Regular abelian Banach algebras of linear maps of operator algebras

Hegula:-c abelian Ban.aeh algebras of linear maps of operator algebras

1. Introductiono

Erling Bt0rm.er

Department of Mathematics, University of Osloa

In recent years it has become apparent that spectral theory for linear maps of von Neumann algebras is intimately connected with J!.,ourier analysis., The present paper is an attempt at obtaining

a deeper Ullderstanding of this relationship. If B(H) denotes the von Neu,·nann algebra of all bounded linear operators on the Hilbert space H into itself, we shall study abelian Bc:JJ.ach subalgebras of B (B(H))- the Banac~1 algebra of bounded linear :naps of B(H) into

itself ^o Thus in tl.1e process we shall obtain some insight into the extremely complicated Banach algebra B(B(H)). The main difficulty encountered in this Banach algebra is the bad behaviour of its norm.

Recall that a theorem of Grothendieck [4] identifies B(B(H)) as a Banach space with (B(H) '~ ;) )

* ,

where g- is the trace class ope-

rators on E with the trace norm, and

0

is the projective tensor product of Banach spaces. We shall therefore try to avoid the norm as much as possible and shall restrict attention to maps which are ul trav:eoJ.dy contim1ous and which map the Hilbert-Schmidt operators d{ into themselves, and as operators on C:t{ are normal operators ^o Such maps will be called operator normala Furthermore we shall have to require thc:J.t our abelian Banach algebras will have a well behaved Gelfand theory. We have partly for this reason and partly because this case .contains most of the interesting examples,

- 2 -

restricted attention to regular abelian Banach algebras of operator normal maps o The:r1 the restriction to

If.

is a concrete represent a- tion of the Gelfand tr3llsformo Ir' particular it should be noted that since abelian C*-algebras are semi-simple our abelian Banach algebras will automatically be semi-simpleo

With these preliminaries we are now ready to give an outlil1e of the papero If G is a locally compact abelian group represented as

*-automorphisms of a von Neumar~ algebra, Arveson, Borchers and Com1es ["l, 2, 3] developed the theory of spectral subspaceso In § 2 we shall generali:.>;e to regular abelian Banach algebras acting con- tinuously on a locally convex topological vector space, much of that part of the theory of spectral subspaces 1!Vhich does not depend essen- tially on the group structure of the dual group of G •

In § 3 we prove the basic general results on operator normal maps ^o \rJe assume ti1e operator normal map cp is contained in a regu- lar abelian Banach subalgebra of B(B(H)). Then lv ^•-t- follows from § 2 that tb_e s-pectru...'U of cp in B (B (H)) is the same as the spectrum of in B( X) • A consequence of this is that if the spectrum of

~o in B(B(H)) is contained in the ur1it circle and cp(1)

=

1 tt ... en cp is either a *-automorphism or a *-anti-automorphismo

In § 4 we give examples of regular and nonregular abelian Banach subalgebras of B(B(H)) ⁰ If a, b E B (H) ilve denote by La and Rb the ill'1.pS X ... ax 2nd ^X-> xb respectivelyo Then L _a. maps every C*-subalgebra of B(H) isometrically into B(B(H)) o If we denote by a ⁰ b the map LaRb' we can imbed the algebraic tensor product to two abelian C>:'-algebras A and B into B(B(H)) • The norm is a cross--norm, so t:r_~e closure A ® B is a regular abelian Banach sub- algebra of B (B (I-I)) consisting of operator normal maps ⁰

If G is a locally compact abelian group and a ~ continuous

- 3 -

representation of G into the automorphism group of B(H) , and

iJ. E l'1(G) - the bounded Borel measures en G ~ al-l E B(B(H)) , where a (x)

= l

^at (x)dv ( t) " Then the image of L 1 (G)

h-~s

as closure

iJ. '-'G ,

in B(B(H)) a regular abelian Banach algebra consisting of operator normal maps o However, the image of .M( G) · need not have regular

closure ^a

If f is a co.n-tinuous complex function on a product space X x X , we say f is positive definite if the n x n matrix

(f-. ( '\' . . ^!--. h v

,xi,Xj)) lS posl-vlV8 Vhenever x₁,oao,xn E "''-" In § 5 we prove a Bochner theorem for such functions when X is a compact Hausdo~ff

spaceo

The results in §

5

are applicable in the study of the algebra A 0 A when A is ar.1. abeliail C*-algebra, because the spectrum of A 0 A ca.""l be identified with SpA x SpA ^o If cp E A 0 A i t is

shown in § 6 th2.t cp is a positive map if and only if cp is com- pletely positive, and that this in turn is equivalent to

ep

being a positive definite fu~wtion on SpA 0 A ⁰ A consequence of this and our general Bochner theorem is that if cp E A® A is extreme among all positive iclenti ty preserving maps in B(B(H)) then cp is a '"-automorphism of B(H) ⁰ The section is concluded by noting that the case A :2! A includes the examples a(L1 (G)) exb.ibited in

§ 4, so that the above results for cp E A 0 A are applicable to maps of the form

note that if H is finite dimensional then every map in A ® A has a complete set of eigenvectors in the Hilbert-Schmidt operators

X

consi,sting of rank 1 operatorso In § 7 we shmv an infinite dimensional result which is a converse to this assertiono

·- 4 -

Let X be a locally convex topological complex vector space Let A

be a regular abelia:J. semi-simple Banach algebra over the complex numbers with ffil appro:CLmate unit consisting of elements whose Gelfand transforms are real a...11d with compact support, cf ⁰

[9,

10] ^o We assume X is a left A-module via a map Ax X ~ X, (a,x) ... ax , 1-'lhich is separately continuous and faithful in both variables ^o Our typical example \..rill be when A is continuously represented into the algebra of continuous linear operators on Xo

If S c A a...11.cl Y c X we 1 et

s.:..

'

y.l.

Clearly

= ^{x

= ^[,:J E ^X EA

ax=O for all aES}

ay = 0 for all y E Y}

is a closed ideal in A ^o \rJe let Sp A denote the maximal ideal snace in A , identified with the set of continuous characters on A ,,

If a E A we denote by

Z (a ) == { y E Sp A : y ( a) =

a (

^{y ) =} 0 } If F c Sp A is a. closed subset we let

j (F) -· {a E A : Z(a) contains a neighborhood of F , and support

a

is compact} ^«

\..Je recall from.

[9,

25D] that j(F) is the smallest ideal in A whose hull is We denote by

I

X(A,F) == j(F).~- ^o

Then X(A ,F) is a closed subspace of X , called the specti'&

Finally if x E X we denote by

-- 5 -

Sp (x)

=

h( [x} _;) ,

the hull of the EUYlihi.letor of x in J1.., Sp(x) is a closed sub- set of SpA o Furtl'lermore Sp (x) = 0 if and only if x = 0 ^o

Indeed, h( {x} , ) _... = 0 if and only if

[ 9, 25

D Corollary] , if and only if ax -- 0 for all a E A , if and only if x :::: 0 , since the representation Ax X .... X is faithful in both variables ^o

~n Let A and X be as above" Ijet F be a closed subset

of A ~ a E A and x E X n •:rhen we have

(i) If Z(a) contains a neighborhood of Sp(x) then ax ·-· 0 "

(ii) X E X(A1F) if and only 1-^"f' Sp(x) c F ^o If supp Ct ^... cF then ax: E X(A,F) ^o (iii)

Proofo (i) By assumption h([x}_{1 )} is contained in the interior of Z(a) ^Q By the asSl..un.Dtion on approximate unit in _~- ^A there is b E A such that

b

^has compact support ar..d 1\ab-

all

^{< e:} for given e > 0 ..

Then h([x} 1 ) is contained in the interior of Z(ab), so

ab E j(h( (x}_{1 ))} ^o By

[9, 25

^D] ab E (x}L, ioeo abx

=

^0.. Since

e: > 0 is arbitrary and c _. ex is continuous on A , a_x

=

0 ^o

(ii) Suppose Sp (x) c F " If a E j(F) then Z(a) contains a by (i)" Thus x E j(F) l

=

neighborhood of Sp(x) , so that e.x

=

0

:l(A,F) ^o Conversely, let x E X(A,F)" Then [x}₁ ::> (j(F)L)l ::> j(F)o Thus h( (x} 1 ) ^c h(;j (l~')) ::: F

[9, 25

D] ^o

(iii) Suppose y [I. supp

a

^o Then, sinGe A is regular, there is b E A such that b(y)

I

0 while ab = 0 ^o Thus b(ax) = ba(x)

=

0.

But then y

t,

Sp (a.x), so we have shown Sp(ax) c supp

e,"

Now use (ii) ⁰

·- 6 -·

,.._,

We denote b:T A the algebra A loJith the identity map of X

,..._,

adjoined, and we consider X as a..'1 A-module as wello Note that

,.._,

by [101

2o7.3]

A is regular, and we can consider S:p A as a sub-

"' set of Sp A ⁰

Lemma 2 .. 20 Let F be a compact subset of SpA., 'I' hen (i) :X(A,F) ^::J X(A?F) ^o

(ii) If a E A and a(y) ₌ 1 for all y in a neighborhood of F then. ax = ^J{ for all ^X E X(A,F) ^o

£>roo:[.. ( i) J.1et

i(F) = {a E A : Z(a) contains a neighborhood of F} ..

Then i (F) => j (F) , so i (F) l c j (f.) l " Let x E X(A,F) 9 and

I I

El E i (F) " The:J. ax

=

0 by Lemma 2 ^o 1 so x E i (F)..__ , and i (F)-

= j(F)l.,

However~

^{X(A,F) :J i(F).L since sup:p}

a

is compact in

"" "'

Sp A for all a E A ⁰

(ii) Let ^t denote the ic1enti ty in A o Then is zero in a neighborhood of F ^o Let x E X(A,F) ., ^By (i) x E

X(A,F),

so by Lemma 2 ⁰ '1 (a-t )x = 0 , i ^o eo ax == x ^o

\ve say a subset Y of X is bounded if for each absorbing neighborhood V of 0 in X there is ^E: > 0 such that r:;y c V ..

~he following result is a generalization of

[3, 2a3o5] ..

Let

v

be an ab.sorbing neighborhood of 0 in X , and let Y be a bounded subset o.f X such that a(Y) c

II

^{a\\ Y}

for all a E .J:>. .. ^1', Let and Then there

is a compact neighborhood N of y ₀ in SpA such that

- '7 -

a . X -

a . (

^{y ) X E}

v

^{f 0 r 3.ll X E}

y n

X (A 'N ) ' i = 1 ' ^{0 0 0} 'n ⁰

l l 0

~~of o Since V is an absorbing neighborhood of 0 in X and Y is bounded, there exists e > 0 such that eY c V o Thus by our assurnptim: or.1 Y , a(Y) c V whenever i\ aJJ < e ^o

be a compact neighborhood of y₀ and a E A such that

Let N₁

a(y) __ 1 For each i E (1, .. o .. ,n} let b. E A

l

be defined by bi(y) ~

£

1.

c

^{y) =}

a.

_l

c

^{y )} ·~

a.

_l

c

^{y )} ₀

(ai ( y) - ai (

y

_{0 )} )a( y) .. Then

bi (

^y 0 )

= o ,

^and

on N ₁ ^o From the regularity of A there is

c E A such that c(y) = 1 for all y in a

ueighborhood H2 ^{OJ~ y 0 0} Let N be a compact nei~~borhood of Yo contained in the interior of N1 (l N2 " Let J: E Y

n

X(A,N) .. Now c(y) _·- 1 for y in a neighborhood of N ' ^{ar1.d N} contains Sp(x) by Lemma 2 .. 1 .. Thus ex = x by Lemma 2 o 2, and simi:!.arly ax

=

x ~

lve thus have b.cx -- b.x = a.ax-a.(y )ax= a.x-a.(y )x .. Since

l l l l 0 l l 0

!!b.c\¹1 < e, b.c(Y) c V" Thus a.x- a. (y )x E V for all

1' l l ' l l 0

x E Y(l X(A,N), Q.oE .. Do

If E is a Banach algebra we denote by aE(x) the spectrum of x as an element in E ..

9orol~ar~_?o4 .. Suppose X is a Banach space and that the identity operator is in A.. Let a E A • Then aB(X)(a) = [a(y): y E SpA}

EE£of. Given e > 0 let V = [x EX~

\Jxl\

^< e} , and let Y be the unit ball in X ⁰ If y ₀ E Sp A , then by Lemma 2 .. ·1 (iii)

- 8 --

Yfl X(A,N)

I=

(0) for e'=.tc.h compact neighborhood N of y_{0 "} Thus a(y_{0 )} E aB(X)(a) by Proposition 2,30 Si!lce [a(y): y ESp.A} = aA(a), we have shown aB(X)(a) ^:::::> aA(a) ^o The converse inclusion is imme- diate, since we can consider A as a Bans.ch subalgebra of B(X) eontaining the ideatityo

It should be rem.s.rked that just as in the theory of spectral .subspaces of automorphisms we can introduce the auxiliary c::mcept R(A,E), cfo [1] and then prove that X(A,E)

=

nR(A,V), t.vhere the intersection is t~:en over all closed neighborhoods V of E , see the proof of [1, Proposition 2.2]o However, we shall not need this

3nC. shall therefore not include the proofo We shall rather prove

another result which vve shall not need technically, but which is of importa."lce for our understanding of spectral s11bspaces.

Proposition 2.~.20 T_;et B be a Banach subalgebra of A satisfying the same assumptions as A ⁰ Let r : Sp A .... Sp B be the restriction

map y _, Y

I

^{B o Suppose F} is a cmnpact subset of Sp B .such th&t

"''

-r-'(-::i11 . t . S A

~ "'"'/ lS compac 1n p ^o Then vie have X(A,r-1 (F)) == X(B,F).,

Proof ^{0 1}

ro

our previous notation add the subscripts A or B to distinguish between A and B ^o Let :x: E X(B ,F) ^$ IT·hen by Lemma

hence r-¹ (hB ( {x} l

n

B)) c r-1 (B') o There-- fore we hav~ that if J is the ideal in A

X

then

r-1 (F) :J r-1(hB([x}₁ nB)) - [ y E Sp A : ker y ::.J J x}

-- h(Jx) :Jh({x}l), since J_," _{~,_} c (x} 1 ..

generated by {x} l

n

^{B ,}

- 9 -

x E X(A,r--1 (F)) ₅ and v•Te have shown X(B,F) c X(A,r-1 (:B')).,

Conversely let x f X(A? r-1 (F)) , hence hA ( [x} 1) c r-1 (F) o

IJet b E jB(F) ^o Then ZB(b) =>

~3'

^{o If y E r-}1 (1i') then r(y) E F .so

b

^{(r( y)) = 0 ,} hence b E ker(r( y)) = (ker y) n B ^o Therefore b(y)=O, so yEZA(b), and111rehaveshown r-'\.B')cZA(b)~

Since F is comp::tet and contains a neighborhood of ^}i'

'

there is a compact neighborhood of F contained in

z

_{1) (b)} ^o

Since r i.E; continuous by the above argument

r- 1 (N) is a neichborhood of r-1 (N) c ZA(b) o Thus ZA(b)

r-1 (F) 1 and is a. neighbor- hood of hence by the definition

From the proof of that Since x E X(A,r-1 (F))

/1 I

lemma iA(r- ¹(F))-

.it thus follovm from Lemma 2o 1 that bx

=

0 ^o Since b was arbitrary in jB(F), we have sh01lm X(A,r-1 (F)) c

jB(F)l = X(B,F), and the proof is completeo

Let H be a Hilbert space and .J ^r- and

d-f.

the trace class and Hilbert-Schmidt operators on H respectively" We denote the inner product on ·a{ by (x9y) = Tr(xy*) and the norms in

'T

and d{ by

\I

l!

^{1 and \I 11}₂ respectively.,

We say cp is operator normal if cp is ultraweakly continuous and the restriction ~ ja{ ^is a normal operator in B (;{) " If moreover cp

16(

is self-adjoint we

S:=-y _~ r·r. _{'t' l} • s _{QP --}e-rat . • _{. O.l} "h _.u.e r ... _{illlu l} an _{. o cp} is said to be a reg~~r opar~or

normal map if _~ is contained in a regu.lar abelian Bana~h subalge- bra of B(B(H)) consisting of operator normal maps.,

We denote by \' cp\1 _.I ·2 the norm of

cp I

Cr( whenever cp

laf

^E B((T{) "

Note that v.Jhen cp is ul tra;,veakly continuous then its adjoint map restricts to a map cp* E B(<CI) with norm

l\cp*\1

= \\cp\j ^a

~a 3.,2o Let cp E B(B(H)) be regular and O-;?erator normal, and

denote by

*

^the adjoint in B(of. ) of cp

Ia(

^{Then ¢ I~}

=

^{(f)* ,}

a."ld iicp'\ _{1i ·} 12 _ IICf.\" < !I II

,..-.__

l2;:'oof_., Let x E ·J a.nd y E 2Tt ⁰ Then (¢(x),y)

=

(x,cp(y))

=

(cp* (x) ,y) , so ~ (x)

=

cp* (x) o Let A be a regular abelian Banach subalgebra of B(B(H)) consisting of operator normal map such that cp E ^A., Le-e r denote the restriction map _~ ^... ¢

1-o-e

^{of A into}

B(d"()o Then r ^{.J_I:.J ~;-} continuous" Indeed, if ( $ )

n is a sequence in A converging to

y

'

^{a11d r(¢n)} converges to ^{~ I} in

B(K)

then clearl;y \f (x) = ~' (x) for each x E

o-t .

Thus the graph of r is closed, so r is continuous b;y- the closed graph theorem. Since r is an iso211or];hism of A into B(&e) i t follows that uA(cp) :::::J

uB(.:;{) (cp) , hence t'l1.e spectral radius of cp in B(trt) is not

- 11 -

larger than the spectral radius s of cp in A ^o But !lcrll2 equals the spectral radius of cp in ^B( _'"-' >f') _{\... ?} so \lcp\\2

-

< s 0 By the minimality of the spectral radius norm in a regular abelian

Theorem 3n 3. Let A be a regular abelian Ban.ach suoalgebra of B(B(H)) consisting of opeJ..'ator normal maps ^a Then the map cp ...

cp!X

is an isometric isoinorphism of

f.fp :

cp E A} onto [cp

!a-e :

^{cp E A} , 1Jvhich}

extends to an isomorphism of C(Sp A) onto the closnre of { cp

I

^{d{ : cp E A} in B ( a{ ) •}

for cp E A " Then clearly a is an

isomorphism of [~: cp EA} onto [cp

ldf. :

^{cp EA} Q Let r(cp)} =

cp IH •

By Lemma 3~2 r is norm decreasing on A , hence if

x

is a char- acter on the norm closure of r(A) in B(d{) then

x

or E Sp A ^o Thus for ~ E A we have

Thus a is norm decreasingo However,

I

I ... II.

i cp:; is the spectral radius of

cp

in A, so b~r the minimality of the spectral radius [10'l3o7~7J,

Thus !ICf>!! =

!lcp\\

2 , and the theorem follov:s ..

.Qorollar:v 3 .. -40 If cp is a regular operator normal map in B(B(H)) then aB (B (H)) ( c~)

=

aB (d-t) (

cp!

ift.) ^o

cp is contained in a regular abeliru1. Banach subalgebra of B(B(H)) consisting of operator normal :naps and containing the iden-·

tity mapa Thus the corollary follm\iS from Corollary 2 .. 4 and Theorem 3.,3 ...

~ 12 -

Pro_posi~ion 3...::...2.., Let cp be a regular operator normal map in the unit ball of B(B(H)) such that q(1) = 1 ar:_d such that its spect- rum in B(B(H)) is contained in the unit circle" Then cp is either a *-automorphism. or *-a:ati-automorphism of B(H) ..

£<?.£!"

By Corollary 3e4 the spectrum of cp

I

^d-{_ in B(:j{) is contained in the unit circle, so cp

ld-t

normal implies cp

I Oi_

^is

unitary" In particuls.r, since cp-'1 E B(B(H)), cp-"l

!2rl

^{is the}

an.joint of cp

I

^{d-( o Since}

\lcp1\

= 1 and cp(1)

=

1

cp

is positive (i .. e~ a > 0 in B(H) implies cp(a) ~ 0 )" Thus if x,y E ~ + - the positive cone in

7 -

then cp*(y) E r.T c 0{ , so

-1 + + -1

hence cp -1 ( ) ,y > _{0 "} Thus cp

c:;

^_

..

_{T .. Since cp is norm}

<-

continuous on B(H) --1 C(H)+ ... C(H)+, whore C(F.) denotes the

'

^cp

compact operators on H, using that ^~+ is norm dense in C(H)+"

Let B be the C*-algebra €:1 + C(H) " Then cp ^-1 is a positive linear map of B carrying 1 on itself" Since cp is operator

~wrmal ~ cp : , hence by continuity, cp: C(H) ... C(H) ^o Thus cp is also a. pcsitive linear map of B into itself preserving the identity" Thus ~') is an order-isomorphism of B onto itself, hence is ei·cher t:J. *-automorphism or a *-anti-automorphism [5]"

By ul travJeak continuity of cp the desired result follows., We shall need the next result in the sequelo

~emm~ 3~6" Let (~~)~EJ be a uniformly bounded net of regular operator normal m2ps, which converges pointwise ul trawea.ldy to a map cp E B(B(H)) " Then we have:

(i) cpid<. EB(~)o

(ii)

(iii) (iv)

cp

l

^{X ... q}

I

^(.r( weakly in B(;1{) , so in particular v

(cpv LX)* ... (cp

I

^{X)* weakly o}

If the rpv pairwise com..-rnute then cp

I

^X is normal ^o If cp ... cp in norm then cp is ul trmJealdy continuous ^o

v

Consequently~ if (J.ll) and (iv) hold then cp is operator normalo Proof" Choose K > 0 such that

II cp)\

_s K for all v E J ^o By

Lemma 3.,2 \\cp)\_{2 .::_}

\lcp)\ .::_

^{K, so}

Ccr)

^iY,)vEJ is a uniformly bounded net in B(?rt) " Thus there is a subnet (cpa)aEI such that

( !

cpa (} -^Jf) aEI converges weakly to an operator ,,, '~' t: B(o-f.";" , i.,e., (¢(x),y) = lim(cpa(x),y) = lim Tr(cpa(x),y*)

ex. a

for all x,y E ^\Y .a...._ ⁰ Now ( cpa ) etEI , b · e1ng a su ne b t o _ 1.e con f tl verg·ng 1 net r ^{m )}

''t"v vEJ ' r;on,rerges point11\fise ul travmakly to Thus if y E Cj

( ¢ (x) ~y)

=

ltm Tr(cpa (x)y*)

=

Tr(cp(x)y*) = (cp(x), y) .,

Thus ~~ (x)

=

cp(x) for all x E 6-( , so cp : d-{ ... 2r{ ⁰ Furthermore

cp) a< ... cpl

^K weakly since each converging subnet does" .Since

\lcr)l

₂ < K for all v ,

J\cp1!

2 .::_ K, hence cp

l

^{d-e E} B(d{) " This proves (i) and (ii)o

Nov1 assume all the co commute, and let l'1 c B(?{ ) be the abe- v

lian von Nemn.aru1. algebra generated by all the maps cpv

I ^{df ..}

^Since

cp

jo-{

v 1veakly,

weakly, we have by Lemma 3" 2 that

Since

(cp) 1rn

* ... (cp

I

^J-()

*

(cp

I

^X..)*

I

^T = ^{cp* .. Since}

Ccrl

^{x)* E :r-1,} is normal, a.i'ld (iii) follmvs., If ^cp ·v ^{-;. ['(\} 't"

hEnce wocp is ultraweakly continuous for each w E B(H)*, and cp is itself ultraweakly continuous., This concludes the proof of (iv) a.i'ld therefore of the lemm·J..,

- '14 -

4a Exam~les of re~J-~r_algebra~o

The most easily obtained example of regular abelian algebras of operator normal maps are of the form x .... ax == L X

a and x _, xa = Rax 1 v.rhere a belongs to an abelian C*-algebra A .,

and R

a are isometric isomorphisms sin0e A is abeliano When A is not abelian La is still an isometric isomorphism, so that every C*-algebra A c B(H) has a canonical isometric imbed- ding in B(B(H)) •

vie denote the map LaRb = ~La by a ® b for a,b ^E B(H) ^o Taking linear combinations we can in this way consider the alge- br·':lic tensor product B(H)

0

B(H) as a subset of B (B(H)) con- sisting of ul trm.·rea2dy continuous maps, which restrict to bounded operators in B(M) ^o If x,y E _~

'

then (Lo.x,y)

=

Tr(axy*) =

Tr(x(a *y) *)

=

(x, a *y) , so L* a

=

_La* and similarly R* ^d = R * ^{a o} Thus the restriction map B(H) (i)B(H) ^_, B( x._j is * -preserving when B(H)

G

B(H) has the *-operation (I: a. ®b.)* = I: a~ ®b~ ^o

l l l l

Note that since Rb is anti-isomorphic in b the 1.mbedding of B(H)

0

B(H) into B(B(H)) is not an algebraic isomorphismo However, if A and B are abelia_r:t subalgebras of B(H) , then the imbedding of A

0

B in B(B(H)) is a *-isomorphi~3ffio

Lemma 4o1o The norm on B(B(H)) restricts to a cross norm on B(H)

0

B(H) ^o

~.. Let a,b E B(H) ^o Then clearly \\a0b\! _:: \\a\\1\b\!.. To show the converse inequality let e > 0 and choose unit vectors

s,'ll

^E H such that J\asll ~ 1\al\- e and 1\bn\i > !\b\\- e, Let v

• /1

'I

^{II .}

be a partial lSometry of rank ' such that vbTl

=

1 br}11

s ..

^Then

\\avb-rl\\

=

\\bn\1\la.s\\ ~ (\\bl\-·e)(\\a.\\-e:), henee 1\a®bJ\

.?:

l\a.\J\lb\1 ^a

- 15 -

Pro_I>osition 4oSo Let A and B be abelian C*-subalgebras of B(H) .. Then the closure A ® B of A(;) B in B(B(H)) is a regu- lar abelian Banach sub algebra consisting of operator normal maps, Proof. By Lemma 3,_6 each map in A G B is operator normal. The

rest is immediate from Lemma 401 and a result of Tomiyc:ma [13], Remark 4.30 By Propositiun 4a2 each map of the form a® b with a and b normal, is regular in the sense of Definit~on 3"1" It ce,n be shown that even more is true, namely that the Banach subalge- bra of B(B(H)) generated by a® b is regulara

If G is a locally compact abelian group we denot~ by M(G) its measure algebra, consisting of all bounded Borel measures with convolution as multiplication and *-operation

0'CE) =

~(-E) .. We write multiplication in G and its dual

G

additivelye I am in- debted to GaKo Pedersen for discussions which led to Proposition

Lemma L~o4,. Let G be a locally compact abelian gruup and t .... ut a continuous unitary representation of G on the Hilbert space H ^o Let r1.t(x)

=

u~xut, x E B(H)" Then for each fl E M(G), al-l d~J-

c

fined by a (x)

=:

at(x)dtJ.(t), is an operator normal map such that

\J. ^J

c

^a

I 21--)

* = a."'

I

d-C. ^o

fl l-l

fsoof.. It is easy to see that t _.. c:.t

I

j{ is a continuous nni tary representation, cf", [12]a Thus a~IX E B(Cr{)"

we have

r

(a\J.(x),y) =

J

(at(x)?y)d!-l(t)

= J (

^{x , a_ t ( y) )} d1-4 ( t ) ,,

=

J

(x,at(y))d\-l(-t)

= ( x ,

J

^{at ( y) dfjt-f))}

=

(x,ar.t(y))

If x,y ^Ed{

- 16 -

Thus (a.

!

>.-t)

*

= 0."', Since

Jl • '~ tJ. a. ^00."-' = a. ,..., ::: 0."'"' ::: a,...,oa a. com-

11 f-1. f-1. *f-l. f-1. *11 f-1. f-1. ~ f-1.

mutes with its adjoint, so af..lio<. is a normal operator .. Finally, i t follows .from [1] that au

is operator normalo

is ultraweakly continuous~ hence

Lemma 4~ 5.. Let G be a locally compact abelian group.. Then the map T : l'1 (G) -• B (Leo( G) ) defined by

is an isometric isomorphism int0 ..

T (f) = ^f..l*f

f-1. for f E L (G), ^CX)

;rrqQ.f... It is vJell known a11d easy that T is an isomorphism into B (L⁰⁸( G)) .. Moreover, i t is shown in the proof of [ 8, 3 .. 4 .. 1] that T is a continuous multiplier of L:::o(G) endowed with the weak-*

11 1

topology induced by the elements in L (G) , and furthermore that the adjoint map T* is a continuous multiplier of L 1 (G) ^o By

11

[8, 0 .. 1 .. 1] \1'1'~!1 = 11~11

,

hence _l\T11

11 = !lull ..

Proposition 4 .. 6.. Let G be a locally com?act abelian group and H = L 2 ( G) .. Then there is a canonical isometric isomorphism a of l'1(G) into the operator normRl maps in B(B(H)) such that

a

\.I

I ¹ --.~ v'- = (rv "'"f-1.1 ! ' c'-. P)

*

..

Proof.. Let A. be the regular representation of G on H , and let S be the *-isomorphism of L⁰⁰(G) into B(H) defined by Sfg

=

fg for g E L²(G) .. Let a.t(x)

=

t..txA._t for x ^E B(H) o

By Lemma L~ ^.. 4 a.₁₁ is operR.tor normal, and a.;;P-f

=

(a.f-lldf)* for each f E L⁰⁰(G).. Indeed, let g E L2 (G) and s,t E G.. Then we have, with ~(u) = g(u-t), u E G,

(a.t(Sf)g)(s)

=

(A.t(Sf(t.._tg)))(s)

=

(t..t(Sfg-t))(s)

=

f(s-t)g(s)

= (ftg)(s) -- (Sf g)(s)

t

- 17 -

hence Let g, h E L- (G) ; then 'vve have, using the ^;:;, Fubini theorem,

r

(a.~(Sf)g,h) =~ {at(Sf)g,h)d~(t)

= J J

(a.t (Sf)g) (s)

ETSJ"

ds dll(t)

r r

= jj

f(s-t)g(s) h(sj ds d!l(t)

= :

g(s)

h1sT ( J

^f(s-t)

^d~--L(t))

^ds

= J

^g(s) il(S)"

(~·.*f)(s)

ds

=

(SI-l*f g,h) ,

and a. f./Sf)

=

⁸

~-t*f

as asserted .. From the definition clear that \Jai-l

!I --

^{< I 1~--t I 1.} However, \ve have just shown

of _al-l tl:at

a : S ^{..-v- _,}

s '

and since

s

is an isometry, we have

Lco(G)

!.l L""~( G)

By Lemma 4 .. 5 we thus have

I,\ a !I > SU1)

Po.

(Sf) \l

=

1-l -- 1: -, IT -1 ,, !-l :iof 1-

=

11~--tll

^'

Corollary 4 ..

7..

Let G be a locally compact abelian group and

it is

H

=

L'2(G) .. Then there is a canonical isometric isomorphism of L1 (G) onto a

reg~lar

abelian subalgebra of B(B(H)) consisting of operator normal maps ..

/1

£E_oof.. Restrict a in Proposition 4 .. 6 to L ' (G) , and use that L1 (G) is regular ..

If H is a finita dimensional Hilbert space it is obvious that every operator normal map in B(B(H)) is regular., However, if H is infinite dimensional this appears to be false ..

~- 18 -

Corollary 4a8o If H is a separable infinite dimensional Hilbert space, there e::cists an operator normal map cp in B(B(H)) such that the Banach subalgebra of B(B(H)) generated by cp is non- regular ^a

Proofa Let G be a nondiscrete locally compact abelian group

such that L2 (G) is sepa.cable, and identify L2 (G) with H ^o Then M(G) is a nonregular abelian Banach algebra, since ^" G in its

Patural imbedding in Sp M(G) is nondense, while t~e vanishing of a Fourier transform ~ , p E 1'-'I(G) , on

A

G implies 1-l

=

0 ^o :Wet A be the isometric image of M(G) in B(B(H)) constru_cted in Propo- sition L~o6o Then A is nonregular, so by [10, 3a7~4j there exists an element cp E A such that the Banach subalgebra of A generated b;y cp is nonregularo

-· 19 -

5o

A general Boclu1er ^theorem~

Let X be a locally compact Hausdorff spaceo We say a complex:

valued fu_nction f on X x X is .:QQ.§i ti ve de.fini te if for each

finite subset [x₁,"""'xn} in X the nxn matrix (f(xi,xj)) is positiv3., In the next section ·~e shall need a generalized Bochner theorem for such functions., I am told that such results may exist in the literature, but I have not found any references"

Proposition 5.,1., Let X be a compact Hausdorff spRee with a count- able base for its topology., Let G denote the group of continuous functions of X into the unit circle, equipped with th3 discrete topology" For ^X E X, y _{E G '} denote by (x,y) the value of y at Xo Let f be a continuous positive definite fur1ction on X such that f(x,x) ^::: 1 for all ^X E X., Then tbere exists a posi- tive Baire

( 1)

measure l.l of total mass 1 f(x,y)

= J

(x,y)(y,Y"Ydt..t.(y),

G

on G such T.JJ.at for all x,y EX ..

Conversely, every function f defined by (1) with ~ as above, is continuous positive definite satisfying f(x,x)

=

1

J:.roofo With pointwise multiplication G is a discrete abelian

" ^,., p(x)(y)

groupo J.Jet G be the dual group" Imbed ^X in G by = (x, y) ^o Then p is continuous, for if ^X n ^{... X} in ^X then

(xn,y)

....

(x,y) for all y E

G ' since y is continuous" Define a

"" "'

function f on G by

'""

f(z) if z = p(x)- p(y)

if z §!, p(X)- p(X)

y,rhere we denote multiplication in

G

addi ti vely" We show f is well defined" Suppose z

=

p(x)- p(y)

=

p(x')- p(y') .. Then for

... 20 -

all y E G we have (2)

In :particular, if x

=

y then (x', y) = (y', y) for all y , so x'

=

y' ^o Thus by assumption f(x,y)

=

^{f(x' ,y')}

=

^{1 = f(:3) ..}

Now assume x

I=

y.. We show x = x' , y = y' ⁰ Suppose not, say

... v-1 0 Then there is y E G such that (x,y)

f

(x' ,y), hence

by (2) (y,y)

I=

(y' ,y), and y

I=

y' .. Now at least -':;hree of the elements x, y, x', y' are distinct, for if not then x = y' and y = x' ^o But then by (2) (x,y) 2 = (x' ,y) 2 for all y, which is impossible since x

I=

x' • Say x, y, x' are all distinct ^o If y' is equal to one of them, y'

=

x, so by (2) (x,y) ²

=

(x' ,y)(y,y) for all y , which is also impossible. Thus they are all distinct.

But then y can be found so that (2) is violated. Thus X

=

x' and y

=

^y' as asserted, and we have shown f is well defined.

Since X is compact and p is continuous, p(X) is compact.

"-'

Since f is continuous we have thus in partir:!ular that f is measurable, so ^,.__, f E L (G). ^co We show f is positive definite ..

,.,

z_{1 ,} ^{0 .. o ,} zn E G , Hi th, say, zi

=

p (xi) i = 1, ... ,m , zi ,t p (X)

i=m+1,o .... ,n .. Then if z.

J

z.

l J and zi- zi

=

p(x)- p(y) , i t

u

Let

follows from the previous argument that Z.:

=

p(x), z-;

=

p(y), because

.L u

"'

by thP- Pontrjagin duality theorem G =

G

⁰ We therefore have

"'

f(z.-z.)

l J

r

f(xi ,xj) '

= '

1

i , j E [1, .... ,m}

i = j E (m+1, o," ,n}

otherwise

Lo

Since the m x m .matrix (f(x.,x.))

l J is positive, so is the n x n matrix (f(z. -z.)) , and f is positive definite as asserted ..

l J

By Bocl1...11.er' s theorem [9, 36 A] there is a finite positive Baire measure ~ on G such that

- 21 -

,...,

f(z)

= j

^['

<

z ' y ) d\-l ( y ) G

In particular

""

f(x,y) = f(p(x)-p(y))

=J

<x,v>(y,vYdllCv) G

a., e .. on

a ..

Let that

g(x,y)

=J

(x,y)"{y,y)d~'.(Y) .. Then it is trivially verified g is G positive definite., g is also continuous, for let (xn,yn) ~ (x,y) in XxX .. Let gn(y)

=

(xn,y)(y₁₁

,y)

^and

g(y)

=

(x,y)"Gl-,r:;., Then g_{11 _,} g pointwise, and all functions are

lh~iformly bounded.. Sir.ce ^1-l is finite the Lehesgue bounded con- vergence theorem 3Tields

r ~

g ( xn, Y n)

= J

^{gn ( Y ) dll ( Y ) - >}

^j

g ( y ) dll ( y )

=

g ( x, y) ,

G G

and g is continuous ..

By assumption f is continuous, and f = g Thus f

=

g, and (1) holds for all x,y EX., In particular 1

=

f(x,x)

::::; s

^{diJ ( y) ' so ~} has mass 1 Finally, the last statement .in the G

proposition follows from the previous paragraph ..

- 22 -

6o The algebra A 0A .,

In this section we ane.lyse the regular abelian Banach subalgebra A 0 A of B(B(H)) in more detail when A is an abelian C*-algebrao Our results indicate that its relationship to abstract harmonic ana- lysis is quite prof01m.do Recall from [13] that 8pA0A can be identified with SpA X SpA • We shall therefore write elements in SpA 0 A as pairs ( y, y ^{1 )} with y, y ¹ E SpA ⁰

We denote b;y- C(Sp A ®A) the continuous complex functions on SpA 0A , and by a the canonical isomorphism of C(Sp A ®A) onto the nor::n closure of {alo-e, : a E A® A} , described in Theorem 3o 3o

\ p+ \JJ

We denote by ~ and o~soa the positive ~~u self-adjoint Hilbert-Schmidt oper8.tors respecti velyo We say an operator a E B(d-{) is JlOSitivity !)reserving (respectively hermitian pre- servigg:) if a : ^d{

+

^-7

act.+

(respectively a :

df __, d-{

r) ⁰

Soa SoC:I.

Theorem 6o1o Let A be an abelian C*-algebra acting on the

Hilbert space H ⁰ Let notation be as above, and let f be a con- tinuous function on SpA ®A o Then f is positive definite if a:.r1d only if a(f) is e. :positivity preserving operator in BO{).

\~e shall need the following result on hermi ti~~ operators in B( o-() ⁰

iJemma 6o2o Let f E C(SpA g;A) ^o T:t.en if a(f) is hermitiEm preser\riing, then f(y,y')

=

f(y' ,y) for all Y,Y' E SpA ⁰ In particular, if y "1, .. ^{o o ,} yn E SpA then the n

x

^{n matrix}

(f(yi,yj)) is self-udjointo

Proofo Assume first a(f) is the restriction to ~ of a map cp E A(;:JA, say n

cp

=

I: a.®b. ~

• /1 l l

l = l

a. , b. E A o

l l Then for x ^E d-( we

- 23 -

have I:a.x*b.

=

rn(x"')

=

rr\(x)'*= I:b~x*a~, so that 'Ea.®b. ⁼ I:b~®a~

l l 't' 't' l l l l l l

on

0\

^{0 But then}

(y,y')(Z:a.®b.) = (y,y')(L:b'!'Ga'!')

l l l l

=

I: y (b. ) y' (a. )

l l

so that f(y,y')

= illv' ,y)

in this caseo

In the general case choose a sequence (cpn) in A

Q

A such that the restrictions to (){ converge to a(f) in B(d{) ^o Say cpn =I: a. ®b. ^o

l n lil Let cp+ = L: b'!' ®a'!' , so that ~ = t(co +cp+) E P_(!)Ao

n 1n 1n n •n n ·

If x E d-<_ then llcp+(x)-a(f)(x)!Jn ₂ = Jlcp (x*)*-a(f)(x*)*lln _{2 =}

l!

cpn (x"") -a (f) (x*) 1!2 .... 0 uniformly for in norm in B ( d{__) ^o By Theorem 3 ^o 3

'I ,, _,

^{T (f)}

1 X! 12 ^{~ 1 .,} hus ¢n _, a

~n _, f in supnorm, so f(y,y') =lim ~n(y,y') =lim $n(y' ,y) = f(Y';Y"T ~

n n

r£2of of Theorem 6o'1o Assume a(f) is positivity preserving, and let y _{1 ,} o ^o o, ^y n E E~p A ., If B is the weak closure of A then every character on B restricts to a character on A , and

A® A c B ® B as subalgebras of B(B(H)) ^o Thus in order to show that the n

x

n matrix (f(y. ,Y .))

l J is positive, we may assume

A = B, ioGo A is a von Neumann algebrao Let ;:: > 0. Now a(f) can be approximated in 1l

\!

_{2 -} norm by restriction of maps in A

G

A , and each operator in .A can be approximated in norm by linear com- binations of mutually orthogonal projectionso We can therefore

find mutually orthogonal projections in A

vdth sum '1 such that y.(e.)

=

1

l l i

=

·1,ooo,n, and constants

i, j E [1, ^{o o o} ,m} , such that if ^1Ji denotes the restriction of

A. •• ' lJ

- 24 -

I: A. . e . ® e . to

?-<

^then

l.J 1.

a

( 1) 1\a(f)- ~i!2 ^{< e: o}

li'urthermore, if we replace ¢ by t( 1jJ +$ +) , cf. Lemma 6o 1, we may by that lemma assume ¢ is hermitian preserving.

Let V. be the closed subset of SpA

1. correspoonding to e.

1.

under the Gelfand transform. By Proposition

2.3

there is a com- pact neighborhood N ..

l.J of (y.,y.)

1. J in

I,'. . c V. x V . , i, j E [ 1 , ••• , n} , and

l.J ^1. J

(2)

!I

^W (x) -

$

^{C Y · , Y ·}

)x!l

₂ < e:

1. J

SpA~A such that

fer all x E X(A ®A ,N .. )

l.J with

llxl!

2 _:: 1 , vrhere X

=

a{. ⁰ Choose compact neighborhoods W.

1. of y.

1. such that

w.

x\tl. c N .. ,

1. J l.J

i, j E [1,.,. ^o ,n} , and let fi be the projection in A correspon- ding to the characteristic function

xw.

1.

of

w.

_1. ⁰ Let no 't-IT pk be one of the projection-s f . , e . -· f . , i

=

^{1 , • o o ,} n , and

1. 1. 1. e.

1. for

i

=

n+1,. ^o., ,m , and renumber them so that We can thus write

where

SincG that

(3) n . )-( p . c X (A ® A, W. X W . ) ,

-J_ - J ^1. J

i , j E {1,.o.,n}.,

n. =f.

"'"l 1.

Let q. < p. be a 1-dimensional projection,

1. - J..

and as above adding p.- q. for i == 1, ^{o • •} ,n, and

1. J..

n+1, • .,. ,m to the gi , we can write

for i

=

1, " .. ^o ,n •

• "'1

1.

= . , ....

,n , for i

=

-- 25 -

1ir _'~'

=

2: p _{rs r} q ® a _~s

l )-( ,

_·

vJhere P.,,~ E [A. . : i, j E (1, ... ,m}}

.L~ J.J

sional. Choose po.rtial isometries qrs of rank one with domain qs and range qr such that (qrs)₁ <r,s<n is a set of matrix

n - -

units, qrr ~' qr o Let q = I: qr- , and let I1 denote the factor r=1 -

B(H)q of type In spanned by the qrs. If I1n is the n x n

complex matrices then the map I: a rsqrs __, Cars) is a

*

-isomorphism, hence an isometry of l'1 onto 1'1n ⁰ Let e be the 1-dimensional :projection e =

n

1 2: ^qrs in 1'1 .. By ( 1)

\Ia (f) ( e )-ljr (e)

ll2

^< _{E: '}

hence, since ~(e) is self-adjoint,

llxll

< \lx\1 2 for ^X E 21{

-

^{s .. a}

and a(f)(e) _{~ 0 '}

By (2) and (3)

! 1,•, ( o )

ili (

^{y y ) q II} < E:

li 'P :J.rs -" r' s rsii2 "

Thus vre have

(5)

< ne ..

By Lemma 6o2 the operator 2: ~ ( y , y )q

r s rs is self-adjoint. Thus by (4) end (5)

(6) 1 -:'

- I: 1li ( y , y ) q > (-ne-e ) q "

n · r s rs - If

a

=

(a .. )

J.J is a matrix in 1'1n then its norm is majorized by Ij a .. 1 "

J.J Indeed, Ia. -i < j11all for all

lJ - i, j , so we have

llali

²

= \la*all

_.:: ^{- T (} _{r a} ^{*. )} _{a =} ^'I

_I,

_{a 2}

112

_{= ....} ^)'I _{j aij} ¹² _~ ^II _l•,a ^11..-.1 _I L, _I a.·!" l.J, ^I we have

1\(f(yr,ys))- (~(yr,ys))!l ~ 2:lf(yr,ys)- ~(yr,ys)j

_:: n2!\a(f)-

~11 2

^{< n2e "}

Thus from ( 1 )

- 26 -

If we combine this \"lith (6) we have since (f(yr,ys)) is self- adjoint,

2 2

(f(y ,y )) > (-n -n --n)e ..

r s -

Since e is arbitrary (f(yr,ys)) ^~ 0, and we have ^sho~ f is positive definite ..

Conversely, assume i is positive definite., Let B denote the weak closure of ^A and let y be the restrictio~ to ^A of y ^E SpB .. Thu.s (f(yi,yj)) is positive for all ~1 ^.. o .. ,Yn ^E SpB ..

In order to show a(f) is positivity preserving it suffices to show a(f)(p) > 0 for each 1-dimensional pro,jectio':l p in

at

⁰

For this it suffices to show that for each nnit vector

s

in H

and e > 0 there is a nonnegative real number a such that

(7)

l<a(f)(p)s,s>- al < € 0

We let p ,

s ,

and e > 0 be given ..

Choose mutually orthogonal projections e_{1 , ..} o.,en in B and t..ij , ^{i, j E}

I:A. .e.®e.

l J l J

(1,oao,n} such that if is the restriction of to X then

that y.(e.)

=

l l 1 .. Since

Choose a is an isometry we have

y. E Sp B

l

Let ^ljl' be the restrict1on of L: f(y. ,

y.

)e. ^® e . to J-(_ o Then

l J ^l J

l!a(f)-lJ1'11

₂ ~

lia(f)-vll

_{2 +}

ll$-lJI' 11

₂ ^< ~+ ~ = e ..

such

Let

s. =

e.s and let ~ be a unit vector such that p is

l l

the projection on the subspace it spans., Then we have

- 27 -

<

^{w t (p)}

s'

^{S) =?.::} f(y.,y.)(e.pe.s,s)

l. J l. J

= I:

f

(Y'

i,

Y'

^j)

<P

~j ^,p

s{

= 2: f(y.

,Y'.)<<s.,'ll)YJ,(s. 3YJ>YJ>

l. J J ^l

=

⁼⁼ f(y.

,Y' .)<s .,YJ>Zs. ,YJ>

l. J J l.

?:

0 '

since (f(yi,y)) :::_ 0 .. If q is the 1-dimensional proje~tion on the subspace spanned by

s

^then

l<a(f)(p)s,s>- <~'(p)s,s>l = I<Ca(f)-w')p,q)j :: \la(f)-~

'11 211PII2llqll2

^{< 8 ..}

Thus with a= (v'(p)s,s> the proof is complete ..

Recall that if cp is a linear map from one C*-algebra M

into another ^N, t:L:;n 'P is said to be positi_;~ if cp(x) > 0 for each x > 0 in i'I ., cp is said to be com:R_letely po~ if cp® t n· • M®r1 n .... N01.'1 n is positive .for each n, where

identity map on I"'n.,

Corol~~X. 6 .. 3.. Let A be an abelian C*-algebra acting on the Hilbert space H.. Let cp E A® A.. Then the following conditions are equivalent:

(i) cp is pos~tive.

(ii) cp is completely positive.

(iii) ~ is positive definite on SpA ®A ..

Proof.. (ii) =>(i) is trivial.. Since cp is ultraweru~ly continuous, cp

I

~ is positivity preserving if and only if cp is positive .. Thus

(i) <=> (iii) is immediate from Theorem 6 .. 1.. To show (iii) => (ii)