to the
theory of normal states. Too many of the results and techniques appearing in
that section reappear in the later chapters. Of course, all omissions affect the
exercises and groups of exercises that can be undertaken.
As noted in the preface appearing in Volume I, certain exercises (and
groups of exercises) “constitute small (guided) research projects.” Samples of
this are: the Banach-Orliz theorem developed in Exercises 1.9.26 and 1.9.34;
the theory of compact operators developed in Exercises 2.8.20-2.8.29, 3.5.17,
ix
X PREFACE
and 3.5.18; the theory of b(N) developed in Exercises 3.5.5, 3.5.6, and
5.7.14-5.7.21. There are many other such instances. To a much greater extent,
this process was used in the design of exercises for the present volume; results
on diagonalizing abelian, self-adjoint families of matrices over a von Neu-
mann algebra are developed in Exercises 6.9.14-6.9.35; the algebra of
unbounded operators affiliated with a finite von Neumann algebra is
constructed in Exercises 6.9.53-6.9.55,8.7.32-8.7.35, and 8.7.60. The represen-
tation-independent characterizations of von Neumann algebras appear in
Exercises 7.6.35-7.6.45 and 10.5.85-10.5.87. The Friedrichs extension of a
positive symmetric operator affiliated with a von Neumann algebra is
described in Exercises 7.6.52-7.6.55, and this topic is needed in the develop-
ment of the theory of the positive dual and self-dual cones associated with
von Neumann algebras that appears in Exercises 9.5.5 1-9.6.65. A detailed
analysis of the intersection with the center of various closures of the convex
hull of the unitary conjugates of an operator in a von Neumann algebra is
found in Exercises 8.7.4-8.7.22, and the relation of these results to the theory
of conditional expectations in von Neumann algebras is the substance of the
next seven exercises; this analysis is also applied to the development of the
theory of (bounded) derivations of von Neumann algebras occurring in
Exercises 8.7.51-8.7.55 and 10.5.76-10.5.79. Portions of the theory of repre-
sentations of the canonical anticommutation relations appear in Exercises
10.5.88-10.5.90, 12.5.39, and 12.5.40. This list could continue much further;
there are more than 1100 exercise tasks apportioned among 450 exercises in
this volume. The index provides a usable map of the topical relation of
exercises through key-word references.
Each exercise has been designed, by arrangement in parts and with
suitable hints, to be realistically capable of solution by the techniques and
skills that will have been acquired in a careful study of the chapters preceding
the exercise. However, full solutions to all the exercises in a topic grouping
may require serious devotion and time. Such groupings provide material for
special seminars, either in association with a standard course or by them-
selves. Seminars of that type are an invaluable “hands-on” experience for
active students of the subject.
Aside from the potential for working seminars that the exercises supply, a
fast-paced, one-semester course could cover Chapters 6-9. The second
semester might cover the remaining chapters of this volume. A more leisurely
pace might spread Chapters 6-10 over a one-year course, with an expansive
treatment of modular theory (Section 9.2) and a careful review (study) of the
unbounded operator theory developed in Sections 2.7 and 5.6 of Volume I.
Chapters 11-14 could be dealt with in seminars or in an additional semester
course. In addition to these course possibilities, both volumes have been
written with the possibility of self-study very much in mind.
PREFACE xi
The list of references and the index in this volume contain those of
Volume I. Again, the reference list is relatively short, for the reasons
mentioned in the preface in Volume I. A special comment must be made
about the lack of references in the exercise sections. Many of the exercises
(especially the topic groupings) are drawn from the literature of the subject.
In designing the exercises (parts, hints, and formulation), complete, model
solutions have been constructed. These solutions streamline, simplify, and
unify the literature on the topic in almost all cases; on occasion, new results
are included. References to the literature in the exercise sets could misdirect
more than inform the reader. It seems expedient to defer references for the
exercises to volumes containing the exercises and model solutions; a signifi-
cant number of references pertain directly to the solutions. We hope that the
benefits from the more sensible references in later volumes will outweigh the
present lack; our own publications have been one source of topic groupings
subject to this policy.
Again, individual purposes should play a dominant role in the proportion
of effort the reader places on the text proper and on the exercises. In any case,
a good working procedure might be to include a careful scanning of the
exercise sets with a reading of the text even if the decision has been made not
to devote significant time to solving exercises.
CHAPTER 6
COMPARISON THEORY OF PROJECTIONS
We take up the detailed study of von Neumann algebras in this chapter.
The principal tool for this study is the technique of “comparison” of the
projections in a von Neumann algebra W relative to 9.B y these means we
develop a notion of “equivalence” of projections in 9 (meaning, loosely, “of
the same size relative to 9’’A)ss.oc iated with this equivalence, we have a
partial ordering of (the equivalence classes of) projections in 9-with
corresponding notions of “finite” and “infinite” projections relative to 9%’.
In these terms, we can separate von Neumann algebras into broad types
(algebraically non-isomorphic) and show that each such algebra is a direct
sum of algebras of the various types (the so-called “type decomposition” of
von Neumann algebras). The simplest of the types (“Type I von Neumann
algebras”) is analyzed and examples of some of the other types are studied.
6.1. Polar decomposition and equivalence
In the discussion following Lemma 2.4.8, we observed that each bounded
+
operator Ton a Hilbert space .# can be expressed as H iK,w ith H and K
self-adjoint operators. We referred to H and K as the “real” and “imaginary”
parts of T-noting the analogy between this representation of T and the
corresponding representation of a complex number in terms of its real and
imaginary parts.
If we pursue the analogy between representations (decompositions) of
complex numbers and those of linear operators, we are led to consider the
possibility of a “polar decomposition of operators analogous to the de-
”
composition of a complex number as the product of a positive number (its
modulus) and a number of modulus 1.
With the function calculus for self-adjoint operators at our disposal, there
is no problem in producing a “polar decomposition” for an invertible operator
T. As modulus, both (T*T)’12 and (TT*)’” suggest themselves. At first
guess, we might expect the number of modulus 1 in the polar decomposition
of a complex number to correspond to a unitary operator in the case of an
operator. The non-commutativity of the operator situation introduces a
399
400 6. COMPARISON THEORY OF PROJECTIONS
complicating factor. Shall we multiply the modulus of Ton the left or right by
the unitary operator (if it is, indeed, to be a unitary operator); and which of
(T*T)’”, (TT*)’12s hall we use as modulus? A small amount ofexperimenta-
tion shows that writing T = U(T*T)”’ (somewhat hopefully), and, then,
“solving” for U as T(T*T)-’’’ produces a unitary operator U (while
”’
T(TT*)- will not, in general, be unitary-nor would (T*T)-”’T). The
computation involved in this is
(T(T*T)-” ’x, T(T * T)-‘ “x) = ((T*T)-‘ 12T*T(T*T)-‘ ”x, X)
= (x, x).
If WH is another “polar decomposition” of T (with W unitary and H
positive), then H = W*T so that H2 = H*H = T*WW*T = T*T. As
H 2 0, and the positive square root of a positive operator is unique (see
Theorem 4.2.6), H = (T*T)’/’ and W = T(T*T)-’” = U. Of course,
T* = (T*T)”’U*, while T* has its own polar decomposition, T* =
V*(T**T*)”2= V*(TT*)1’2T.h us T = (TT*)’l2V;a nd this last equality
provides a “polar decomposition” for T with the positive operator factor
appearing on the left. This, incidentally, redresses the balance between the
two candidates for “modulus” of T. Combining T = U(T*T)’” and T* =
(T*T)’/’U*, we have TT* = U(T*T)U* (so that TT* and T*T are
unitarily equivalent, when T is invertible). Since U(T*T)’12U*i s a positive
square root of U(T*T)U*, (TT*)’12 = U(T*T)’/’U*.B ut V*(TT*)’/’ =
T* = (T*T)’/’U*, so that UV*(TT*)’l2= U(T*T)1’2U*= (TT*)”2;a nd
V = U. Thus the same unitary operator appears in the “right” and “left”
polar decomposition of T.
For the polar decomposition of the general bounded operator, we must
replace the unitary operators of the preceding discussion by operators that
map one (closed) subspace of a Hilbert space isometrically onto another and
annihilate the orthogonal complement of the first subspace. Such operators
are called partial isometries. The first subspace is called the initial space of the
partial isometry, and the second subspace (its range) is called its jinal space.
The projections with these subspaces as ranges are called the initial and Jinal
projecfions, respectively, of the partial isometry.
6.1.1. PROPOSITIONTh. e operator V acting on the Hilbert space X is a
partial isometry ifand only if V*V is a projection E. In this case, E is the initial
projection of V, VV* is thejnal projection F of V,a nd V* is a partial isometry
with initial projection F andfinal projection E.
Proof. Suppose, first, that V is a partial isometry with initial projection
+
E. Then IIVxll = IIVEx V(1 - E)xII = I(VExI1 = llExll 5 llxll; so that
II V (1 I 1. Ifx is a unit vector in the range of E,t hen 1 = (x, x) = (Vx, Vx) =
6. I. POLAR DECOMPOSITION AND EQUIVALENCE 40 I
(V* Vx, x). From Proposition 2.1.3 (the “Cauchy-Schwarz equality”),
V*Vx = x. Ify is in the range of I - E, V*Vy = V*(O) = 0. Thus V*V = E.
Suppose, now, that V* V is a projection E. Then for each x in the range of
E, (x, x) = (V* Vx, x) = (Vx, Vx);w hile, for y orthogonal to the range of
E, 0 = (V*Vy, y) = (Vy, Vy). Thus V is isometric on E(R) and 0 on
(I - E)(R). It follows that V is a partial isometry with initial projection E.
In addition, V = VE = VV*V, and VV*VV* = VEV* = VV*. Thus VV*
is a projection F and FV = V. Consequently F(X)c ontains V(X). But
F(R) = VV*(X)E V(X).H ence F is the final projection of V. As VV* =
(I/*)* V* = F, we conclude, from the foregoing, that V* is a partial isometry
with initial projection F and final projection E.
6.1.2. THEOREM(P olar decomposition). If T is a bounded operator on
the Hilbert space #, there is a partial isometry V with initial space the closure
r(T*) of the range o f T* and final space r(T) such that T = V(T*T)’” =
(TT*)’/’V. If T = WH with H positive and W a partial isometry whose initial
space is r(H),t hen H = (T*T)l12a nd W = V.I f neither T nor T* annihilates
a non-zero vector, then V is a unitary operator.
Prooj. Recall from Proposition 2.5.13 that r(T*) = r(T*T) so that
r(T*) = r((T*T)l/’). Since
((T*T)’”x, (T*T)’”x) = (T*Tx, X) = (Tx, Tx),
there is a partial isometry V with initial space r(T*) and final space r(T)s uch
that T = V(T*T)”’.T hus T* = (T*T)”’V* and TT* = VT*TV*.N ow
[V(T*T)”2V*]2= VT*TV* = TT*,
so that V(T*T)”’V* = (TT*)’I’. Hence
T = V(T*T)”’ = V(T*T)’12V*V= (TT*)’/’V,
(Note, for this, that V*V = R((T*T)”’), from Proposition 6.1.1, so that
(T*T)’/’ = V*V(T*T)’I2= (T*T)”’V*V.)
With W and H as described, W*W H = H, so that T*T = HW*WH =
HZ.H ence H = (T*T)”’ and W = V.
If T and T* have (0) as null space, their ranges are dense in X.H ence V is
a unitary operator, in this case.
Note that (T*T)’l2 and (TT*)l/’ are contained in each C*-algebra
containing T. However, V may not lie in such an algebra. If T is a positive
operator, V is R(T). With 2I the algebra of multiplications by continuous
functions on L,([O, 1)) (relative to Lebesgue measure) and H multiplication
by a positive function that vanishes on [0, $1, R(H) is a projection different
from 0 and I. Since ‘?coIn tains no projections other than 0 and I, the polar
402 6. COMPARISON THEORY OF PROJECTIONS
decomposition of H cannot be effected in ‘ill.If T is invertible, (T*T)’12a nd
U (= T(T*T)-”’)l ie in each C*-algebra containing T. The critical informa-
tion concerning the possibility of polar decomposition within a C*-algebra is
found in the proposition that follows.
6.1.3. PROPOSITION. If T lies in a von Neumann algebra 9 and UH is the
polar decomposition of T, then U and H are in 9.
Proof. As noted, H = (T*T)’I2€ 9,sin ce 9 is, in particular, a C*-
algebra containing T. If T’EW‘T, ’UHx = T’Tx = TT‘x;w hile UT’Hx =
UHT’x = TT‘x. Thus UT‘ and T’U agree on the range of H. Since T‘
commutes with H, both the range of H and its orthogonal complement are
stable under T’.A s U is 0 on this complement, both UT’a nd T’U are 0 there.
Thus UT’ = T’U and U E 9”= 9.
If T is normal, (T*T)’” = (TT*)”’(=H). Thus UH = T = HU (from
Theorem 6.1.2). Conversely, from uniqueness of the polar decomposition
(‘‘left’’ and “right”), if UH = HU, (T*T)”’ = (TT”)”’ and T*T = TT*.
To compare the dimensions of the ranges of two projections E and F
acting on a Hilbert space, we compare the cardinality of orthonormal bases
for each of these subspaces. Another (equivalent) technique for comparing the
dimensions of the ranges of E and F to see if they are the same would be to
seek a partial isometry with one as initial projection and the other as final
projection. If E and F lie in a von Neumann algebra W and we insist that
our partial isometry lie in 9,w e are demanding a stricter comparison of E and
F-a comparison relative to 9.T he structure of 9 would seem to exert an
important influence on the possibility of comparison; and, consequently, the
structure this comparison process imposes on the projections of W will reflect
the structure of 9.
Elaborating this idea leads to the Murray-von Neumann comparison
theory of projections in a factor and its extension to a comparison theory of
projections in a von Neumann algebra.
6.1.4. DEFINITIOTNw. o projections E and F a-re said to be equivalent
relative to a von Neumann algebra W (written, E F(9)) when V*V = E
and VV* = F for some V in 9. H
In view of Proposition 6.1.1, the operator V in 9i s a partial isometry with
initial projection E and final projection F. Since E = V*V and F = VV*,
both E and F are in W.M ost often, the von Neumann algebra W relative to
-
which the equivalence of E and F is being asserted will be clearly indicated by
the context. In this case we say that E is equivalent to F and write E F.