WORDS WEAK

Internat. J. Math. & ^{Math. Sci.}

VOL. 15 NO. 2 (1992) 409-412

409

SEMI-SIMPLICITY OF A PROPER WEAK

H*-ALGEBRA

PARFENYP.SAWOROTNOW

Department

ofMathematics The Catholic University ofAmerica

Washington,

D.C.

20064 U.S.A.

(Received January 23, 1991)

ABSTRACT. A

weak right H*-algebra is a Banach algebra

A

which is a Hilbert space and whichhas adensesubset

Dr

^with the property that for each xin

Dr

^{thereexists x such that}

(yz,

z)

(y, zz

r)

for all y,z in A. It is shown that a proper

(each

^{z is unique) weak right} H*-algebraissemi-simple. Also thereis anexampleof weakright H*-algebrawhichis notaleft H*-algebra.

KEY

WORDS AND PHRASES. Hilbert algebra, H*-algebra, weakright H*-algebra, ^{weak left} H*-algebra, complemented algebra, right complemented algebra,left complemented algebra.

1980 AMS

SUBJECT CLASSIFICATION

CODE. Primary: 46K15. Secondary: 46H15, 46H20, 46K10.

1. INTRODUCTION.

Assumption of semi-simplicity plays animportant role in the theory ofcomplemented al- gebras. It was noted in the author’s last paper

(Saworotnow [1])

^{that it is} rather difficult to deduct semi-simplicity from axioms of a

(proper)

weak right H*-algebra.

However,

there is a different story forthecaseofatwo-sided

(weak)

H*-algebra. Hereit isnottoo difficult to show that eachclosed two-sided ideal has anidempotent which, inturn, impliessemi-simplicity. But it wasestablished inSaworotnow

[1]

^{that eachproperweak right H*-algebrais also aweak left} H*-algebra. ItfollowsthateachproperrightH*-algebraissemi-simple

(Theorem

²

below).

^This

isthe central result ofthis paper. Weincludedalso important consequences ofitandanexample ofanalgebrawhich isaright H*-algebrabut notaleftH*-algebra. Thealgebrainthe example is alsoanexample ofaweak right

H*-algebra

which is notaweak left H*-algebra

2. PRELIMINARIES.

A

weakright g*-algebra

(Saworotnow [1])

isaHilbert algebra

A (a

^Sanachalgebrawhich isa Hilbert

space)

^{whichhasadensesubset}

Dr

^{with the}property thatfor eacha E

Dr

^thereis

410 P.P. SAWOROTNOW

amember^a of

Dr

^suchthat

(xa,

y)

(z,

ya

r)

for all x, yE

A;

a is called thefight adjoint of

a. It^issaid to be proper ifa isuniquefor every^ain

Dr;

this isequivalenttothe condition that the right annihilator

r(A)

^x

e A Ax 0}

^of

^A

^{consistsofzeroalone}

(A

is properif and only if

r(A) (0)).

Wedefineweakleft H*-algebrainasimilar way. Weaktwo-sidedH*-algebraisaweak right H*-algebrawhichisalsoa

(weak)

^leftH*-algebra.

THEOREM 1.

Every

weakright H*-algebrais aright complemented algebra

(Saworotnow [2]),

^{i.e., the}orthogonal complement

R

^pofanyrightideal

R

in

A

isalsoaright ^ideal.

PROOF.

If x

R

and a

A,

then

(xa,

y)

lim(xa,,

y)

lira(x, ya,)

^{0 for some}

sequence

{a,, }

^C

Dr

^{convergingtoaand eachy} R. Thisimplies that

R

^pisalsoaright ideal.

PROPOSITION 1. Theorthogonal complement

P’

of eachtwo-sided

I

inaweakright H*- algebra

A

isagain^aweak right H*- algebra.

(Note

^{that wedo not allege}

^I

itself to bea weak right H*-algebra.)

PROOF. First notethat

P’I

C

I

f31

(0),

i.e.,xy 0 for all x

I ,

yE

I.

Nowconsidera

P’

and let

>

0bearbitrary. Take b

e Dr

^{sothat a b}

I1<

^eandwrite

b=

bl

+b2, b =c1+c2 ^with bl, c I^v andb2,c2 I. Then

a-b I1<

^{eandwehavefor}

each x,yEI:

(zb,

y)

(xb +

^xb2,y)

^(xb,

^y)

^{(x, yb")} (x,

^ycl

+ ^{yc2) (x,}

^ycl),

(2.1)

which simply meansthat _cl is aright adjoint of

b.

Thus: every neighborhood^ofa contains a vector havingaright adjoint.

PROPOSITION2. Each closedtwo-sided ideal

I

inaproper weakrightH*-algebra

A

is a proper weakrightH*-algebra.

In

fact,it is alsoaweakleftH*-algebra.

PROOF. ItwasshowninSaworotnow

[1]

that

A

isalsoaproper weak leftH*-algebra. This meansthatP’isalsoaleftideal

(we

^{canusehere the}proof ofTheorem 1

above).

^Thus:

I

isthe orthogonal complementofatwo-sided ideal. Proposition 2nowfollows’from Proposition 1

(I

^is

theorthogonal complement of thetwo-sided ideal

I);

the fact that

I

is properisalso easy to establish.

3. MAIN THEOREM.

Nowwe can proveourmainresult.

THEOREM

2.

Every

proper weakrightH*-algebra

A

issemi-simple.

PROOF.

Proposition 2 imphes that the radical

(Jacobson [3]) R

of

A

isaright H*-algebra.

Hence

it contains anon-zero vector a havinga (unique) right adjoint ^a O. Then Re" 0

(otherwise

^xa

112= ^(x,

^xaa

^r)

^{0 foreachx}

^{e A)}

^{andasin27Aof Loomis}

^[4]

^{one canshowthat,}

forsomescalar

A,

the sequence

{

Aaa

}2’*

converges tosomeidempotente

R.

Thisisimpossible since everymember of

R

isageneralized nilpotent

(Theorem

16, page 309inJacobson

[3]).

An

importantconsequence of this theoremis the fact thatwe can nowapplytothealgebra

A

thetheoryofcomplemented algebras developedinSaworotnow

[2]

and

Saworotnow [5] (more

SEMI-SIMPLICITY OF A PROPER WEAK

H*-ALGEBRA

411

specifically: Theorem 1 inSaworotnow

[2]

and Theorem 3 inSaworotnow

[5]). ^We

summarize it asfollows:

THEOREM

3.

Every

proper weak fight H*- algebra is a direct sumof simple weak right H*- algebras,eachof whichisasemi-simple.

THEOREM

4.

For

each proper simple weak right H*-algebra

A

there isa Hilbert space

H

and a positiveself-adjoint norm-increasing operatora on

H

such that

A

is isomorphic and isometricto thealgebraof all Hilbert Schmidt operatorsaon

H

such that actisalso ofHilbert Schmidt type.

Thismeansthat eachsimpleproper weakright

(as

^wellas

left) H*-algebra

isof thetypedescribed inthe Exampleonpage 56 ofSaworotnow

[5].

4.

AN EXAMPLE.

To

conclude the paper,wegiveanexampleofaright

H*-algebra

whichisnotaweakleft

H*-

algebra. Thisexample shows thatourassumption ofanalgebratobeproper israther essential.

EXAMPLE.

Let

A

bethealgebraofall2x2 matrices andlet

0) (0 0)

el= 0 0 ^e21 1 0

Consider thesubalgebra

A0

^of

A

generated by_el andel,

A0 {.e

+pe2

,p complex}.

Then

A0

^isaright

(as

^{wellas a}weakright) H*-algebra

(note

that

Xe

^isarightadjoint of

te +

^pe2

^).

But A0

could not bealeft weakH*-algebrasincetheorthogonal complement

L ’ ^{{e }}

^oftheleft

ideal

L {e2 }

is notaleftideal

(here {x}

denotes the 1-dimensionalsubspaceof

A

generated by

x).

Note that

r(Ao) (0)

^and

e(A0) ^L (here g(A0)

denotes the left annihilator ofA0,

e(A0) {x" xA 0}).

REFERENCES

1.

SAWOROTNOW, P.P.

Relationbetweenrightandleftinvolutions ofaHilbertalgebra, Proc.

Amer.

Math.

Soc.

102

(1988),

57-58.

2.

SAWOROTNOW, P.P. On

ageneralization of thenotionofH*-algebra, Proc.

Amer.

Math.

Soc. 8

(1957),

^49-55.

3.

JACOBSON, N.

The radicaland semi-simplicity forarbitrary rings,

Amer. J.

Math.

(1945),

300-320.

4.

LOOMIS, L.H.

Abstract HarmonicAnalysis,

Van Nostrand,

^1953.

5.

SAWOROTNOW, P.P. On

theimbeddingofaright complementedalgebraintoAmbrose’s H*-algebra,

Proc.

Amer. Math.

Soc.

8

(1957),

^56-62.

6.

AMBROSE,

W. Structure theorem for aspecial class of Banach algebras, Trans. Amer.

Math.

Soc.

57

(1945),

^364-386.

7.

SMILEY, M.F.

Right H*-algebra,

Proc. Amer.

Math. Soc. 4

(1953),

^1-4.