A STRUCTURE THEORY FOR SEMI-FINITE QUASI-U[NITARY ALGEBRAS

A STRUCTURE THEORY FOR

SEMI-FINITE QUASI-U[NITARY ALGEBRAS

By Eishi HoNGo

(Received November 23, 1960)

l. Introduction

The concept of the unitary algebra has been introduced by R. Gedement , [4] as an abstraction of group algebras on unimodular loeally compact topolo- gical groups. A generalization of the concept, that is, an abstraetion of the case that unimodularity is removed, has been formulated by J. Dixmier [2]

as the quasi-unitary algebra, and the theory has been developed by L. Pukanszky [7] and others under the restriction that the left ring is of semi-finite class.

In the previous paper [5] the author has proved for every semi-finite quasi- unitary algebra that there exists a quasi-unitary algebra, which is dense in the completion of original one, and which can be renormed in such a way that it be- comes a unitary algebra. Then many problems on semi-finite quasi-unitary algebras are reduced to the cace of unitary algebras. Recently, J. H. Schue [8]

has introduced the eoneept of the L" algebra which is a genralization of the H"

algecra in the,viewpoint of the Lie algebra with emphasis on the infinite di- mensional one, and has obtained the Cartan decomposition for semi-simple alge- bras, and a classification for simple algebras. In the theory of finite dimensional semi-simple Lie algebras, a seemingly indispensable tool is a symmetrie bilinear form, the Killing form, which is non-digenerate on them, and for L' algebras the connecting property

([x, or], z) ==r (pt, [x*, z])

plays a crucial role instead of the Killing form. As is easily seen, every unitary algebra holds the same connecting property with respect to the ordinary Lie pro- duet (commutator) [x, or] ==xy --yx.

The purpose of the paper is to investigate the structure of semi-finite quasi- unitary algebras in the framework of eomplex Lie algebras.

2. Preliminaries

Definitions and known results concerning quasi-unitary algebras and L"

algebras are presented in this section, and the treated materials are limited to

1

those which are necessary for later sections.

(1). Quasi-unitary algebras [2], [7].

A quasi-unitary algebra is an associative algebra R over the complex num- ber field, which is a pre-Hilbert space with an inner product, and which has an involutive anti-automorphism s and an automorphism J' satisfying the following conditions :

(i) (xS) xS)=. (x) x)) (ii) (x, Xj) l}lll; O,

(iii) (xy, z)=(or, xSiz),

(iv) the left multiplicationy--Åray is continuous for fixed x,

(v) the linear combinations of the elements of the form ay+(xy)j are dense in R.

It follows from the definition that the involution (involutive anti-automor- phism) s is conjugate unitary ((x", yS) == (or, x)), and the right multiplication pt-Åryx is continuous for fixed x.

(2). Unitary algebras [4].

A unitary algebra R is a quasi-unitary algebra with paj---x.

(3). Left rings [2], [5], [7].

Let R be a quasi-unitary algebra and let tfoR be the Hilbert space obtained by the completion of R. Then there exists a bounded operator U. (resp. V.) (x cR) on {foR satisfying U.y==ay (resp. V. y =yx) for everyy(R. The left ring Rg (resp. right ring Rd) of R is the weakly closed self-adjoint algebra of bounded linear operators on &R generated by the set of all operators U. (resp. V.).

The minimal closed extentionJ of the automorphism i in •foR is a positive, self-adjoint and inversible operator, and let S be the continuation of the involu- tion s over {bR, then J'i = SJS.

An element a6foR is said to be left bounded (resp. right bounded) if there exists a bounded operator U. (resp. V.) satisfying U. x-- V. a (resp. V. x = U. a) for every x E R. If R is a unitary algebra, then x E 8.)R is left bounded if any only if it is right bounded, therefore such element is merely said to be boundeel. The set Rb of all bounded elements is a unitary algebra containing the original R, and it is called the bounded algebra of R.

Let IV be a ring of operators, then a projection P( IV is said to be finite if

there exists no partial isometry VEN with V*V==P, V*V=QÅqP. The ring IV

is called finite if every projection Pc N is a finite projection, and is called semi- .finite if every projection P E IV contains a finite projection. A necessary and suf-

ficient condition that the left ring Rg be semi•-finite is that J is the minimal closed

extension of the operator M'M-i ,where M' is a positive, self-adjoint and inver-

AStructure Theory for Semi-Finite Q;uasi-Unitary Algebras 3

sibleoperator belonging to Rg in the sense of von Neumann, and M'==SMS. If Rg is semi-finite, then R is said to be semi-finite.

(4). Semi-finite quasi-unirary algebras [5], [7].

Let R be a semi-finite quasi-unitary algebra, then the automorphism J is re- ' presentable in the form 1=[M'M-i] ([ ] is the minimal closed extension pro- vided it exists), and there exists a canonical trace g on Rg defined on a strongly closed two-sided ideal in of Rg. mg denotes the two-sided ideal formed by the elements TERg for which T'TEm. Let I;dEK (EKERg) be the spectral repre- sentation of M', and let Ri (mg be the "-subalgebra of Rg consisting of all oper- ators XE in"`, for whieh there exists a projection E'(A)---E'x, -E'x,, A -- ()ui, ÅrL2), OÅqNiÅqN2, with X=E'(A)X=XE'(A). For X, YE Ri it is evident that there exists an interval A satisfying the above condition. Now let RA be the set of all ope- rators in Ri satisfying the above condition for a fixed interval A. Then Ri be- comes a semi-finite quasi-unitary algebra with respect to the following opera- tions:

(x, y)=op((xM'-' (A))(YM"i (A))") for X, YERA,

xJ == M'(A)XM"i (A) for XE RA, xS .= M'-i (A) x*Mr(A) for XE Rzi,

where M'"(A) =j.NndEK. The image of Ri under the isomorphic and is-

ometric mapping U.-Åra is a dense subset of foR, and the image is a quasi-unitary algebra with respect to the extended oparations and the inner product of the ori- ginal algebra. The set Ri beeomes a unitary algebra with respect to the involu-•

tion S and an inner product defined by ÅqX, YÅr== q(x yS), 'and this algebra is denoted by R2.

The quasi-unitary algebra R is a continuation of R', if R' is a subalgebra of R, the inner product, the automorphism and the involutive anti-automorphism in R' are the restriction of the corresponding notions in R, and finally R' is dense in R. R is said to be maximal, if it has no proper continuation. Every quasi-unitary algebra is containd in a uniquely determined maximal algebra.

In paticular, if R is a unitary algebra then the bounded algebra Rb is the maxi- mal algebra containing R.

(5). Boundedalgebras[9].

Let R be a unitary algebra and let Rb be its bunded algebra. An (two-

E. HONGO

sided) ideal of R is a linear subset I satisfying (i) xIor(I for every x,y(R,

(ii) I is selfd•adjoint, (iii) I is closed in R.

If R is a maximal algebra, then the condition (ii) is superfluous, beeause of the maximal property of the algebra, and this definition and the usual definition of (two-sided) closed ideal completely correspond to each other. An ideal I is said to be simple if it contains no ideals other than {O} and I itself, and said to be puTely non-simple if it eontains no simple ideals other than {O}. The center of R

--

is commutative algebra Z of operators definen by Z---RgARd. Then R is simple if and only if the center Z consists of only the constant multiples of the identity operator I, and R is purely non-simple if and only if Z contains no minimal pro-

• jection operators.

Let Z be the center of the bounded algebra Rb, then there exists a family of mutually orthogonal projection operators {PA} in Z with the following pro- perties :

(i) PARb is an ideal of Rb,

(ii) PxRb is a maximal unitary algebra, (iii) PKRb is simple or purely non-simple, (iv) Rb is the direct sum of all P"Rb.

Now leV2{ be a purely non-simple maximal unitary algebra. If the underlying pre-Hilbert space of at is separable, then there exists for each tc [O, 1] such a maximal simple unitary algebra ut(t) that ut is isomorphic to the generalized direct sum (in the sense of von Neumann) of ut(t), namely ut-vut(t). Thus in the separable case, many problems on the bounded algebra are reducible to corres- ponding probrems on maximal simple unitary algebras.

(6). L' algebras [8].

An L' algebra is a Lie algebra L over the complex number field such that the vector space of L is a Hilbert space and for each x c L there is an x" E L with the connecting property

([x, or], z) == (pt) [x*7 2])

for all y, zEL. A Cartan subalgebra H of the semi-simple L* algebra L (L--- [L, L]) is defined as a maximal self-adjoint abelian subalgebra. If follows from the definition that

(i) H is neeessarily closed,

(ii) H is not only a maximal self-adjoint abelian subalgebra, but also a maximal abelian subalgebra,

(iii) L== H+ [H, L].

AStructur'e Theory for Scmi-Finite Quasi-Unitary Algebras 5

3. Simple maximal unitary algebras

This section is devoted to the study of some spacial class of infinite dimensi- onal Lie algebras defined on simple maximal unitary algebras with respect to the ordinary Lie product [x, pt]---xy-ptx. Every unitary algebra is contained densely in a maximal unitary algebra, and in the separable case every maximal unitary algebra can be decomposed into the generalized diret sum of simple maximal unitary algebras. Therefore, in the separable case, the study of unitary algebras is redueible to the study of simple maximal unitary algebras, so we shall begin with the study of the Lie algebra defined on a simple maximal unitary algebra.

Let uti be a separable maximal unitary algebra, and let ut be the set whose elements are consisted of all the elements of eq doubly indexed by integers, na- mely the matrices (fij) with 7. IIfi,•li2Åqoo. Then ut is alinear space with res- e] pect to the following operations:

(i) ct (fii)+B(gij) =(afi,•+Bgij) for any complex numbers a, B, (ii) Åq(fi,•), (gij)År == 2Åqfij, gijÅr•

iJ'

Now let pto be the set of all such elements a--(aij) of or that for any choice of

finitely many integers J'i, j2,•••J'k, and any xi, x2,••-, xk E 9Ji

kk k

ÅrÅíl ll 12] aimjn xn li 2 l:S; eÅrt2 ÅrS li x. Ii 2,

mml n:ml mpt1

where 7 is a constant and depends only on a. Then uto beeomes a maximal uni- - tary algebra with respeet to the following operations:

oo (iii) (aij) (bij) = (2aikbkj), th =. 1 (iV) (aij)S == (aijS).

If the original uti is simple then uto is simple too.

After the prototype of H" algebras [1], O. Takenouchi [9] remarked that any simple maximal unitary algebra can be brought in the above form, futher- more the original algebra can be chosen as a simple algebra with an identity element. In other words, any simple maximal unitary algebra is the total matrix algebra over a suitably ehosen simple maximal unitary algebra with an' identity element. In the same manner, uti is the total matrix algebra over a suitably chosen maximal simple unitary algebra ut2 with an identity element.

Since uti contains an identity element, it must be a matrix algebra of fininte order. Then there is a sequenee of matrix algebras

uto(==R), ut1, ut2,''', utn,'''

where ati(i=1, 2,•••) is a total matrix algebra of finite order

Let D(PIo) be the set of all diagonal matrices of ?{o, and let ai(e) be the diago- nal matrix such that the i-th diagonal entry is the identity element of g2{i, and the other entries are all zero, namely

di(e)= (F O•..o oX'

i) e

o

o '•.

K ••.

Now we shall denote by {(p,(i;, 1')} (i, 1', k, =1, 2, •••) the orthonomal basis of ?(o defined by

j v (Pk(i, i) ==

i)

o -•-•-- ' o •--•-•-

O •••••• O (Pk O "'

where {(Ak} (k=:1, 2, •••) is an orthonomal basis of ?.ti.

Let R be a unitary algebra, then R becomes a Lie algebra with respect to the ordinary Lie produet

[x, y] == ay-orx.

We shall denote this algebra by L(R). In this case the connecting property is written by the inner product Åq,År as follows:

Åq[x) pt]) zÅr =Åqor) [xS) z]År.

Let I be a closed Lie ideal of the Lie algebra L(R), then it follows from the connecting property that IL.AR is a closed Lie ideal of L(R). Therefore the Lie algebra L(R) is, in a certain sense, a eomplementeal algebTa.

An element ac S2R is said to be centTal if it satisfies Åqa, ayÅr==Åqa, orxÅr

for every x, y E R. The set C of all central elements in R is evidently the eenter of the Lie algebra L(R).

A Lie algebra L(R) is said to be semi-simpte if it satisfies

R =- [R, R].

A Structure Theory for Semi-Finite Q;uasi-Unitary Algebras 7

Let D. (xER) be an operator defined by D.y= [x, pt], then the operater Ddi(.) is diagonal with respeet to the orthonormal basis {qk(i, i)}. In fact,

Ddi(.) {nk (p, g) =: [ai (e), qk (p, g)]

== qk(p, g)Sip " qk(p, g)6ig•

It is evident that Dd,(x.) and Dd,(.).di(.) are diagonal too. The closed span of all di(.) in wro is denoted by D(e).

The normalizer of a subset Ar of L (R) is the set of all elements xEL(R) such that

[N, x] ( IV,

and the centTalizeT of ZV is the set of all elements x of L(R) such that [N, x] == O.

Lemma 3.1. D(9Jo) is the noTmalizer and the centralizer of D(e).

Proof. If aER has the following matrix form - all a12 '"''''"

a- a21 a22 "''"'''

i)

then it follows that i

v

[aj (e), a] =:( o -' ai,i o '

-- a2)i

i) ai)i ai)2•..aiJi-i '-'3i'-i)ia,)i+i...

-ai+11i

o :o

:

therefore [D(e), a](D(e) if and only if the matrix a is diagonal. Furthermore, [D (e), a] =:O if and only if a is diagonal. Thus the proof is completed.

Let A be a maximal self-adjoint abelian subalgebra of D(uto), then from the lemma 3.1 it follows that A is a maximal self-adjoint abelian subalgebra of R.

D(e) is the center of the Lie algebra L(D(E}Io)) which is defined on D(2tto) with re- spect to the ordinary Lie product. Therefore D(e) is contained in every maxi- mal self-adjoint abelian subalgebra of D(uto).

1

Lemma 3.2. Let R be a Lie algebra whieh is also a pre-Hilbert space with an involution s, ana which has the connecting property

Åq[xp y]7 zÅr == Åqy) [x") z]År.

Let A be a self-ado'oint smbset of R, and let C(A) be the centralizeT of A, then R can be decomposed into the form

R =C(A) + [A, R].i)

Proof. If x E R, y E C(A), z E A, then it follows from the connecting property that

Åq[g) x]) orÅr ---- Åqx) [zS) y]År = O)

therefore [z,x]cC(A)SAR, that is, [A,R](C(A)LAR. Conversely, if xER, yc[A, R]SAR,zEA, then

O ----` Åq[z, x], y]År == Åqx, [zS, y]År,

therefore [zS, or]=:O, hence [A, RILAR(C(x4), that is, C(A)J-AR([A, R]. Thus the proof is completed.

It follows from the above lemmas that the Lie algebra L(R) can be decom- posed into the form

R == D(uto) + [D (e), R].

Let C be the center of L(R), then the centralizer of R is evidently C, therefore R == C+ [R, R].

In other words, the Lie algebra L(R) is necessarily red2ective.

Let R be a simple unitary algebra, then the left ring Rg is a factor, that is, the eenter Z =RgARd cohsists of only the constant multiples of the identity

operator I. Factors are classified into five types (I.), (I..), (IIi), (II..) and (III).

But in our cace only types (I.), (I..), (IIi) and (II..) ean occur [3].

The left ring Rg is said to be pTopeTly injZnite class if it is not of finite class.

Now let R be a simple maximal unitary algebra, then whose left ring Rg is of finite class if and only if R contains an identity element [6]. Therefore in this case the matrix algebra uto is of finite order.

If R is a simple maximal unitary algebra whose left ring Rg is a factor of type (I), then its matrix form is the total matrix algebra over the complex num- ber field for a suitable choice of E)Ii. Then it is evident that D(SJo) and D(e) are coincide, and D(uto) is a maximal self-adjoint abelian subalgebra of L(R), there-

1) If R is a linear space and L, M( R, then L+M means the closed span of L and M in R, In what

follows we shall use the notation + in this sense.

A Structure Theory for Semi-Finite Quasi-Unitary Algebras 9

fore D(uto) is a Cartan subalgebra of L(R). The Cartan deeomposition follows from the above decomposition. This is the case which is ineluded in L* algebras.

Lemma 3.3. IfR is a simple maximal unitaTy algebTa whose left ring is of p7"operly infinite class, then the Lie algebra L(R) is semi-simple.

Proof. It follows from the lemma 3.1 that the center C of the Lie algedra L(R) is contained in L(?Io), therefore c E C has the following form

Cll

C--"' C22 O

--- - e-

o '•.

e-

k ']

Let e(i, ]') be the matrix in uto such that the (i, j)-entry is the identity element of

"Ji and the other entries are all zero, namely t

e(i) i)=: f/ O ... ) i o •"".."."".

Ill

i) ii 8 111111.9..g..9.•''

j' l.

v

then [c, e(i, i)] =O implies cii==cjj, therefore the diagonal entries of the matrix c are equal to one another. From the fact that the left ring is of properly infinite elass and from Zllciill2Åq oo it follows that c= O. Sinee L(R) is reductive, then i C==O implies that L(R) is semi-simple. Thus the proof is completed.

If L(R) is semi-simple then the mapping x.D. is one-to-one. In fact, if D.

==D, then from the conneeting property it follows that Åq(D.--- Dpt) u, vÅr =Åq[x--y, u], vÅr

="ÅqX-`- Ye [v) uS]År)

for every u,vcR. Since L(R) is semi-simple, then the set of all elements of the form [u, v] is dence in R, therefore x==pt.

Let A be an associative algebra, and let L(A) be the Lie algebra defined on A with respect to the ordinary Lie product, then every two-sided ideal of A is evidently a Lie ideal of L(A), but the converse assertion is not necessarily true.

Now we shall consider the following lemma:

Lemma 3.4. Let A be a simple associative algebra, ana let L(A) be thq Lie al-

gebra deJined on A with Tespect to the eTdiuary Lie predzzet [x, y]---ay-yx. If L(A) contains no non-neTo abelian Lie iaeats an(l if A =[A, A], then L(A) is simple as a Lie algebra.

Proof. Let I # {O} be a Lie ideal of L(A), and let K be the set of all ele- ments x( A such that

[x, A] ( I, then it is evident that I(K.

The set K is a Lie ideal of L(A). In fact, since I is a Lie ideal, then it fol•- lows from the Jacobi identity

[[x) pt]) z] == -- [[y, z], X]-[[z) x]) or]

that [[x, y], z] is contained in I for every x( K, y, z, E A, therefore [x, pt] is con- tained in K. Hence K is a Lie ideal of L(R).

Now we shall show that the set K is a subalgebra of A. If x, ptEK, zEA, then from the definition of K it follows that

[x, orz]([x, A](I, [pt, zx]E[y, A](I, therefore the identity

[x, orz] + [or, zx] + [z, ay] =O

implies [ay, A](I, so that ay E K. Hence K is a subalgebra of A.

By the assumption the Lie ideal K is not abelian, then there exist [x, y] E K such that [x, y] #O. Since K is a subalgebra of A and also a Lie ideal, then [ec, yz] EK and y[x, z] E K. Hence the identity

[x, yz] == [x, or]z+or[x, z]

implies [x, y]z ( K for all z ( A. Then it follows from the identity t[x, y]z = [t, [x, y] z] + [x, y] zt

that A[x, y]A(K. If A[x, y] =0, then the set of all elements such that Au==O is a proper associative ideal of A. But from the simplicity of the associative al- gebra A there exist no such ideal in A, therefore A[x, y] "L O. Now let zcA be such an element that z[x, y] * O. In the same manner there exists an element t that z[x, y]t 7E O. Therefore A[z, or]A AF O, The associative ideal of A spanning by A[x, y]A coincides with A, because of the simplicity of A. Therefore we have A== [A, A] ( I.

Hence L(A) contains no proper Lie ideals, that is, L(A) is simple as a Lie algebra.

AStr,pcture•Theory for Serni-Finite Quasi-Unitary Algebras 11

Thus the proof is completed.

Lemma 3.5. Let R be a simple unitary algebra. If the Lie algebTa L(R) is semi-simple, then it is simple as a Lie algebra.

Proof. Since the Lie algebraL(R) is semi-simple, then R=[R, R]. Let I be a non-zero abelian ideal of L(R), then it follows from the connecting property that the orthogonal complement of I in R is an ideal of L(R), therefore we have R=\ [R, R], and this contradicts the semi-simplicity of L(R), hence L(R) contains no non-zero abelian ldeals. Therefore by the lemma 3.4 the algebra L(R) is sim- ple as a Lie algebra. Thus the proof is completed.

Now we shall consider a relationship between the simplicity of the Lie al- gebra defined on a simple maximal unitary algebra and the type of its left ring.

Lemma 3.6. Let R be a simple maximal ttnitaTy algebra, then the Lie algebra L(R) is simple if ana only if its left ring Rg is of properly inLfinite class. If the

left Ting Rg is of!inite class, then L(R) is neither simple nor semi-simple.

Proof. If R is of properly infinite class, then by the Lemma 3.3 the Lie al- gebra L(R) is semi-simple. Therefore by the lemma 3.5 it is simple as a Lie al- gebra. Conversely, let L(R) be simple. Since L(R) is reductive, the maximal unitary algebra R contains no identity element, therefore the Ieft ring Rg is of properly infinite class. If Rg is of finite class, then the algebra R contains an identity element. Therefore L(R) has the eenter which is consisted by the scalar multiples of the identity element. Henee the Lie algebra L(R) is neither simple nor semi-simple. Thus the proof is completed.

Lemma 3.7. Let R be a maximal ttnitary algebra. If the Lie algebra L(R) is semi-simple, then every closed Lie ieleal is semi-simple too.

Proof. Let Ibe a closed Lie ideal of L(R), and let C(I) be the center of the Lie algebra L(I). If L(R) is semi-simple, then [x, R]==O implies x==O, therefore in the same way as Il* algebras the closed Lie ideal I is self-adjoint. It follows that the set of all elements of the form

C(I) + [I, I]

is dense in L Since the orthogonal complement I' of I in R is an ideal of L(R), we have [z, I']=O for everyxEC(I), therefore

[x, I+I'] = [z, I] + [z, I'] --- O.

Since the operator D. is bounded in S)R and sinee the set of all elements of the

form I+r is dense in R, it follows that D.R =O, so that z is contained in the cen-

ter C of L(R). On the other hand L(R) is semi-simple, so C=O, therefore we

12 ••• E. HONGO

havez =O. Thus the proof is completed.

Let Ibe a purely non-simple ideal of the maximal unitary algebra R, then I is the generalized direct sum of simple maximal unitary algebras ?I(t), name-

ly I-N•f 2rc(t). If x, )t E I, and if

X-•vX(t), Y--.••:Y(t), then we have

x + v '•- x(t) + y(t), ay j•- x (t)y (t),

therefore we have

[x, iy] •'v [x(i), y(t)].

Hence if I is semi-simple as a Lie algebra, then ut(t) is semi-simple too.

Theorem 3.1. Let R be a maximal unitary algebra whose underlying linear space is separable, and let L(R) be the Lie algebra aefined on R 2vith respect to the oTdinaTy Lie preduct. If the Lie algebra L(R) is semi-simpte, then it is the geneT- alizeel direct szem of simple Lie algebTas.

Proof. The maximal unitary algebra R is the generalized direct sum of simple maximal unitary algebras. Since the Lie algebra L(R) is semi-simple, then in the generalized direct sum every simple maximal unitary algebra is semi-simple as a Lie algebra, therefore by the Lemma 3.5 it is simple as a Lie algebra. Thus the theorem is established.

Corollary. UndeT the same assumption as in the theoTem 3.1, the aegebra R is tlze generalized diTeet sum of simple maximal unitary algebras whose left Tings are of pTopeTly injinite class.

The corollary follows directly from the theorem 3.1 and the Lemma 3.6.

Now we shall return to the decomposition of the Lie algebra L(R). Let R be el separable simple maximal unitary algebra whose left ring is of properly infinite class, then by the theorem 3.1 the algebra L(R) is simple as a Lie alge- bra. For a linear mapping at of a subalgebra Hof L(R) into the complex num- bers let V. be the set of allvcR such that [h, v] =ct(lt)v for allhE H. Then Vat is a closed subspace of R, and will be called a root relative to H if and only if V. \= O. Let xi be the linear functional on D(e) which assigns the eomplex num- bers in i-th diagonal entry to every element of D(e). The operators D. relative to the subalgebra D(e) are diagonal with respect to the orthonormal basis

{qk(i, j)}, and diagonal entries have the form

A Structure Theory for Semi-Finite Quasi.Unitary Algebras 13

ÅrLi (x) - )tj (x).

Let a7 be the set of all non zero roots relative to D(e), and let Vd be the root space belonging to the root ct, then the Lie algebra L(R) can be decomposed into the form

R=Vo+2Vct,

ctE P7

where Vo is the algebra D(uto). If the left ring Rg is a factor of type (I), then R is an Il' algebra. Therefore D(e) = D(uto), and D(e) is a Cartan subalgebra of L(R). The above decomposition is evidently a Cartan decomposition of L(R).

For a root ct there is a unique ddi in D(e) with ct(d)---Åqd, aptÅr, d.----ab', lid.li;:SVM2/llell. In faet, let ac =Xi-Nj and let a. be the element of D(e) of the form

a. pu

i)

j)

0

. ..

.Tem.

I{eil2 o

"' o

pi,te i-

i2

o

..

then ct(d) == (xi -- Nj)(a) == Åqa, a.År, and therefore lixi ---xjll --- lid.Ui$V2/Ueli. If at is

a root then ---ct is also a root and V.. =V8. If ct, B are distinct roots then IV.J-Vp and it fpllows from the Jaeobi identity [V., Vp](V..B. A simple system of roots is given by

• {)L,-)L,.,}: i= 1, 2,....

If the left ring Rg is of finite class, then the Lie algebra L(R) is not semi- simple. Since the Lie algebra L(R) is reductive and the unitary algebra R is maximal, the Lie algebra L(R) is the closure of the direct sum of a semi-simple Lie ideal of L(R) and an abelian Lie ideal which is isomorphic to the Lie alge- bra of complex numbers. But in the same way as treated in the case that the left ring is of properly infinite class, the Lie algebra L(R) ean be decomposed into the form

R== Vo +Z Vct•

ctEM

Now we have the' following theorem:

Theorem 3.2. Let R be a maximal simple unitary algebra, ana let L(R) be

.

the Lie algebra aej7ned on R with respeet to the oraiuaTy Lie pToduet. Then L(R) can be decomposed into the form

R = Vo + 12] Vtu, ctEM

with the following pToperties:

(i) W is the set of all non-xeTo roots Tetative to the abe"an subalgebra D(e), (ii) Vo is the normalizer ana the centralizer of D(e), and is the direct sum (flnitely or infinitely many) of a Lie subalgebra which is a simple maximal uni- taTy atgebra with an identity element,

(iii) if ct f=B then V.-L VB, ifa+/9 isaroot then [V., Vp]( V..p, all V. are isomouphic one another as lineaT spaces, lictll;:SllV2M/llell foT ale Toots ct•

4. Semi-finite quasi-unitary aigebras

This section is devoted to the study of the semi-simple Lie algebra defined on a semi-finite quasi-unitary algebra. The Lie product using here is not ordi- nary one, but some another one having the same connecting property with the L' algebras.

Let R be a quasi-unitary algebra, then there exist the quasi-unitary alge- bra Ri, the unitary algebra R2 and the maximal unitary algebra Rb which is the bounded algebra of R2 defined in the seetion 2. In this section the notations Ri, R2, and Rb are used in this sense.

Now we have

ÅqX, YÅr =q(XM"i(A) Y'M' (A))

== q(XMi S(A)Mi g(A)M, - i(A)Mt - i(A) Y*Mf e(A)M' e(A)) = q(M' ?i(A)xM' l(A)M' -2(A)M' a-(A) y* M' l(A))

==(M'S(A)xMrl(A), M'}(A)yM'4(A)).

Then the mapping

Vr (X) = M' i' (A) XM'S (A) is a unitary mapping of R2 on Ri.

Lemma4. 1. The mapping " ana the involutive anti-automorphism S are com- mntative, that is,

v(xs) .-. (th(x))s.

Proof.

((V (X))S .--. M' -- i (A) (Mie (A) XM'S(A))*Mt (A) == M' "2 (A) Mr -i (A) .X*Mt (A) MtS (A)

=V(xs).

AStructure Theory for Semi-Finite Quasi-Unitary Algebras 15

Thus the proof is completed.

The mapping i;n(X) of R2 on Ri is one-to-one and preserves the inner product.

Let L(R2) be the Lie algebra defined on R2 with respect to the ordinary Lie produet

[X, Y] == XY- YX.

Now we define in Ri a product [ , ], by ["(X), th(Y)], ="([X, Y]).

Since " is a linear mapping of R2 on Ri, then the product is linear is each fac- tors, and evidently it satisfies the following conditions:

(i) ['kPz(X), ',;n(X)],=O,

(ii) [ptkln(.X'), ,P•(IY)], :-[NLe(IY), }In(X)],,

(iii) [['kle(X), 'tlt(Y)],, ',Ie(Z)], + [[A;t(Y), NIn(Z)],, Yr(X)], + [[Nln(Z), i;n(X)],, 'NPn(Y)], =: O•

Therefore the quasi-unitary algebra Ri becomes a Lie algebra with respect to this Lie product. This Lie algebra is denoted by L(RiÅr. It follows from the lemma 4.1 that the Lie algebra L(Ri) has the connecting property

(["(Xr), 'NIn(Y)],, 'tPn(Z)) - (aPf(IY), [Vn(X)S, 'NIn(Z)],).

The mapping ftlr has the following properties:

(i) an isomorphic mapping in the sense of Lie algebras, (ii) an isomorphic mapping in the sense of Hilbert spaees, (iii) a mapping which preserves the conneeting property.

Hence the structure of L(Ri) is determined by the structure of L(R2). In parti- cular, a decomposition of L(R2) eorresponds to a decomposition of L(Ri). Since the unitary algebra R2 is dense in the bounded algebra Rb, then the structure of L(R2) is closely related to the strueture of L(Rb).

Let j be the automorphism of the quasi-unitary algebra R. An element

a E {iÅr/R is said to be quasi-central if it satisfies (a, xjpt) == (a, ptx)

for every x, pt E R. Then we have the following }emma:

Lemma 4.2. An element iln(X) E Ri is quasi-central if ana only if XE R2 is centrat.

Proof. If X, YE R2, then

',1"(X)'aln(Y) --- 'kln(Y) •,In(X) ,

= : Mt (A) M' ,r (A) XM'S (A) M' -- i (A) M'g (A) YM'l (A)

-- M,e (A) YMt"(A) Mtl(A) XIM'" (A) .---: M,l(A) (M'(A)XY -- YM' (A)X) M'S(A) -- V([M' (A)X', Y]),

therefore iln(Y) is quasi-central if and only if[X, Y] =O for every XE R2. Thus the lemma is proved.

Theorem 4.1. Let R be a semi-jZnite quasi-unitaTy algebra whose undeTlying

"neaT space is sepaTabte. If {SÅrR contains no non-zero quasi-central element, then there is a Lie algebra L(RS) which is clense in foR as a "neaT space, and which is the generalizea airect sum of simple Lie algebras.

Proof. Let R be a semi-finite quasi-unitary algebra whose underlying lin- ear space is separable. If &R contains no non-zero quasi-central element, then

SÅrRb contains no non-zero central element. Since the Lie algebra L(Rb) is reduc- tive, L(Rb) is semi-simple as a Lie algebra. Let Rb ;v ut(t), namely the gener- alized direct sum of simple maximal unitary algebras ut(t). Furthermore, every

?{(t) is semi-simple as a Lie algebra. Therefore it follows from the lemma 3.5 that the algebra L(E!I(t)) is simple as a Lie algebra. Hence L(Rb) is the gener- alized direct sum of simple Lie algebras, and the image L(RS) of L(Rb) under the mapping " is the desired one. Thus the theorem is established.

In general, the Lie product in L(RS) can not be extended to the whole Hilbert space tbR, therefore the Lie product can not necessarily be extended to the quasi-unitary algebra R. However, from the ifact that R is dense in tfoJR it follows that the above deeomposition of the Lie algebra L(RS) is closely related to the decomposition of R.

References

1. W. Ambrose, Structure theorems for a special class of Banach algebras, Trans. Amer, Math. Soc,, Vol. 57 (1945), pp. 364-386.

2. J. Dixmier, Algebres quasi-unitaires, Comm. Math, Helv., Vol. 26 (1952), pp. 275-322.

3. J. Dixmier, Les algebres d'oPerateurs dans•l'espace hilbertien, 1957, Paris.

4. R. Godement, 7hecrie des caractt}res, I. Algebres unitaires, Ann. of Math., Vol. 59 (1954), pp. 47-62.

5. E. Hongo, On qttasi-unitar2 algebras with semi-fine'te left rings, Bull. Kyushu Inst. Tech., No 3 Åq1956),

pp. 1-10.

6. R, Pallu de la Barriere, Algebres unitaires et espaces d'Ambrose, Ann. 1'Ecole Norm. Sup., t. 70 (1953), pp. 381-401.

7. L. Pukanszky, On the theor] ofquasi-unitarl al.gebras, Acta Univ. Szeged, t. 16 (1955), pp. I03-121.

AStructure Theory for Semi-Finite Quasi-Unitary Algebras 17

8. J. H. Schue, Hilbert sPase methads in the theor7 ofLie algebras, Trans. Amer.Math. Soc., Vol. 95(1seO), pp. 69--80.

9. 0. Takenouchi, On the structure of masvimal Hilbert algebras, Math. Jour. Okayama Univ., Vol. 1 (1952), pp.1-31.

Kyushu Institute of Technology

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