Finally,. I am grateful to the Carnegie Trust for the. Universities of Scotland for its kind financial support while this research was being conducted. J. Moffat. ýI v ...
(i)
GROUPSOF AUTOMORPHISMS OF OPERATOR ALGEBRAS
By
J.
MOFFAT
.4
Thesis the University
submitted
for
the degree of Doctor
of Newcastle
upon Tyne.
1974
of Philosophy
in
BEST COPY AVAILABLE
VARIABLE PRINT QUALITY
PREFACE
is
This thesis
submitted
or the degree of Doctor upon Tyne.
Newcastle by the to
author
be original
being
Chapter
for
of Philosophy
No part
a university
except i which
in accordance with
where is
of
the regulations
in the University
has been previously
it
degree. otherwise
The contents indicated,
the
of submitted
are believed main
exception
introductory.
ý3ou, A4-J 4
Newcastle
upon Tyne
September,
1974
N
0
\\
\N.
\\
ýI
(iii) ACKNOWLEDGEMENTS
It
is
Professor three
a great
J. R. Ringrose,
thesis Finally,
Universities this
for
to
thank
all
my supervisor,
his
help
during
the
past
years.
I would also this
pleasure
research
like
to thank Mrs. E. Fitz-Patrick
for
typing
so well. I am grateful of
Scotland
was being
to the for
its
Carnegie
kind
Trust
financial
for support
the while
conducted.
ýI
J. Moffat
v
(iv) ABSTRACT
An important
is the study of their the
we study
topological
of the theory
part
unitary
representation
groups,
of locally
representations.
In this
such groups,
and more general
of
types of continuity
tations
(which we call
and measurability
automorphic
in certain
continuous
a, of an abelian
representation,
group G as a group of automorphisms space U.
The topology
of such represen-
We consider
cases.
is
of a C*-algebra,
on a(G) is that
unitary
shown to be equivalent
operator
In the case of a locally \
cient
that
condition
representation a(9)(A) If
there
exist
with
(AE2I,
projections of projections. of the tensor more general
relation
unit)
can be defined,
We show that
situation.
continuous
and suffiunitary
Ua E 21 such that
of a von Neumann algebra in
in R. which extends the usual
product
a necessary
9E G).
G is a group of automorphisms
an equivalence
Ug in the weak
(when Z is a factor
a strongly
g-+ U9 of G by unitaries
= UgA U*
Such a
to a norm continuous
a, we obtain
simple C*-algebra
or a separable
as a
compact group G and a weaker continuity-
on the representation
condition
derived
on
(A) (g A Ug U9 E G, AE 2I). = a(g)
of 21, such that
closure
acting
on'21.
g-4 Ug of G by unitaries
representation
a
connected topological
subset of the Banach space of bounded operators representation
are defined
representations)
and shown to be equivalent
thesis,
of C*-algebras.
as groups of automorphisms
Certain
a Hilbert
compact groups
certain
terms
of G, on the
definition
results
of von Neumann algebras
of equivalence
concerning carry
the type
over to this
"
(v)
Ergodic formations space, certain
of
L"(X,
µ)
is
von Neufnann algebra condition
for
invariant
under
gebras
the
the G.
G-invariant (G R acts of
the
existence
transformation
group R.
(X, µ).
G of
We ar'so states ergodically
of groups
If
is
X
of
of
group
a locally
show that and the if
of
ergodic
that
measure
over
to
the
a general and sufficient
normal a link
compact
We prove
carry
a necessary
a faithful
trans-
of
an equivalent
automorphisms
This, gives
existence
study
von Neumann algebra.
an abelian
under
an amenable
extremal
space
concerning
results
the
essentially
a measure
on X invariant case of
is
theory
state
exists action
0 and I are
the
on R-
between
normal
of G on subalonly
invariant
projections).
N
S
N
(vi)
CONTENTS "
PAGE
CHAPTERI CHAPTERII
Introduction of Automorphic
Continuity
Representations
Covariant
CHAPTER IV
On a G-twisted Equivalence Relation von Neumann Algebras
CHAPTER V
Connected
CHAPTER VI
Implementing
Groups Topological von Neumann Algebras a Group
a Unitary CHAPTERVII
Ergodic
3G
Representations, Permanently Automorphisms Spatial
CHAPTER III
Acting
Z01,
for
on
of Automorphisms
by
Representation
Theory and von Neumann Algebras
20
APPENDIX A
1.4 8
APPENDIX B
158.
APPENDIX C APPENDIX D
0
1ýý0
,
-1CHAPTER I
INTRODUCTION Chapter,
In this for
required
the the
assume that analysis, texts
for
complex
IzI
= 1)
to
'
all
and
Z
the
C*-algebras
Banach algebra
IP\
are
functional
of
Three
theory.
basic
algebras
are the
book
([44]).
Throughout,
defined
over
denote
will
We
the
elements
the
set
of
field real
z E(C such that
integers.
of
A linear
normed linear
spaces
follows.
concepts
operator
(i. e. those
circle set
of
and Sakai's
C. denoted
unit
the
and measure
[8])
and results
work which
with
subject
linear
our
numbers, the
the
Q7],
of Dixmier
we assume that
numbers,
topology,
reference
the
familiar
is
reader
definitions
those
of
understanding
elementary
two books
of
we introduce
associative
algebra
which is also a
to the norm 1I.1[ is
space relative
said to be a
if
(i) IIABII s IIAIIIIBI) (A,BE21) (ii)
21 is complete relative If
21 has a unit,
to the norm topology.
we denote it
by %,
or simply
I if
no
11IýI We = 1. This assumption confusion arises:,,, assume always that \ B(SI) will denote the centre involves no essential restriction, , ofW. Throughout will
this
denote the unit
thesis, ball
if
b is a normed linear
space, ýA1
of jD i. e.
L, =,{x E£, lIXI!s1} Let * be a map from 2i to 21. * is'called N
N
an involution
if
the
-2following
(i) (il).
(iii) If
are
conditions
91 is
(XA+µB)*=RA*+µB*
(X> µE
(AB)*
(A, BE
g#p*
=
A#*
(A
=A
an algebra
*,
involution
with
is
A Banach *-algebra
C*-algebra
satisfied
a Banach
is a Banach *-algebra
A) BE2)
,
21)
E 21)
.
13 s called algebra
a *-algebra. involution.
with
A
such that
IIA*All= JIA112(AE21) ý
If 21 is a C*-algebra,
and
is a closed
is also a C*-algebra
then
with
of !U,
ideal
the involution
A* +3.
A+ý ([B],
two-sided
ß. s. 2). and ßc
Suppose 2I is a C*-algebra,
21.8
is
said to be
if
selfadjoint
II {A*;
8=
AE
BI
=
3 is
If
S is also a subalgebra
of I,
If
8 is closed,
a C*-algebra,
8 is also
B#
called
.
a *-subalaebra
called
of 21.
a C*-subalgebra
oft. Let denoted
a(A)
91 have unit a(A), 11
be the
to
is a non-void
oýfi .I containing
For
I.
I.
set
without
fit, we define
of XEC
such that
compact subset of C, and if " and A. then. (A) a
by ([B],
AE
=
the A-
spectrum AI
is
of
A,
singular.
03is a C*-Subalgebra
QU(A)
1.3.10).
Thus we may write
ambiguity
as to the containing
the spectrum of A as a(A) C*-algebra.
An element A
-3of 21 is said to be (a)
normal if
(b)
unitary
(c)
self-adioint
(d)
positive
real
numbers)
AA* = A*A A#Aý AA*
if
A ='A#
if
Let 2S+denote the set of positive set of self-adjoint
[8],
1.4.2)
1.3.9).
thus we can define
if
2l# be the
Banach
linear
functionals
continuous
be positive,
of 21 if
a state
(! I*)1
ahd 1.1.10). topology
and the
convex
subset
By the
Krein-Mil'man
hull
of
its
of
set
([]
a, by saying on 2Is,
A-BE91
dual
of 2I i. e. Z# is on U.
fE
If
\. the
. I#,
set
f is
of
all
said
to
(A E 21+) f(I)
also
= 1.
is
in
compact
E(2º) of
all
so E(2i)
weak*
is
weak* the
Such extreme
2.6.4
(i. e. a(. 1*, ý1))
of 21 is
states
of WI' can be
of U ([$],
states
the
E(21) is
theorem, points.
Each element
of
combination
(21*)1,
extreme
ordering
Z0
as a linear
expressed
a. convex cone in Zs.
f 2: 0, if
written f(A)
f is
Ws. a. AE then
If
a partial
AiB Let
elements
a. s. the of 91 and 91
R.
c
2i+ is a closed
\\\
set of non-negative
I.
elements of a(A)
(by
R+ (the
A= A* and v(A)
if
a weak* ([8],
compact.
weak*-closed points
are
closed 2.5.5). convex
called
pure. states.
If
!K and ß. are C*-algebras,
is a linear
a homomorphism it from 2I to 8",
map from 21 to 8 such that'
-4-
(1)
n(AB)
(ii)
n(A*)
n is an isomorphism and ß(4)
the
(A, B E 21)
=
(A
n(A)* {0}.
Ker n=
if
C*-algebra
determined
tion
(B) =* t(A) n
of
all
y)
=
E 21)
.
Let 9 be a Hilbert
bounded
operators,
with
space the
involu-
by (TX,
(x,
(TEE(N),
T#Y)
x, yE9)
A homomorphism n: 91-" ß(ßt) is called If
n is an isomorphism
If
LcßO),
. of 9Son X.
a representation
we say n is a faithful
representation.
let
ýc9,
[ev;] denote
the
closed
linear {Ex;
Let it there
of ä# generated
subspace EE&,
xEYý
also denote the projection is an xE$
by
.
onto this
subspace.
Suppose
"
such that
U= [n(91)Xý then vector
it is
said
for
the
to
representation
03' = W* is
be a cyclic
7 is called
a generating
x is called
a cyclic is
8 is`a
If
{A E 8(11); AB = BA
[BY]
7c4
it.
the commutant of S.
called
and x is
representation
If
subset
a cyclic
of 3(),
let
(B E 6) } 9cU
is
such that
=u
set for
or generating
B.
If
{x}
is generating
for
8,
vector.
said to be a separating
set for 63 if
Bx =0
(x E F)
-5-
imply
and BES
B=0.
Let 8 be a *-subalgebra 8 if
and only {x}
If
Y is
is
and if
vector,
x is
the
at
same time
is
For x. ,yEN,
let
with
U, with
place
and fE
E(21),
there
Conversely,
vector
manner.
and vector.
there
is
a countable
S.
functional
If 2I is a C*-algebra of wX x s is a cyclic representation of
on a Hilbert
f=
Wxfoof
space Uf,
Suppose {ndaE
in this is ä family
A' {pia}«E spaces A"
The
91 on E®9 is defined E® na of , a
sum representation,
=
of
such that
of 2I arises
representation
and 2.4.1).
ýE®xa)(A) ý
if
of !S on the Hilbert
of representations direct
for
xf,
every cyclic 2.4.4
a separating
(Tx, y)
wx in
cyclic
both
denote the linear
w
and write
unit,
said to be a separating
decomposable
Ton B(14),
Theorem 3, p. 27)
separating-generating
separating
for
is generating
([ft],
B'
8, x is
we say x is"a
of If which
subset
for
be countable
to
said
for
separating
separating
vector,
generating 8 is
if
of 03(U), then
as
(A E 2I)
E®xa(A)
vww 'h'uri, ýJQ.
Every-representation
of 21 is ([8],
representations \said to be irreducible
is
equivalent
this
sense
ä direct'sum
A representation
2.2.7).
of
cyclic
n of .I on N is
if
n(ii). This
in
=
CIS
to saying that
)\ there
are no closed
subspaces
-6-
F of 9 such that aF.
n(21)F
([B],
Let i1 and n2 be irreducible
2.3.1).
1 on Hilbert to n2 if
representations
spaces U1 and 92 respectively. is an isometric
there
linear'
of
n1 is equivalent
map U from U1 onto N2
such that 1= U, (A)U If
f is
f is
tion,
then
([$],
2.5.4).
n2(A)
of 91 and of
a state
a pure
(A E 21)
the if
state
Moreover,
corresponding
cyclic
if
and only
any irreducible
.
is
of
([$],
to one obtained
in this
The universal
representation,
n, of 91 is defined fE
The Gelfand-Naimark sentation
of 21 as a norm closed 3=
Q+Q,
1.16.6).
proper
closed
E®{klf;
The C*-algebra
2.4.6,2.5.4).
as
it is a faithful
that
fE
of
. repre-
of ß(U) where
C*-subalgebra
E(21)}
2 is
ideals.
two-sided
manner.
E(`I)}
theorem states
irreducible.
representation
is equivalent
E®{ltf;
representa-
simple
In
this
if
21 contains no ý knýia. wsw case allLrepresentations
of 21 are faithful. U with
For a C*-algebra of
unitary
morphism
elements from
of 21.
unit,
let
U(21) denote the group .
An automorphism,
y,
of 2U is
an iso-
to 21.
WARNING: . Note that (A#) Y
_
(A)# 'y
(A E 2I)
(t)
-7-
Such maps are usually
Let
However,
automoxphisms y not
(t).
satisfying
be the
aut(9J)
composition is
to consider
have no occasion
we shall
to as *-automorphisms.
referred
([ß],
of
Since
of maps.
isometric
group
1.3.7),
between
isomorphism
every
of 91 under
automorphisms
all
aut(91)
of
each element
C* -algebras
is' isometric,
thus
'Q (Z) (2I) C3 auf (B(i)
= set
aut(o)
all
of
has identity
bounded
Ls If
operators the
element
A-
of ß(U)
) U. ! space
automorphism
(A E 2f)
A
UEU (ö3(kt) ), where 9 is a Hilbert
C"-subalgebra
Banach
on the
.
space, and 1 is a
AE 21 implies
such that
UAU* EW then
the map % is
an automorphism
confusion
arises).
automorphism
If
A -º UAU*
y:
In
this
of. 21, and y is
U, VE U(3())
(or
simply
ad U if
case we say that
y is
a spatial
of 21, denoted
ad U1.
by the
implemented
unitary
U.
(9i) auf are such that
and ?E
l,, V ad
ad U' .= =
VAV*
(A E 2I)
=
A(V*U)
(A E I)
then
UAU*
so
(V*U)A
thus V*U E V.
Hence there U=
is 'a unitary
QE 2I" with
QV
N If
also U. VEý,
then QE U(((2I)).
,
If
UE U(2I), we say
no
-8-
ad U is an inner inner
automorphisms If
X is
autbmorphism of %.
valued
algebra
when the
defined
by
Hausdorff
the set of all
+ µ9) (z)
=
xf (x) + µ9(x)
fg(x)
=
f(x)
f*(x)
=
f(x)
is a weak*-closed [$],
1.4.1,
A(p)
We see from of pure
states
transform. phism,
[0],
Since
for
AEI.
the
and norm are
X}
C*-algebra
with
let
unit,
§ be 9.1
homomorphisms from 21 to C. hence weak*-compact.
By ..
to C(§21) via the map A -+ A
(A) = -p
Ch. V, §8, of U.
C*-
an abelian
(x)I ;xE
subset of(2I*)1,
!Y is isomorphic
where
all
C(X))
gE
non-zero
'N
of
g(x)
{f sup
2I is an abelian continuous
C(X)
set
involution
structure,
(f, if
on X is
algebraic
ýýfLI=
Conversely,
the
space,
functions
continuous
(xf
of all
of U is a subgroup of aut(o).
a compact
complex
The group inn(s)
(p Eý).
Lemma 6 that
The mapping Gelfand
A-A
is
just
is, called
transform
is
the the
an algebra
set
Gelfand isomor-
we have
6 (A)
=v
(A)
_
JA(p)9
pE
_
{P (A);
pEe}
N
;
ja}
N \\\
N
\
N
'ý
-9-
Suppose A is is
now 9 is
a normal
by %(A).
it
with
The C*-algebra
of 21.
element Denote
abelian.
C*-algebra
an arbitrary
identity,
and
by {I,
generated
A,, A*}
The map
A=P'"P(A)
is from 4ý onto ß(A) 21(A) Hausdorff If:
isomorphism
=
2I(A)
from the compact
so A is a homeomorphism.
space §O(A), - denotes
bijection
a continuous
of C*-algebras
C(A (A))
=
we have
C(a(A))
The isomorphism f-,, -+ is
(A)) C(o t
21(A)
-"
functional-calculus.
the
called
f (A)
Note
that
if
A is
normal
IIAII {IA(P)I IIAII = = sup ;PE §Z(A)} {IP (A) I; =a sup
pE
§21(A)' ý4
= N
N
Suppose 2 is projection onto
{1%1 sup ;XE and 8 is
a C*-algebra,
of
norm one from
91 onto
a C*-subalgebra ß is
a linear
of U.
'A
map a from 91
8 such that
(i)
n(B)
(BEý3)
=B
(ii) Iln(AýI IIAII Topologies
00; on
von Neumann Algebras
Let )I be a. Hilbert (a)
a(A)}
The strong
If
xEU,
operator
space toboloav
the equation
(? 5ý
(A E 2i)
ý\ý
- 10 IITxII Px(T) = defines \, {px
]#} is
sxE
For\f the
a seminorm
px on Q3(Ji).
The topology
the
operator
called
fixed S, the
strong
maps T -º ST,
T -» TS
?
topology,
by the
family
denoted
7'S
on B (11), and
are
continuous
the
map T -. T* is
map-,,
(S, T) is
defined
on B(N)
continuous
x 3(H).
ST -+ However
not
s-continuous.
S(If) zable if
is complete for 1 ä# is
Given Ilxill2 E
([7],
separable.
The ultrastrong
< co. the
operator
{x}
X=
Ch. 1, §3, p. 30).
of
elements
=
a seminorm on U3(ä#). The family
defines
the ultra strong
Qs
For x, yE91
define
defined
called
os
such seminorms or as.
is the same as for
Ch. 1, §3, p. 34).
(9) S by a seminorm on I (Tx
y>I ,
by {Pxly
is
of all
topology
pxIY(T) The topology
operations
([7], B(II) on 10
coincide
14
denoted ?
topology,
of algebraic
The weak operator
of ki such that
iE JITxi112)*
defines
Ts and ?
and metri-
equation
px(T)
Continuity
topology,
T
topology
a sequence
the strong
the weak operator
; x, Y E ü} topology.
denoted 7
w
?
s.
11
The map (S, T) -+ ST" is not 7w-continuous, fixed
S the maps T °'-+
are
however for
w
as is
continuous,
ST
the
-º
,T
ST
map
T _, T* 8(14)
is
?w compact
The ultraweak
If
X=
([-7],
{yi}
and Y=
are sequences in 4 with ý.
2) 11y1
