are ,, . ,, ., .. •• .., ab(t)=fGa(ts-1)b(s)(is, ' i tz'(t)==a(e-1).

Bull. Kyushu lnst. Tech.

(M. & AT.S.) No,1, 1955

N*-Algebras'' and -,-Finite Clasg Groups(t) '

,- By-. Kaz6 Tsun , . ...l r-. I., lll -.L 1 .. Lt

(Received Oetober 23, 1954) 1. Introduction.

Since tlie character theory of locally compact (abbreviated to LC) abelian group. and theory of harmonic, analysis on it "rere deve]eped, the center of the investigations has moved in tbe nen•-abelian case. Thenceforth, the study of tlie direct produet of LC abelian . .crroup and compact group by I. E. Segal. i6],(?) Lana various stuaies by many authors are found, but unfortunately .we'"cannot yet find the conclusive' results. Recently, J. Dixmier[2]

defiped a linear operation it of 'W*-algebra onto •its center, aftenvards studies ,on the opera.tion ig were acted,. qnd R. Godement {31 discussed the haniionic analysis on the LC

, .t L/ ":1-.'1.

group-his eentral..crroup.-which hqs som'e kind of finiteness. And the same author [4]

developed the character tbeory of LC ttnimod"Iar grdnp$. In t}iis country, under the term tecentering", the inttoducing of tlie operation lt to C*-a]gebras "'as attempted by Y. Misonou and M. Nakamura 'l51. And while bur Jinvestigation was going on, we came to knoW that H. Umegaki introduced the operation h to sesreral a]gebras ahd studied the representation theory oE algebras. His resultsC3) are expected to appear in a shert time.

'

Now the objects of our paper-are to verify the relation between rnaximal txvo-sided ideals in finite class N*-algebra which is defined on finite class group and its characters and ' Boclmer's type theorem with respect to finite class N*-algebras.+

In bq2, definitions a`nd Preliminary results whicli ttre neceSsary for our objeets are men- ' tioned without proofs. And in g3 we define AT*-algebras and Constitute a finite class N*- algebra 9-(. In S4, we prove the relation betsveeh maximal" tS,o-sided ideals in the finite class-N*-algebra.9J and its characters.. And finally-•in• g5i theorems xvhich involve Bochner's

type theorem with respect to the finite class N*-algebra P•( are proved. . ' , Our special thanks are due to II. Umegaki for his,helpful advices concevning this paper.

-••,2• Definit'iQns.and preliminary regults. L

Let G be LC unimodular group. Then -S'denote the a]gebra of complex varued continttong " funetions which,are defined,on G'and have compact sttpport, and•its algebraic operations

are ,, . ,, ., .. •• ..,

ab(t)=fGa(ts-1)b(s)(is, ' i tz'(t)==a(e-1).

' If Radon mea$urferl' it bri 'G•satisfies that ' '

(1) This paper was ivritten at Kyushu Un.iversity in Feb- 1953. rt is •now published i,n tlie ]953 foTm.

(2) The numbers in bracke.ts refer tq the references qt the end of the paper- , ,.

(3) Tt is IIj Umegaki: Operator algebTa of finite 'elass, S'o-dai Math. Seminat'y Rep. No. 4 Deccml}er 195-", pp• 123-i129.

1

L

K, TSUJI

' ' ' fGa'aÅqt)ap(t)lllO for eve'ry 'a E 2,

we cal} it positive tptpe, and if it $atisfies that

'

JGatb(t)dpÅqt) =J(;ba(t)d/{(t) for every a,b E Åí,

it is called central.

DEirms'rrlol (1). Let G be a LC group.

(a) 'Hilberti space fi)., ' '' ' •

(b) Two 'unitarJr representations sDUs, s-Vs are cDntinuouds one ofG te ttp, aird

satisfr that

UsVt=VtUs for evety s,t EI G, .

(c) •An •invelua"on S of tÅé sach that Vs==SUsS for everrs E G,

II7e call the s)tstenz {pC.,Us,Vs,S} consiste(l of them double unitLu" representtttion of G.

'

If nosv l- is arbitrary central positive type Radon measure on LC unimedular group G, 'we can define the double unitary representation {Ii-År.(iw),Us(p),Vs(p),S} of G by mp• Theri,

'

DF•1•'INiTToN (2). if there enists a bouncletl operator Vx (or Ux) on tb(ib) sttch that Vxf==

U,x(or, Uxf.=V,.pt) for everpr f e S, we sacr titat tlie elempnt x. EI. tSÅr(l-) is bou.ndecl.

DEiq'iNrr]o.N' (3). •If zve have i -

UsV sx' ---- .x -- fer' every s E' GÅÄ

tae saJt tlmt tlte'element .r E {fo. is 'centrali and tire seei ofall'central element'is denete(l bpt ,b.#.

r1

Then, the follossfing` thedrerrl is verified; , . . ,

THEoRF"M (1), if .x-,is arbitrar}t elemene in tb, and if Kx+ ,is tlie lease closed•conven' subset

' of •b which contains UsVsx for every s (E G, then Kx meets with IiÅrlt at itn;gue element sc4, and x'b, is 4he orthogqhal projection of x onto 04.

Let glS be W*-algebra generated by operators Us of a double unitary representation '

-

' {Lb,Us,Vs,S}• Then, .' . "

TrtEoREM (2)• Let .r be a boundecl.element in {ilÅr, dnd let K= be tlie least weaicl)r closecl conveu set in JJ7*-algelra fRS wltich centains operdtors UsU.eUs'1. T/ten Kx meets hoith center ER4 of SIS at tsniguq elenzent, cmd the elernent is Uxij. i: . .-

`DEFfts'TTToN '(4) Let Ib be a central positive'trpe ]Racloit nzeasure on LC ma'n'zodular group G. ancl Iet SIS be M'-algebra generatecl by Us{)b) in clbuble umitcz,)t representation {•b(l-),UsQb)i Vs(pÅr,S}. If glS ts finL'te alass, then we call the measure pa firzt'te class. '

DEFrNITIoN (5). if every ceiLtral positive type Rczdbn nzeasure.on LC t-nintorlular:tg7T.ottp G is fint'te cLczss, tiuin G is callecl a fint'te class group.

TffEoREM (3). LC uriimbdtilar group G is finite class group if twtcl enlor if tltere exists tlLe fumslanz7ntal s)tstene ol" bonvpact neigltbourhbocls ae unis e in G zvbic.h are invariqnt, bpt

ever)r inner antomerpltisms of G. /

N*..AI.GEERAS AND-FINrTE CLASS GROUPS '3

i-i- DE,FiisiJ'rToN (6). 'ij JIJr*-algebra generatecl by Us and Vs (s.G G) cire' irreducible,•t7te dbuble uindtcu)t representatiore {O,Us,Vs,S} is saicl irreducible. if central positive type IZEulen ineasnre defines tlte irreducible doubte 'u•nitary •representatton of G, tlten we cael'the Radbn measure character of groizp G.

THEoREiv (4). xct necessai st and sitfficient condition Lliat character #b is fi•nit.e class is that is htts elte form: dlL(t} =X{e)dt, "ivltere IX(e) iLs central posi.tive t)tpe.contlnuerts function deJiineol

,1

THEoREM (5).., LeS {•{År,UF,Vs,S} be a deuble unitai}r represeneati.on definecl b]e measure /b offinite class, A necessaiv cuul suf/flcient condition that pa is cltaracter ts that fo.lt ts

dimension 1.

r

,TH,EOIÅqEM,(6), A fini'te .class, groitp G has sttffi'cient mart)t chartncter i.e., tlteTe exists aliaracter X of G such thaq X(s) --L-" X(e) for each s =i: q.

Above Definitions and Theorems originate in R.Godement [4]. We 1iope that reader

see the ori.oinal as for :detailed verificationG etg.. i-

!- 3• N'-algebras. .r •

--/ -

IFL

DEFrNrToN I. A compleee normed *-algebra A tvith norm ll . Il ove) the complav number field is catleal iV"C-aigebra, of it ltas the operation ig ivith tlte fellotving properttes :

'

(a) the operation ig ts a continuotts lira?ai mdPpi,ng ofA bnto its center Aij, artcl is ' ideizlity on tlte center A#. • • tr

(b) if a E A is,'hermetian (resp. herniitian[positive), a4 ts also ttermitian (resp. Itemtitian pesitive); ivhere thas a is lterniition (resp. Iterntitituz positive) mean tiiac the carTespondiag

ee7g.!)OrtO.ain.COilil.iin)"ljq,.i`,S,(bO..Pe)'4"tO'li4e,P,',Fi,e,n,riatt.-9,nb,O(ff.fl,AIS,Jterntittan,(resp-•itermitianposieive)

" L pdrticalarlOhi (ab)ij=ta.bg "' ' for ev`)7J, it''e!'A4}' b E 'A.

:- ]- p[ t t- tL '- t' e] ll 't -- :] :: '- '

,A' simple ,examplg. ,of' N*-algebra is given ,by a suitable C*-algebra which is contained in,W*-glgebra of finite class, We,shall consider qn N*-algebra with some hnportance.

-- , in bounded algebrq iFI, of L:-,systeni H (ipr"the. sense df W. Aml)rose Il]) .of LC uilim-

odular grpup G? if we define La .and Ra such th4t / , : ,

: ,. ,:, L4x=iFzeut•Rax ;..va for gverya E.A,x Ei H,-

then Lq, and Ra are bounded pperatoTs on H,•,apd N}te have that ..

.. Lq.'=(La)', B, a'=ÅqRa)'• . -

Henceforth, fov the sake,of convenience of notations we rrtay use La,Ra insLead ofa E A,

tL, p.Syi if we pu,t , , . . ,, :.

• Ua 1!t= lll La ,Ill i= sgp.EH, " .. lle $i ll Lcar ll2=sup 11 a,v lh for every a+. E A,

llailr= lll Ra [ll =sup..eH,".ll,si 11 Rthxll:=sup ll -anII: for everÅrt a EI A.

then we have that 11aIIt== IIaIIr for eVery a t[E A.rT]ie reason is. that

ll a1!i= ln La lll = [ll (La)" gl i= Llll La*lll =sup ll-a'•x ll2=sup ll a'a 1le= ll allr

4 •- , ' K, 1-SVJI

(in the general ll .r*H= 11x1le). Then in algebTa Aiwe introduc'e a new norm ll.[1 suc}i

lla ll =Max{ ll alli == ll a i'r, 11 allt} for al1 a E A•

In that case we have clearly that . .-. i T,

(1) [l a] ll ll; O and llall = O if and on]y if a == O.

(2) ll a+b ll ;sS 11 a ll + ll b 11 •, (3) ll cta ]1'- 1 pt 1 ' ll a ll •

Then A is a Iinear norined space.

Secondly if {an} ( A is 1'1•am-an 11 --,tO (as m,n.o6), then-we have i

'

' ]l am-an Pl= lll La. '- La. lli .O (as m,n' co) (i)

ll ain-an il:,--O (as in,n--co) (ii)

and there exists a bounded operator L on H as uniforrzi limit of {Lah} by (i) and on the other }iand there exists a E H such that li an--all 2-O (as n.ee) by (ii). Then we

1iave

ll (La.i '-" L)b H2EIII ill Lan-L "1 . Il b l!e.O (as n 'J oo )' for all b (E A,

[i (an-a)l, iin-.-S ll an-all2• lllRblll.O (as n.co) for all b Ei A, - therefore, lve- },lave that Vi=ab for all b E H,•and a,E A and L.==La.

Consequently, ,

' . (4) A,is complete with respect .to norm 11.Il.

Then A is a Banach space. Moreover we have

(s) " ab ll lssM ax I : :2 il'E: Il : :ii ll bb ill] .K-. ll a [t . :l }, " foi every a,b E A-

Then A is a complete normed *--algebrat

We slial] nosv show that the algebre'A is N*-algebra. The definitien of elements in bounded algebra is equivalent to the d-efinition ef bounded elements of Hilbert space H

in Definition (2). If KP," is a convex set consisting of UsLaUs-1 (UsVsa) for all" s EE G, the clos"re K: of KP, with respect to norm]l,lll rneets with center A4 of A at the

unique element ah by Theorem (2) and weil-known J. Dixmier's Theorem. On the other hand it follows from Theorems (1) and (2) that the closuTe K: of K9 wi'th respect to

norm ll .Ii2 meets with the center A# at ' the tinique ' element any. Then sve know blearly that the closure Ka ef K: lh'th respect to norm 11 . Ii meets with tiie cent-er A" at th,e

L unique element alj. And it follows from Theorem (2) that the above ,defined mapping lt ef A to A4 satisfies all cenditions of Definition-1, Then A is N*-algebra.

N'evertheless it is'not alw'ayS that ceTiter Ah =l= {O} in general-

In order tliat AI =l= {O}, it is suercient that there exists at least one coi[rtpact neighbou- rhood of unit e of group G vihich ig invariant by every inner automorphisms in G. And i

in order that'we have the following:,

(3.1) If (e'a)k==O, then a=O for everyaE A, i - •'':

it is clearly suMcient that group G ig a finite class group. '

.

N*-ALGEBRAS AI:i,P I;1,.NITE CLASS GROVPS

5 RI•:rLvtARK. ,The autbo.r has,indicated FgLus'that H. Umegaki's report mentioned in Sl,

popl4ins. the folloxying Theoreips•: •.,` -.•'.,., . . , J..

L. t"

;.

L

,(i) •'"CTe have diat A4 =l= {O},,if and only if- there exists at least one compact neighbour-, hood of e which is jnvariutlt by every, inner automorphisms in grotip G. (weak centering).

,,(ii) We have the'formular '(3.1), if and, only if G is a finite elass group. (centering)•

:.

Henceforth }ve shall,mainly consider tlie,case which G is finite class group. Now sve supplement ideal unit 1 in iN'-algebra A of finite qlass .qToup G and call briefiy it finite class N`-algebra and desplay by 9.[. Then .general element a of 9vt has the form a == al

+ aiwherc at is compiex number and at E. A, and its algebraic operatiens are defined l)..y,the natural.,ways• MgreosreF its norm is llall = 11 ct1+ai ll =1al+ Ila' 11, and it is

11 mapped to a4=ct]+atk by the operation h. '

4. The relation between maximal two-sided ideals o'f finite class N*-algebra

'• :' and its characters. ' • .• ' '-

Now a trace of finite class ilg*-algcbra QJI is central positive continuous'lihea"r functional a l.e,, -

cr(ab)==cr(ba), a(a'a);.llP , foT eveTy a,b (! a(• , ,, t-

' Then in the first place we have the following:

i . -t 1.- --

-

LE:mfA 1. A jCXnite class IV'eC-algebra 9.( has tlte cemLpleee s}rstem of traces.

i

'

PRool". Let a be a continuous .positive Iinear functional on SIlt. Then in virtue of '

Definition I if we extend c te al1 9.t by a(a)=ff(aly) for every a E el, then it is clear that cr('a) is.a trace of Åí!(; Conversely ]we shall prove that this process gives every trace

defined bn E}I: As•ttace cr 'is•eentinuous,,if-we have that . , 11 Z"a:LUsi Vsta-alt ll r•h'.:"6 (aiÅrO,i ]E] cts=1, sE E G),, ' - . '

then iVe have ' '''' ' ' "'' '' i` i'i : ,

. tr-

--

]a(= cti Usi VsE a) --a(ab)1Åq E. ''' L' '''! ' i

'

.-

But by centrality of frace e-, it is invariant with respect to' everY'inner automorphisms in G. Therefore we have that• .l•aÅqa)-•cr(a#)iLi-. E, and/that o'(a)=ir(aij). - • i' Now we suppose that cr(a'a);O fo.r qrbi,tvpry fixed element a E 9.t and all trace c.

'

Then ifaLive introduce hermitian positive b==(a*a)h, we have that cr(b)=O for all conti•

ppous positive lipear functio.nal a on 9v,r" by tl}e. qbove-rpentioped fa, ct.. Therefore the spectre of Lb is O and b==O. But finite ciass N'-algebr'a 9J has propertY (3.i). Then '

'

a=O. Therefore Dt has the complete systern of traces• Q.E.D.

.t

-

Considering proof of Lemrna 1, continuous'central linear functionals of finite class TiSl*-

a;gebra 9.( are determined by •their values on center ?1lj• L"Therefore the set of continuous

ceutral linear functionals on 9.( coincides with the set of continuous 1inear functionals'on

commutative algebra QIij. Tlien we can utilize results obtained for commutative algebrat

Particularly ?(4 is isomorphic in a)gebra C(SL) of continuous functions which are defined

6 ' - '- K TSUjt '• ' t-

on a suitable compact space a (spectrum of VIn), and" point pa Ea is homornorphism a

-p' (ai) of 9.(g onto the complex nllmber field. And that is one tvhich call chataCter of commutative algebra Plij. Now we call every continuous central linear functional p on

?t which coincide with character of P(lt on 9.(4 eliaracter'of finite c]ass' N*--algebra 9J. Then

there exists one-to-one correspondence betsveen Chavacters of'?( and chm'acters of' its c' enter S?.Cij i.e., peints of .O- •And charaeter•of 9J is continuo,us positive central linear functional

on 9.r i;e., trace of A, which is characterized by that '

IL(ab)=:t-(a,)pt.(b) for every a,b E Ptn.

In 'fact it has only stronger property, that is: "

- LEMbftx 2. A trace ti of .ifinite cltzss IV"`-algebra st;I is a cl!aracter, tf and only Llie ts

satisLIVes tJtat

ib(ab) =p(a)l-(b) for every a E 9J4, b E P.L

PI{ooi--. It is clear that die condition is sufficient. Let pa be a character of sti(. Then

by Definitionl(c), we have ,

li (bb) = )-((ab) #) '= = tb (a.b#' ) == pa (a) IL(b ij ) = A• (a) IL(b) for every a e 9.1 lt , b ( 9J.

Q.,E.D. -- .tt t

And the folloiving Lemmas'aria Theorem are proved. ' ,, , 1- '

Li•;rfNtftN 3. Eveir t,race oT offini,te class N*--algebra corresLponcls ivilh tmigtte regular Borel nre(rstire fi) on the compacc space .ÅqL of cltaracters 'of e(, itnd we liave

' a(a) =J-. p(a)dol(pa) for atl ae ?•L

- '

PRoor By continuity and centrality of ff, it is determined by its values on "J4. And compa'ct space SL is simultaneously the spectrum of SU,i and at# is isomorphic to algebra C(.q). Then ff considered on atU gives a linear funetional of C(S)), and if we fix a E Z!l#, then p(a) is continuous function on S]t. Therefore by Riesz-Mm'koff-Kaktttani's Theorem, a corresponds with'unique regular Borel measure o on a such that

a(a)=a[pa(a)I=: !Rpi(,a)dtu(pa) for all a E 2V[4.

Then by the mentiori in the first part of proof, we obtain that

.(.)=f-.p(a)dto(pa) fot alla E! E!f' , Q.E.D.

LEMbfA 4. Let K be a compact convex set consistea of traces a sttcls tfi(zt ff(1) -$ 1.

Tlteh non-nuU eL'sreme points of K are cltaracters of 2(.

' -

PRooit. It follows immediately from Lemma3., Q•E-D•

TH-oRErvf I. In finz'ee'class N*-algebra ?(, tltere e.fuises one-to•-one correspondence among,

(a) maxi'maZ two-sftleci ideals of Åíe[; i

(b)r cltaracters of Åí!I; ' ' '

(c) ma`Mimal ideals of center ?I4 ,of E}I. •

N*-ALGERRAS AIhjD FTNITE CLASS GROUPS 7

.Tltenltla's correspondence ts tlte ftitlmving: elte maxima'l tivo]-sided ideal •ivliicli chatacter rv bf 9.( clefines i•s given b)' egttatio,i p(a'a) = O; elte maxinial ideal of st.k ivltielt i•s ca3fined b)t

nn-imal ttvo-sidecl ideal ni of Qt is ,m A 9.(4. , , -

PtÅqc?o• 1?. Proof Qf, this, Thep,repi is almost similar .in proof of [R. Godement 4, Lemma 151, but we give it for convenience of readers.

(i): Cori'espondence betsveen (a) and (c). '' l'

,Let m be a mELyimal two-sided ide41Lof .9J. Then we'.shall consider in4. As we have thet (ab)k==a,.be anq/b:E m for every, a;E:i "el4, b4 E mij, then ab E m therefore qfb4 (E nr4. Similqrly ,sve have •bij.a E nilt. Then mh is a non-null ideal of QII. '•Now

putti ng th at - ': , . •. • .., rt

, , FlniF{al(ab)g E ,m# for every b E 9LI},

i.t is clear nlt contains, ni4 and m.and nit i=l= E}[-for in4 =l= 9J4. Then nii is two•sided ideal of 9.(. But pi is max'imal, tlierefore we ,have that m=mi. Therefore inig ( m A 9Jli, and eonvetseLy sve know tliat m A PI4 C-.InU because the opera.tion h is identity on 9.t4. Then we obtains that m4=m A E)I#. As,,above mentiomed we know that maximal two-sided:ideal m.of,P-( is characterized by the follosving:

(4.Z) nT={ai(ab)it E m4 for every b (E! ?r},

Therefore different maximal two-.sided ideals of 9.( ineet with different ideals of 9y(l.

Now we shall prove that mij is maximal ideal of 9.ln. "lf it is not so, then mlt is

. contained in mat imal' ' ideal ' n of 9J 4'. ' Put ting th ti t'

(4.2) m'={al(ab)# (E n for ever)T b Ei gL },'

' then m/ is non•trivial two-sided.ideal of st.{, and contains ln and n. But lt differs from :n4. This contradicts to that nr is maxirnal. Therefore :n# must be maximal.

Conversely if n is `maxinial'ideal of et4I itheri '"it is clear that nii d'efined by (4.2) is maximal two-sided-ideal of SU shch that in' A, 9-1# Å}' il. : ,, 'i ' '

Llt .-' J r.

'

(ii): Correspondence between (a) and (b). ,

+, 1 1: '1

If m is maximal two-sided ideal of 9J, then by (i) there exists a corresponding maximal ideal mij'6f`9JU. Then natura!ly there existsi unique charactei pa of 9Jlt such that a'Ei mig is ,equivalent to ,p(a) = O. If we extend .pa to character ef 9.I,.then we know by . (4.•1).that .i-Ji

(4.3) ' m={a ,l pa {( ab) ig] = p(ab) ;O for every• b E QI }.

In virtue of Cauehy•Sehwarz's Inequality, we obtnins that

(4.4) m ={a]pa(a'a)==O}.

-t

/t

Conversely let pa be chardeter' of S2(, ana xve consiaer the two•sided ideal m defined by (4.4).'-: Clearly that is also rdefined by (4.3)• Therefore if we introduce the ideal n of

2![ij stEtch that, .-, .- ,. -• , ,, '

nny={aIp(a)=O for a E "Jlj}, then we havc that

i,n ={al(ab)aj E'u for every b E D,1}.

On the other hand since tb is character of Vln, IT i.s maxiinal ideaL- Therefore bY'(i)"'m

must be muximal two-sided ideal of 2(. •L • Q•E•D.

CoiÅqoLLARy. Fim'te class IV*-algebra st.( is senti•simPle.

PRoo]". 'lt fotibws iminediutely from Lemma1and above Theorem. Q.E..D.

r

4:

5. Applieations.

Now mod==A is clearly a maximal two-sidea ''ideal of- finite class N*•algebra st.{. The

c]iaracter l- ss,1- iich correspeDds with a maximal two•sided ideal of 9.I, -different firom moo, has the forM dpa(t)=:X(t)dt svhere X(t) is a continuous central positive type ftrnction on G, as the continuous central positive linear funetional of algebra 2 (CÅí[4,I). Then we shall denote by X the set of continu'ous' c"e'ntr'al "positive type functions X(t) on fmite class group G whic}i coTTespond wiih maximat/ two-sided ideals of finite class N*•a]gebra 9.f, aifferent from m•6). If we'identify corresponcling functions with niaximal two-sided ideals, thent X• is tOpQlegiZed•as the set of ma,icimal two-sided ideals of YI. Then X coincides with . 1,

S}-nice, andXbecomesaloeallY 'cbtnPaCt space. '

Now if p EEi n -- mco, as a continuotiE centrhl positive linear functierial of'2, has the

fo Tm dli (t) == X(t)dt, th en we have th at -L ' `' ' :' .

:- r-

. , 'lb(a)=J(,a(t)X(L)4,t, fgr everyaE 2. I

J '

. Since if we fix a E 2 in "(a), then pa(a)iis a continuous function on -("?,-111oo, i.e,, on

X, we shall put lt(a)=a(X) for eyery a.--E Åí. Then we have ,

'

- a(X) =fGa(t)X(t)dt for everya Eii Åí. J -

tI

-

- -t -.

THEoR!•;M ll. (Unigueness of Feun'er.Transforms) if a,b E 8 nd -•

i

-.

J'Ga(t)x(t)dt== J7Gb(t)x(t)dt :fer eve,pt x Ei x, ' .

1- then as elements of ut a= b.

-t

' ' '

-PRool;. It follows from Corollary of Theorem I. QE•D• .

THEeREM M. (Bochner's Tsrpe Theorem) if trace ff oflinite class IV*-algebra 9J has the fornz cla(t)==rp(t)tlt, as linear fanctional on Åí, where {p(t) is a contintteus central positive t)'pe ftmction en finz'te class g7-oup G, then {p(t) inapt be represertied in tlte form: : rp (t) == S . X(t)dto(x), i ' - J, i

tvhere to(x) ts regultzr Borel measure en localZ7 compact space X. , , - PRoolr. BY Lemrna 3, ff cerresponds with uniguq regular Borel measure w on S). 'such

that a(a)=fGIL(a)dto(p) for eyery a,{{E 9L{: But as lhrr:.(a) =O for every a E 2,

if we identify 9-nTee with X, then we have ,, ,

if((e)=Sxa(X)4,to(x.-År . for every aE 2. - .

N*-ALGEBRA.S AND Frt"JTE CLASS GROVPS. 9

On the other hand sinee we have that

oL(a)=J'Ga(t)qJ(e)dt for eveiy rL (E Åí, then we o}}tain tliat

' fc}a(e)r/)(e)dt == f; a(X)d(o(x)

= fxiJ"Ga('t)X(t)dticl(e(X) for every a E fLt.

Therefore using the Fubini's Theorem, we have that

fGa(t)q)(t)de == fGa(L) if.)c(s)dtu(pk:)]dt for every a Ei El.

Then by the continuity of g)(t) we obtain that

q,(t)=J.x(t)d(o(X)• Q.F.D.

References

Ill W. Ambrose, The L?-system of a unimodular group I, Trans. Amer. Math. Soc., 65 (194•9), pp. 26-48.

I2] J. DLwmier, Les anneaus d'operateurs de clas$e finite, Ann. Sci. Ecole iNorni. Sup., 66 (1949), pp. 209-261.

I3] R. Godernent, Analyse harmonique dans Ies groupes cenn'aux I, C. R. Paris 225 (1947), pp. 19t21.

I4] R.Godement, Memoire sur la theorie des caracteres dans les greupes localement cempact unimodulaire, J. de Math. pures et appleques, 30 (1951), pp. 1-110.

[5] Y. Misonou-M.Nakamura, Centering ef an eperator algebra, Tolieku .Math. J., 3 (!951), pp. 243-24,8.

f6] I. E. Segal, The group algebra of a Ioc.ally compact .qToup, Trans. Amer. A'rath. Soc., 61 (1947), pp. 69-!05.

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