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On Left Multiplicative Operators oll H Quasi-Unitary AIgebra
By Eishi HoixTGo
(Rcceived riNlovernber 15, 1958)
1. Introdllction
This paper is a( continuation of previous papers by the author [1] [2]i), which was devoted to the connection between quasi-unitary algebras with semi-finite Ieft rings and unitary algebras, a generalization of a theorem concerning the relationships between complete quasi-unitary algebras and H'k-algebras. We use the terminology of that papers without further comment.
According to the previous paper [1], for any quasi-unitary algebra with a semi-finite left ring there exists a unitary algebra with the same involution in such manner that the underlying set of the unitary algebra is dense in the Hilbert space obtained by the completion of the quasi-unitary algebra. Therefore some problems on a quasi-unitary al- gebra "rith a semi-finite left ring are convertible to corresponding problems on a unitary algebra assotiated to this quasi-unitary algebra. According to the paper [2], Hilbert spaces obtained by the completion ofa quasi-unitary algebra, the associared unitary al- gebra and the unitary algebra mT}' arc closely related to ene anothcr with unitary mappings. Therefore some problems concerning the topological structure of one algebra are convertible to corresponding problems of the other algebras.
The purpose of the present paper is to make a further study of a quasi-unitary al- gebra with a semi-finite left ring by means of the above mentioned manner.
2. Preliminaries
Let R be a quasi-unitary algebra and let S2R be the Hilbert space obtained by the completion of R. The left ring RgofR is the weakly closed self-adjoint algebra of bounded linear operators on aj)R generated by the setof all left multiplicative operaters by the ele- ments of K A necessary and sufficient condition for the semi-finiteness of the left ring Rg is the representability of the minimal closed extensien J of the automorphism i of R in the form [tvrn,Ipi], where n•r is a positive, self-adjoint and non-singular operator beleng- ing to Rg, and where M=SM'S (S is the continuation of the involution s ofR over the space libR). Now let R be a semi-finite left ring, then there exists the maximal extension op of the corresponding canonical trace defined on the strongly closed tssto-sided ideal ln of Rg . It is easily seen that the set ml formed by thc elements TE R"r for which T*TEm becornes a unitary algebra with the inner product (Ti,T:)i= q)(Ti T2*) and with the involu-
1 Numbers in brackets refer to the referenccs at thc end of this papcr.
19
tion T--)-T*. We denotc this algebra again by the same notation m'l. Now let I:)LclEK
(ETk Ei RiD be the spectral representation of A•I', and we denote by Ri the *-subalgebra of Rg consisting of the operators .ISC"E ml", for which there e.xists a pi'ojection E(A):=:: EA,-Eh,, A=(Ni, ,N2), OÅq]'LiÅqÅrtL2, "Tith X=E(A)X=E`]Y'1-(A). For such an interval .ttt we say that it cent/ains the eperators X, and we denote by Rn the totality of operators ofR] contained by A. Ri becomes a quasi-unitary algebra with the fo11owing definitions:
(x, y)=(p(Åqxn•r-i(A))(yn•r--i(A))a:), xs =. M' -i (zt)x*n•f,(A) ,
.X ' =Mr(A) .:YtlM"i (A) ,
where M'n(A)=I., X"dE'A. The left ring Rt' is *-isomorphic svith the left ring RS [3]. Now
we introduce in the underlying setef Ri an inner preduct with the relation ÅqX, YÅr
=(p(XYS), then the set becomes a unitary algebra with the involution X.XS [1]. We denete this unitary aEgebra by R2. The connection between the i: ner productofRi and ,the inner product of R2 is given by the follosving relatien:
ÅqX, Y)=q}(XAdir":(A)Y*Mr(A))
= q, (M' l` (A) JL'M' i-' (A)M'-2(A)M' l (A) Y'AI' // (zt)) =(Milf(A)xMtlt(A), MX].t(A) YAf,)-s(A)) .
Then the extension of the mapping X--ÅrM'i`(A)XM'lf(A) is a unitary mapping of !ÅrR, on
,X')"
R,. We denote by Rl and RS images ofRi andR2 respectively uncler the coriespon- dence U.-a, and we introduce in [hese sets the same algebraic and topological structure as Ri and R2. Then Rl and R:- become a quasi-unitary algebra and a unitary algebra respectively. The underlying setof the quasi-unitaryalgebra Rl isdense in [SbR, and the IÅÄIilbert space tS.')' Rg coincide with the Hilbert space ,S5R.
,i..,k:il,':S-;:i-'",Oi2dE?P.el)aj,atl.2.,e}7."g'riiÅí['i.E•:.a-.-"?,iSl.I{:,V.".,b;il';O.il;E.e:'?i2"a2:
note by Lb the totality of operators 7' belonging to Rg for which T. E nl" and lim q)(T.'T.) exists. Li becemes a Hllbert space with the addition Ti+7'2=:[Ti+T2], and with the inner product (IT", T)=lim qp(T',T.'). We call the elements of LS square integrable with respect to rp.
3. I{esults
In this section we shall prove well known theorems concerning the representability of an operator to a left multiplicative form and give some results concerning the left bound- edness of an element and the spuare integrability of an operater.
The connectien between the inner prednct ef Rt and the inner prodnct of nTe is given by the following eqnality:
On Left Multiplicatk'c Operators on a Quasi-Unitary Atgebra 21
(X, Y) ==q,(-XTnr-"i(A)(XMr-i(A))*)== (XM'-'i(A), YL4ItFi(A)), .
Now "Te censider the mapping X-Xill"-i(AÅr, this mapping is evidently isometric. Since R, sa a linear set is dense in tbRi, and, as it is easily seen, so is true in the space bj)n:l, and Sinee the rnapping carries the set Ri on the set Ri, it ca" be e: tended toa untiary rnapp-
ing P(i bet"'een cS;)R, ancl tS'.:"')ntdi. The inverse rnapping )L":'i is cvidently the extension ef the MTahPeP/LnogntrcTlr/6Yn'iibfe(t'td,)een the inner product of R2 and the inner product of ni'}' is given
by the foIIesving relation:
ÅqY, XÅr == op(XMr '-i(ri) Y*M,(A)) = tp (M' l` (.! t) XM' :' (z() (M'l(A) VM' l' (A))') = (Af'i'(A)xty'-i'(A), ji'i`(A)yJv'-i(A)).
The extension of the mapping X-M']-f(A)X.4f'-/i(A) is evidently a unitary mapping X2 between tS1ft. and rS:1:"t. The inverse rnapping "rill be obtainecl by the extension of the rnapping X'--,AI'-"1(A).XTAI']-'(A).
Now we prove sorne theorems obtained by Pukanszky [3].
THEOREM 1. Let R be a qttasi•-rmitar), al..aebra zvitie a serni-Ltinite tcft n'n.e RS. Su/]Pose that.T =[lll'lttr'-i], and let cp be tite ma.vimal estension of titR correspondin.e canonical tr.ace. DeVTne
aq,):b',.i'til,/1;c:c]/L,l/d't,i7i,in,"iS;f':4,IC;s'g"),iili)liE7:j/(,t,iii]Iib.ecr.'bjY,iaSeofe.,J,e/]';t,nfiic:deia,l,1:,"ii}",jl'iiOl,iS/Z/,",o,l:'i:ei'sa:t,:fl#2ij,'lii)ll'ii/C,t(X,etii,
tLt" and onll TY' the ctosed oPerater [T.nl'] c.k'ists, and then U. =[ T.tl'I'].
PRoOF. Let{X.} beasuitable sequence in Rl which converges to aE bj-)R{.
From the unitarity of the mapping Yn the sequence {ilr(.r.)} converges to T. in the rnetric of •g)mb. From the fact that the rnapping :,NCTi of t15nib on tY')"n, is unitar'y and from "-i(ri.) where A. is an interv:1 which contains the operator U.., the11',"11111i'i•IIIfZ'[li"i2'1('ilei,i(it.2,'l/,,E',E,/Ve,g,,;,iO,Åé.;l/:ii.aco,/kt,O,z,'si'/lp8p,',,",,ti!f.,l,ifi,,116i.i,rg/F,"l,11,;iS,i",ve`,/'.','iLt,t,Yll?7ztAY'l/i•
a2",;'2',g?•;'P,e'ld,'L6Czng,'S':,g."9tge.",.X:li,(Z",',ili(I.]a.2Tk.{?3iOS.";,IE,O,M,,tlteirl•k`1',Z.rne,?l,i"..gf
/Z'rl,oniligh,,liei,/r,ITi",f,ill'i'i,gliji,:o:,$,aliOil'l'lili"llill,e.itiiiolltilke,ilj'tiii,:tli•ilAe.:iini,iii//Md•;i,mlll:,i3/fils]o]'/G',111a/Fs.ii/";,iljilk,il:11ilru,ii
cornpleted.
THEOREM 2. An oPerator T ofnTl' is oftheform U., where a is aleft betmded element
and in the domain of n•1, i( and onlJv ij'f' the minimal closed extcrisi'on of tlte ol)crator TAI'-Hi e.xists and is settare iritegrable acitlt resPcct to g).
PROOE Suppose that the minimal clesed extension of the operator TAI'Vi exists, and is square integrable with respect to g,. Then there exists an element aE tSbn; which correspond to [TAI'-'i] with the unitary mapping Vn. It is easily seen that the operator T is the minimal elosed extension of [TTM"']M'. TheTefore according to the above theo- rem the closed operator T is of the form U.. It fo11ows from the bouncledness of the oper- ator TEiul that theclcmenta is left bounded. The fact that the elementa is in the domain of ?if follows immediately from the inequality !e?lf(A.)a]l2.f.l;. r]p(U.U.'). Conversely, suppose that T= U., where a is a left bounded element and is in the dornain ofnf. iir(a) is an operator of Li, and [Xi(U.)]=: iln(a), therefore the rninirnal closed extension of U.n'r-' exists and coincide with Vp(a). The underlying set of the quasi-unitary algebra Rl is dense in foR and both left rings (R'ig and Rg) coincide, therefore the theorem is true for the quasi-unitary algebra R. Thus the proof is complcted.
Now we have the follo",ing theorem:
T
THEOREM 3. Lel 7' be an operator of nTLf, titen the minimal ctosed extension of the oPtrator TA•f'-i exists z;f and onl7 t:]F' tlte element b tttith X2-i(T) == Ub is a bounded elcment in tbR;.
PRoOF. Accerding to the theorern 2 the minimal closed extension of the operator TA:T'-' exists if and only if T= U., where a is a left bounded element and is in the domain of M. Now "'e have the following equality:
I1 Ua Ur ii 'ny'- ]1]A':1" - " UaAf' l'.rltt" - l` Ui ntt' - IFl11,
where ll •ll and Ilr•]li are norms defined by the inner procluct S;r}R, and ,bR, respectively.
Since U.Eml', X2-i(U.)=7tf'-J-rU.lt't'ti' is square in[egrable with respect to the canonical trace defined on the left ring Rg. Therefore Xi'(U.) is of the form Ub, where b is an ele- ment of tPRf. Frem the inequality
Pl U.U.Ii;slg k il U.If == k Il]iiif'-i' u.nct'-':m ssre have
m uonf' -ij'-i u.n•I, - '•'-s tili:s k lrmi"- tei u.nf' m ]t ae] .
Therefore Ub is a bounded operator in bj)RI. The converse of this asserition ctin be proved in the same manner. Thus the proof is completed.
References
[1] E,HoNGo, On quasi-unitary algebraswith semi-finite left rings Bull. KyushuInst, Tech. 3(1957), PP. 1--1O.
[2] -, On sernc preperties on quasi-unitary algcbras, Bull. Kyushu Inst, Tech., 4 (1958), pp.IT-6, [3] L. PuKANszKy, On the theory ofquasi-unitary algebras, Acta Univ. Szegcd, 16(1955), pp. 103-121.
Kyushu Institute of Technology
