Unbounded generalizations of standard von Neumann algebras

Unbounded generalizations of standard von Neumann algebras

REPORTS Vol. 13 (1978) UNBOUNDED No. 1 PHYSICS ON MATHEMATICAL GENERALIZATIONS OF STANDARD VON NEUMANN ALGEBRAS ATXJSHI INOUE Department of Ap...

633KB Sizes 1 Downloads 129 Views

REPORTS

Vol. 13 (1978)

UNBOUNDED

No. 1

PHYSICS

ON MATHEMATICAL

GENERALIZATIONS OF STANDARD VON NEUMANN ALGEBRAS ATXJSHI INOUE

Department

of Applied

Mathematics,

Fukuoka

University,

Fukuoka,

Japan

(Received January 31, 1977)

The primary purpose of this paper is to study unbounded operator algebras called standard EW#-algebras which are generalizations of standard von Neumann algebras to the unbounded case. The llrst part is an investigation of the weak, e-weak, strong and o-strong topologies on a standard EW#-algebra. It is showed that the weak and c-weak topologies on a standard EW#-algebra coincide. The second part is a study of a locally convex topology called &f-topology on a standard EW#-algebra ZI. It is proved that ‘u is a GB*-algebra under the topology. The third part is an investigation of isomorphisms of standard EW#-algebras. It is showed that if standard EW*-algebras % and b are isomorphic then II and 23 are spatially isomorphic.

1. Some basic theory of standard EW#-algebras We review some of the definitions and results concerning unbounded Hilbert algebras and standard EW#-algebras and show that the weak and o-weak topologies on a standard E W#-algebra coincide. Let 9 be a pre-Hilbert space, with an inner product ( 1 ), and a *-algebra. Let 5 be the completion of 9. Suppose that 9 satisfies: (1) (5lr) = (s*Il*), (2) (&x)

E, 17E 9;

= (G*O,

5,179 5 EQ.

We define n(6) and n’(l) by: 407

= Er

and

1;/E9.

n’(Q>s = rE,

Then, by (2) n(6) and n’(t) are closable operators on & with the domain 9 and n(E)* 3 n@*), 7c’(Q* 3 7r’(5*). We set 90 = @=.%n
denotes the set of all bounded

EW(fj)), operators

on 5.

A. INOUE

26 DEFINITION 1.

If 9 satisfies the conditions

(1), (2) and

(3) $9; is dense in 5, then 9 is called an unbounded Hilbert algebra over ~3~ in $j and z (resp. 7~‘)is called the left (resp. right) regular representation of LB. In particular, if 9,, # 9 then 9 is called pure.

Let 9 be an unbounded Hilbert algebra over go in $j. Clearly .QOis a Hilbert algebra and the completion fi(9,) of gO equals the Hilbert space $3. Let n (resp. n’) be the left (resp. right) regular representation of 9 and let rc,, (resp. nb) be the left (resp. right) regular representation of the Hilbert algebra gO. For each x E $j(%,) we define z&x) and n:(x) by: n;(x)5

n,(x)E = &(E)x,

= %(5)X,

5 E 9,.

Then z,,(x) and n;(x) are linear operators on $(9,) with the domain .QO. The involution on 9 is extended to an involution on $(9,,), which is also denoted by *. Then we have : %(x*)

= %(x)*,

45*) = n(t)*, _ 745) = %(O, _ -n(E)+n(r)

Hence n(9) and n’(9) of strong sum, strong Let %,-,(Q,,) (resp. Hilbert algebra gc, and We set

n;tb(x*> = n;(x)*,

x E wh);

n’(l*) = n’(t)*, l E 9; __ n’(t) = n;(5), 6 E 9; __ .__ __ 46) * n(q) = db/), t,VEg;

= n(E+ $9 __ ___ A * n(E) = Tc(izl),

a E c,

5 E 9.

are *-algebras of closed operators on $(9JO) under the operations product, adjoint and strong scalar multiplication. Y,(9,)) be the left (resp. right) von Neumann algebra of the let pO (resp. wO)be the natural trace on %0(90)+ (resp. ‘YO(9~)+). (%)a = {x E wm;

dx)

E ~(ww>}

-

Then (9&, is a Hilbert algebra containing 5B0. If gO = (9&,, then go is called a maximal Hilbert algebra in $(9,-J. Let m (resp. fm+) be the set of all measurable (resp. positive measurable) operators on $5(9,) with respect to ‘4PO(9,). For every T E %l+ we put PO(T) =

SUP[~&&));

0 G 44)

G T,

6 E (%)62]

and L’(po) = (TEE;

IlTll, := ,~uo(lTI~)“~-= co>,

l
Then IlTjl, is called the LP-norm of T eLP(qO) and ,CQis called the integral on L’(q%). If p = co, we shall identify aO(SO) with L”(& and denote by ljTjj, the operator norm of T E L”(c&.

GENERALIZATIONS

DEFINITION 2.

OF VON NEUMANN

ALGEBRAS

27

We define LT-spaces with respect to pl,, and 9,, as follows:

respectively. For p 2 2 we set L$(vJ

G(%)

= Lp(~O) nL2(9kJ,

(Z,P)

=

44

E JY%)),

(TEE’),

IITII,~

IlTll (2.p) = max{llTllz, llxll

= {x o 5(%);

(x ~G(%I))*

Il%(-9ll~2.P,

From [lo], Theorem 3.9, Lz(CSO) is an unbounded Hilbert algebra over (S?&, in fi(gO). If S? is an unbounded Hilbert algebra over .SJO, then 9 is a *-subalgebra of L;(g,). Hence L”;‘@,) is maximal among unbounded Hilbert algebras containing gO. From [14], Lemma 2.1, Lq(9,) is a Banach space under the norm 11 ]]t2,,,. We shall define the left EW#-algebra of 9. Let n4 be the left regular representation of the unbounded Hilbert algebra Lz(CV,,) and let TIL;@,) be the restriction of T E %&2,) to Lg(g,). Then nT(@ and ‘420(90)(L~(90) : = (T(Lf(~23~); T E %@,J} are #-algebras on L”;‘@,) under n”;(t)* = n”;‘(E*) and (TIL~(9,))~ = T*IL”;‘(CB,). We denote by &(cS) a #-algebra on Loi)(C@,) generated by n;(g) and %O($%,,)lL~(~O).Then C@(Q)is an EW#algebra on L;(g,,) over S&Z,,). For the definitions and results concerning EW#-algebras the reader is referred to [9], [lo], [II]. DEFINITION

3.

Q&(Q) is called the left EW#-algebra

of 9.

An E W#-algebra ‘$I is called a standard EW*-algebra Hilbert algebra 9 such that 2I = %(G#).

DEFINITION 4.

an unbounded Let d(g)

denote the closure of the EW#-algebra t&%)1 Fx

=

(9)

TX,

TE g(9)

Then G(g)

@@), =

{1=;

strong topologies

The weak and o-weak on g(9)

That is, we set

p(m)? x E w%J)l

(W,

TE@@)}.

is a closed EW++-algebra on [i;(g,,)]

THEOREM 1.

Proof:

:=

a(g).

if there exists

topologies

(9) over $YO(gO). on g(9)

coincide.

The strong and G-

coincide.

Let {x”} E 9,(&(g)).

F or each positive integer n we set

c_____ ”

T,, =

nO(xJ

* %(xi)**

i=l

Then, T,, E L1(rpO) and T,, > 0. Hence there exists an element y,, of $(S?,,) such that ~YJ

2

0 and

T, = ndy.>‘.

Then,

IlyAlzZ

=

2

i=l

Ilxdl$. For m > n,

nd~,,,)”

I_ 2 ndvd2

A. INOUE

28 and n,-,(y,,,) Z 0, n&)

Z 0, and so n&,,,) > nO(yn), i.e., ne(y,,,-y.) > 0. Hence we have ----_ (YnlYt&-llY”ll: = (YnlYm -YJ = PO(%(YJn-YJ *MY”)) 2 0 *

It follows that llym-YnII$ = llYmllx-~~Y”lY~~+IIY”ll~ G llymllz?- IIyAlz” In

c

=

IlxillZ.

i=n+l

Therefore,

lim llym-ynllf = 0, and so there exists an element x of jj(LB,) such that m.n-t* = 0. We shall show that x E [~‘@O)J (9). For each a E 9 we have lim Ily.---42 n-bco

= lim p0 (Ico(a)*no(a) - ~c&Y~)~) n-+*

.

= Em Il~o(a>YAli:. “-+CO Hence, x E ~(sz,,(u)) and 00

lim Ilno(4YAIS n** Furthermore,

= lIn0(44lS

=

c-

“=I

Il~o(4xnll;.

it is proved that

x E Gceg@,(4)

= [~3~o)l

(9)

2 (~GnX”lX”) n=1

and

= (zmxlx)

for al1 a E 9. Thus the weak and c-weak topologies coincide. Similarly it is proved that the strong and u-strong topologies coincide. COROLLARY 1.

The following

conditions

are equivalent.

f is a weakly continuous linear functional on g(9). (2) f is a a-weakly continuous linear functional on g(9). (1)

(3) There exist elements

x, y of [iy($So)] (9) fG7

In particular,

if f is positive

= (WY),

such that

T E &W

then

f(T) = G’W).

GENERALIZATIONS Proof

ALGEBRAS

29

This follows immediately from [9], Theorem 4.8, and Theorem

COROLLARY 2. g(9)

OF VON NEUMANN

1.

For each a-weakly (equivalent, weakly) continuous linear functional

there exists an element Df of n,([~~(CSJ] f(T)

= ,UT*

D/>,

(9))’

f on

such that

TE %W.

And:

(1) D/ is uniquely determined by f;

f is positive if and only if Df 2 0; (3) f i ,uo if and only ifDf E no((Q0h)2, where f < p. denotesIf( (A E&(9)+) for some constant r. (2)

Proof:

From Corollary

< quo(A)

1 there exist X, y E [i$(So)] (9) such that --*

f(T) = (WY) = PO(T* no(x) * ~ocY) ) for all T E g(Q).

We set D, = no(x) * no(y)“.

Then, Df o no ([&~o>l

(9)”

and

f(T)= po(T*D,).

It is easily showed that the restriction g off to ‘%o(~o) lZ”;(g,) is extended to a a-weakly

continuous linear functional on %o(go). Using the Radon-Nikodym von Neumann algebra, the conditions (l)-(3) are proved.

theorem

of the

2. The o,L‘$topology on a standard EW*-algebra

In this section let go be a Hilbert algebra and let !jj(go) be the completion of go. We shall define a locally convex topology called ,Zz-topology on ~(L~(~o)) and show that G2(L;(.9,)) is a metrizable complete GB*-algebra under the topology. For the definitions of GB*-algebras the reader is referred to [1], [6]. By the definition of the left BFV#-algebra %(L:(c~~)), every element T of %(Z”;‘(g,)) is represented as T = ToI L;@,)+n;(x);

Hereafter, for each To E %o(90) denoted by To.

To E '4Po(9,), x EL;@,,).

the restriction

To/L;(9,)of To onto Zz(go) is also

NOTATION. For 2 < p -C 00 and T E %(L”;‘(CBo)) we set mllWz,p,

=

~'%,(9&,), x E L;(~o)). ~~f~ll~oll~+Il~ll~2,,,~T = T,+n;(x>,T,,

LJWMA 1. For 2 < p < co ._,I1llc2,Pj is a norm on %(L”;@,)). Proof: Suppose mllTllt2,p~= 0. For each positive integer n there exist T$"E %o(~o) and x, E L";(go) such that T = T&"'+n";'(x,), lITPI QI+Ilx nII(2.p)< l/n.

A. INOLJE

30

Hence, lim IITg)(I 00= 0 “-+a,

and

Iim ((x,,(((,,,) = 0. n-m

For each 5 E ~3~ we have

(T”R%) = llT6”‘~11~+(rr~(T~)xn+(T~)*X”)*+X.*X”)5~r) = ll~6”‘~112’+(~“i’~~~‘x,~s‘l~)+ (EI~~(T(“)*X”)E)+IIX~(Xn>5113 G ll~6”‘11~115112+~1I~~‘ll~ll~~~~“>511211~112+ll~~~~“~5111 G ll~~3ll~ll5ll~+~lI~~~llmll~“llpllSIl~ll5ll2+II~nll~ll~lls2

IIT6ll$ =

(where 1/p -t 1/q = l/2) +o

(n-+ a).

Hence, llT5j12 = 0 for all [E:c~,,, and so T = 0. Let S = S,+z;(x); S, ~?%~(ge) To+@(y); To E +YO(QO), y E L;(gO) be each decompositions of S x E Lz(ge) and T = and T respectively. Then, S+ T = (S, + T,) +n;(x+y) be a decomposition of S+ T. Hence,

coIIS+~ll(2.P~G II&+ Tollm+Il~+~ll~,,,, G ~Il~~ll,-t-ll~ll~z,p,~+~II~ollm+ll~Il~~,~,~~ The other conditions of norm are It followsthat mllS+Tll~z,p,< mll~ll~z,p,+~ll~Il~~,~~. eastly showed. Thus ,I\ (ICz,Pjis a norm on @(L~(C3’,)). DEFINITION1.

Lll Ilw,;

The locally convex topology on &((L”,‘(9,)) induced by the family on %(Lz(90)) and is denoted by 2 < p < co} is called the ,L;-topology

Car;. NOTATION. For 2 f p G co we

set

,JX(%)

= W%)

+ CX%) 3

T = To+ 6 , ToEL:(plo), TI ELiX%)~, ,llTll (2,p)= ~~f~ll~~ll,+ll~~ll~~,p~~ T E,LWo). LEMMA 2.

For 2 < p G w ,Lp(&

is a Banach space

under the norm mjl I lc2,pB.

In particular, (,L%cl);

rnll ll(z,co,,) = (L%o); II IL).

It is proved in a same way as Lemma 1 that ,I1 IICp,pjis a norm on ,Lh(vO). We shall show that (,LX%); ,I1 llcz,p,)is complete. Suppose that (T,} is a Cauchy sequence of mL$‘(9)o). Then, there exists a subsequence {TnCkj} of {T,) such that Proof

rnllTn(k+~)-Tn(k)ll(~,p) < &i By the definition of the norm ,I[ Iltz,p, there exist

(k = 1,2, . ..).

GENERALIZATIONS

OF VON NEUMANN

ALGEBRAS

31

such that T n&+1)- T”(k) = (Tn(k+lj-- T.o,),+ Tn(kjh

il(Tn(k+l,-

Let T,,C1j = (T,,&+

11,

<

(T,,&

$

(T,++xj--

ii(&k+l)-Tn(kjh

titz,pj

; (T,(I))o E L’YvA

(T,&I

<

T&l, (k

f

=

1,2, . ..).

E L%pO) be a decomposition

of

T“(,). We set

(T.&o

= &k-&+

CT,,,,-- z(k-&> (k = 273, . ..)a

(T,ck,), = (Tn(k--l))l+(Tnck,-Tn(k-l))l

Then, (T.&o E L’=‘(%), (cc& E J%?‘o) and c(k) = (T&o Furthermore, for k > r we have

+ (T.&l

@ = 1, 2, . . .>.

lI(Tnck,>o-(Tncr,)olla, = ll(Tn(kj--Tn(k-I&+ ... +(Tn,r+,,-~n&ll, < pi+

... +~. --) 0

(k,r+

co)

(k,r-+

co).

and

++

Hence,

@?q&)

.. .

+&o

> are Cauchy sequences of L”(q+,) and L$(~I,,) respectively. and Tl E LZp(& such that

and {(Tn&

II IIc2.,,) is complete, there exist To E L”(q,)

Since (LX&;

and

Jim II(Tn(&-- TOIL = 0

k-m

Jim Iliad,

k-+m

- TIll~2.,,= 0.

We set T=

T,,+T,.

Then, T E ,L,P(vO) and

= 0. lim colI~(kj-Tllo-Tolla+lI(Tn(kl)l---~~ll~~,,~~ k-+co k-em Hence, limmIITk--Thz.p, k-tm

G

~mm{mllTk-T.~k,ll~~,l,+,~~T.~k~-TI~~~,p~}

Thus, ,Lq(q~,,) is complete. It is easily showed that (La(%); LEMMA3.

II II,) = (&%d;

all llc2.m,).

For 2 < p < q < co we have COLSPO) I3 CcJLi(%) = coL4,(%) = LY%)

and for some constant y

=

0.

A. INOUE

32 Proof:

This follows immediately from [14], Lemma 3.3.

Proof

The inclusion

%(Lz(gO))

c

n 2
For each p E [2, 03) there 42~PCoo ,L,P(gq,).

,L$(&

is obvious.

Suppose

that

T

00

exist T$‘) E LT(q+J and TIP) E L$(v,,) such

that T = T&v)+ Tip). Then, T$‘) - T&2)= Ti2’ - Tip) E L;(v,) Hence , Tj2) = (T$‘)-

Ti2)) + Tip) E L2p(p1,,). Thus, T = Ti2’+Ti2)

That

is, there

= G(%).

exists an element

E Lm(q,)+2<+&Ll(~,,).

x of L;(QJ

such that

T = TA2)+nU;(x). Hence,

T E 4(L;@,)j. THEOREM 1.

(4(LT(9,));

,tT)

is a metrizable

complete

GB*-algebra

over %!O(90)

Im%). Proof From Lemma 4 it is easily showed that (Q(L$‘(Q,)); ,z:) is a metrizable locally convex space. First we shall show that (%(L~(9,)); o3?‘) 2 is complete. Suppose that (T,,} is a Cauchy rom Lemma 4, for each p E [2, co) {F”} is a Cauchy sequence sequence of %(L”;(%,)). F is complete and it follows that there exists Ttp) of ,Lq(y,,). From Lemma 2 ,Lg(v,) E ,LZp(& such that lim,Jl~~-T(P)]] (2.P) = 0. n-to0

The above Ttp) is independent of p. In fact, from Lemma 3, Tcp) E ,Lg(q+,) and Iim ,,,J]T,n-+ca - T(P’ll~2,2, = 0. Hence, T”) = Tc2) for each p E [2, co). Thus, Tc2) E ,
7Ik2. p) =

such that T=

Tt2). Hence, lim,]]T.-

“-+a,

0 for all p E [2, co). Thus, (%!(L~(z%~)); “z”;) is complete.

Next we shall show that (%(L;(BO)); o.7”‘) z is a locally convex +-algebra. Let S, T E %(L;(.C@,)). Let S = S,+nO;(x); So E %!&9,,), x E L;(%,) and T = T,+@(y); TO E %!,(9,), y E Lz(23’,) be each decompositions of S and T respectively. Then, ST = (SO TO)+n~(SOy+(T,*x*)*+xy) is a decomposition of ST. Hence, for 2 < p < co we have oollSU (2.p) G I~~~~~II,+II~~Y+(G~*)*+~Y~I,~,,,

G Il~oll~ll~oll,+ll~oll~ll~lltz,,,+ll~oll~ll~ll~~,~~+ + 11~11~2,~,11Y11~2.~,+ ll~ll~2.2,,llYl1~2.2p,

. GENERALIZATIONS

OF VON NEUMANN

33

ALGEBRAS

G ~Il~~ll,+ll~ll~2.P~~~II~ollm+Il~llt2,~,~+ +wJllm+Il~ll~2.4,)wollm+llYllt2.4,>+ +~ll~~ll,+Il~ll~2.2,,~~ll~ollm+llvll~2,2p,~.

It follows that ,IW’II

Furthermore,

(2.p) G ,IlNl~2,4,

,ll~11~2,4,+,ll~ll~2,,,~ll~l1~2,,,+,ll~ll~2,2~,~lI~11~2,2~~.

we have mIlWl

(2,p) = id =

W,*ll,+llx*ll~2.,A

WlS311mf I141c2 .pj>

= coll~ll~z.p,. Thus (9(L;(90)); ,t;) is a lot a 11y convex *-algebra with jointly continuous multiplication. Finally we shall show that (%(L~@‘,)); _,tz) is a GB*-algebra over %0(9,JlL(;)(90). By A* we denote the collection of all subsets s of %(,Y$‘(9,)) such that (1) %3is absolutely convex, bounded and closed in (@(Lz(9,,));

,t;).

(2) B2 c ‘& 8* = % and ZE!.?~. We shall show that A!* has a greatest member 8, Let %3be each element of A*. Suppose that there IlTll, > 1. Then there exists an element E of (9&, Let T#T = A0 +$(a); A0 E L”&), a E L;(9,) be IIEII;

=

PWO

=

(AAt)+

G Il&5112+11~x~)~112 Hence,

IITtllf

< (1 +ll~~14)m~lT#Tl~~2,4). IlEE”

: = {T E L”(qO)lLt(90); llTllm < 11. exists an element T of !8 such that such that l]E]], = 1 and lIT51j2 > 1. each decomposition of T#T. Then, (@(ME) G llAAo+ll~ll4llL5ll4*

Repeating the above argument, we can get that

G (1 +ll5114)

,llVW’ll~z,4,.

Since ]]T4]2 > 1, lim llT511~”= CO.On the other hand, since (T*T)” E @ (n = 1,2, . ..) “+a3

and % is ,z”;-bounded,

-1

hm ,ll(T#T)“ll~2,4,

n-co

< co. This is a contradiction.

Hence, IlTll,

< 1, i.e., T E &, . It is easily proved that ‘$3, E A’ *. Thus !& is a greatest member of A*. It is immediately showed that (Q&(L”;(g,-J); ,2;) is symmetric. Furthermore, since (Q(Lz@,)); ,t.T) is complete, it is pseudo-complete. Thus, (%(L$(QO)); ,z;) is a GB*-

algebra over 4?&3,)lL;(kz0). COROLLARY 1.

Each positive linear functional on f4?(LG(BO)) is continuous with respect

to the ,L”;-topology. Proof:

This follows from Theorem

8.2.

is an unbounded Hilbert algebra over go in jj(.9,,), is a metrizable GB*-algebra over 4Y’o(~O)~L~(~O).

COROLLARY

,z”;)

1 and [6], Corollary

2.

If9

then (S(9);

A. INOUE

34

It is immediately proved that if SEQ~(~~) then ,IISll~z,P, < IlSll, (p > 2), if x EL%%) then call~Xx)ll~z.p~ G I141~~,p~(P 2 2) and if (9,Jb has an identity then the topology ,z; on %(LT(90)) is e q uivalent to the topology generated by (IJ lip; 2 < p c w}. The ,Lg-topology on ‘2l(L~@,J) is finer than the weak topology on 4(L”;(9,)). TO E %O(Q,), x E Lz(gO) be each decomposition of T In fact, let T = TO+$(x); E @(L%%)). F or each 6, 7 E LT(.9,) we have %,U’)

= lGTl$l

G IGJlr~l f iW~)4dl

3. Isomorphisms

of standard E W*-algebras

Let % (resp. 8) be an EW++-algebra on a pre-Hilbert space ‘33(resp. %). A map pb of ‘$l onto ‘B is called a homomorphism if it is linear, if @(ST) = @(S)@(T) (S, T E 3) and if Q(P) = G(S)* (S E ‘u). If @ is a bijective homomorphism of ‘8 onto %, then it is called an isomorphism of ‘8 onto ‘2.3.Then % and !I3 are called isomorphic. Let di be an isomorphism of fl onto B. If there is a map U of ZDonto E such that lIUt(l, = 115112 (6 E ZD)and G(S) = USU-’ (S E %), then @ is called a spatial isomorphism. Then 2l and % are called spatially isomorphic. THEOREM 1.

If standard EW*-algebras

% and 8 are isomorphic, then ‘% and 8 are

spatially isomorphic. Proof: Let @ be an isomorphism of B onto 9. From [6], Theorem 7.14, we have -@(2&J = !&,. We set Q(T) = Q(T), T E ‘?&,.Then & is an isomorphism of the standard von Neumann algebra K onto the standard von Neumann algebra %$. From [5], 8 6, Theorem 4, d>is a spatial isomorphism of q onto $$, that is, there exist maximal Hilbert algebras SB,,, bO and an isometric map U of the completion !@BO) of QO onto -- the completion $(S,) of 8, such that tBO = bO, 3; = @,,(QO), %$, = %!o(bO), nt(Uf) -= @(@‘(t)) (6 E gO) and s(T) = UTU-1 (T E 2%&B,,)), where PX~(resp. nf) denotes the left regular representation of go (resp. 8,). Let yO be the natural trace on %!o(~o)+. Then, for each positive integer n and 8 E 9,, we have

IIWZZ::= yo(I~2”)

= (u(E*mu(E*~y)

= ((E*Em*v)

From [14], Theorem 2.5, gO (resp. 8,) is dense in (LT(9,); it follows that ULz(g,-,) = L;(B,). We set 9

=

{E E Lx%);

c&G)

E w,

8 = (q E LX8,);

sS(rl)

E Bl,

where 9~4 (resp. 17~:) denotes the left regular representation

$)

= 1151122:.

(resp.(L”;(go);

of L;(Q,)

zZ)> and

(resp. L”;(~o)).

GENERALIZATIONS

OF VON NEUMANN

ALGEBRAS

35

Then it is easily proved that 9 and 6 are unbounded Hilbert algebras over CBeand bO, respectively and 2I = 3!(g), 23 = a(&). Furthermore, we have U!3 = Q and @(&(~)) = &#E) = U&$(5) U-r for all 5 E 9. In fact, .&$(v) E 23 = @(?I) for each 11E 1. Hence there uniquely exists an element 5, of 9 such that @(~$(6J) = &‘($. We define a map V by: q E B -+ 5, E $2. Then it is easily showed that VIBo = U-’ lBo. Hence, V = U-l and @(snz(U-lq)) = ,&(q) for all q E 1. It follows that Ukil = I and @&n;(5)) = &( Ut), F E 9. Furthermore, it is easily proved that ,&(Ut) = U9$'(t) U-l for all [ E 9. Thus, Q(T) = UTU-' for all T E 92(g). That is, @ is a spatial isomorphism of ‘II and 23. Acknowledgement I should like to thank Prof. Dr. A. Uhlmann for his advice and for acquainting me with the work of W. Kunze (: the definitions of GB*-algebras of Dixon and of Allan coincide). REFERENCES [l] [2] [3] [4] [5]

Allan, G. R.: Proc. London Math. Sot. (3) 17 (1967), 91. Arens, R.: Bull. Amer. Math. Sot. 52 (1946), 931. Brooks, R. M.: Pucijk J. Math. 39 (1971), 51. -: Math. Nachr. 56 (1973), 47. Dixmier, J.: Les algtibres d’op&zteurs duns l’espucc hilbertiun, 2e edition, Gauthier-Villars, Paris 1969 [6] Dixon, P. G.: Proc. London Math. Sot. (3) 21 (1970), 693. [7] -: ibid. (3) 23 (1971), 53. [S] Dunford, N. and J. Schwartz: Linear operators, vol. 11, Interscience Pub., New York 1963. [9] Inoue, A.: Pacific J. Math. 65 (1976), 77. [lo] -: ibid. 66 (1976), 411. [ll] -: ibid. 69 (1977) 105. [12] -: Math. Rep. Coil. General Eiu. Kyushu Univ. X (2) (1976), 114. [13] -: J. Math. Sac. Japan 29 (1977), 219. 1141 --: L-spaces and maximal unbounded Hilbert algebras, to appear. [15] Jurzak, J. P.: J. Functional Anal. 21 (1976), 469. [16] Lassner, G.: Rep. Math. Phys. 3 (1972), 279. 1171 Ogasawara, T. and K. Yoshinaga: J. Sci. Hiroshima Univ. 18 (3) (1955), 311. 1181 Powers, R. T.: Commun. Math. Phys. 21 (1971), 85. [I91 Robertson, A. P. and W. Robertson: Topological vector spaces, Cambridge 1966. [20] Segal, I. E.: Ann. Math. 57 (1953), 401.