A CHARACTERIZATION OF B*-ALGEBRAS

Internat. J. Math. & Math. Sci.

VOL. 20 NO. 3 (1997) 483-486

483

A CHARACTERIZATION OF B*-ALGEBRAS

A. K.GAUR

Department

of Mathematics Duquesne University, Pittsburgh,

PA

15282

(Received December II, 1995 and in revised form August 16, 1996)

Abstract.

A

characterizationofB*-algebras amongst all Banachalgebraswith bounded ap- proximate identitiesisobtained.

1991 AMSSubject

Classification:

^46J99,46J15

Key Words andPhrases: Approximate identity; B*-algebra; self-adjoint elements;

Hermitian elements.

I. Introduction.

Werecallthat an approximate identity in aBanach algebra

A

is anet

{ca

^a E

I}

ⁱⁿ

A

where

I

is a directed setsuch that lira

eax

^{x limxea foreveryx inA. If}there is a finite constant

M

suchthat

Ileall ^{_< M}

for all a, then the approximate identityissaid tobe bounded.

Let

A

be aBanach algebra. Foreach x in

A,

let

DA(X)- {f ^e A’ _ll/ll

I

f(x)}.

Byacorollary oftheHahn-Banachtheorem,

DA(X)

^{isnon-empty. Wedenote}

S(A) {x A

For each a

A,

wecall the set

VA(a) {f(ax) f ^DA(X),

^x

^S(A)}

the spatial numemcalrange ofa.

Werecall

[5]

that the relative numericalrangeofainAwithrespectto x E

A,

^isdefined as

x(A,a) {/(ax) f ^{e DA(x)}.}

Thusweseethat

VA(a)=U{x(A,a)’xe S(A)},

^{whichis aboundedsubsetof thecomplex}

numbers bounded by

Ilall.

If

A

has an approximate identity ofnormless than orequalto onethen

A

canembedded, isometricallyandisomorphically,in aunital Banachalgebra

A

⁺ insuchaway that for eacha in

A

V(A+,) Y(),

where

Y(A+,a)= (f(a) f ^e (A+) ’, Ilfll

¹

^{I(a)- Ilall).}

^Fordetails see

[4],

Theorem 2.3.

484 A.K. GAUR

An

element h of a Banach algebra

A

is said to be Hermitian if

VA(a)

C R. We denote by

H(A)

the set of all Hermitian elementsofA. A B*-algebrais aBanachalgebra

A

withan involution,^a

-

a* satisfyingthefollowingconditions:

(1) (a + b)"

a"

+

^b*;

() ()* ,.;

(3) (aa)*

^(a*;

(4)

^a** a; and

() I*1 I1

forall a, b in

A

andainC.

An

elementainaB*-algebraissaidtobe self-adjointif a a*. The setof all self adjoint elements willbedenoted by

S(A).

^Eachelementa E

A

can be written uniquely in the form a h

+

^{ik where}h, k

e S(A).

^Someof thewellknownpropertiesof

S(A)

^{are thefollowing:}

a)

^Theset

S(A)

isarealpartially ordered Banach space,

b)

eachofitselements hasrealspectrum,

c)

ifh,k

e S(A)

then

i(hk kh) ^e S(A),

and

d)

for each h

e S(A),

the spectralradius

p(h) ]lhll.

It

isclear that theset ofHermitianelements,

H(A),

^{ofa Banach}algebrawith abounded approximateidentityofnormless thanorequal toonehas many of thepropertiesof

S(A)

^{in a} B*-algebra.

In

this note we provethat in anarbitrary B*-algebra

A, H(A) S(A)

^{in Theorem}2.1.

Thisresultsmimics aresultby Bohnenblust and Karlin

[2].

In [8],

^{Vidavhas shownthata}unitalBanach algebra

A

withthe followingconditions:

(1) A H(A) + ill(A);

(2)

^{for each hin}

H(A)

^thereexistshi,

h2

ⁱⁿ

H(A)

^suchthat

hl +ih2

^h2and

hlh2 h2hx

is a B*-algebrawith Vidav-involution. Combiningthe results of Vidav

[8],

^Berkson

[1],

and Glickfeld

[6]

^{weobtaintheresultthat}if

A

is a unitalBanachalgebrasuch that

A H(A)+iH(A)

then

A

is aB*-algebraunderthe Vidav-involution.

Here,

weextend thisresulttothe nonunital case intheformofLemma3.1.

Finally, combining the results of Theorem 2.1 and

Lemma

3.1 wehaveacharacterization ofB*-algebraswithboundedapproximate identities.

2. Some Results.

Wenowprove thefollowing theorem.

Theorem 2.1 Let

A

be aB*-algebrawith a boundedapproximate identity

o

normless than orequal toone.

An

element of

A

isHermitian ifand onlyi[it isself-adjoint.

Proof. Case 1. Supposethat

A

hasa unit element 1.

Let f ^DA(1).

^{Thenit isknown}

that such a functional has the property that

f(h*) f(h),

for every h in A. Thus ifh is a self-adjoint elementof

A, .f(h) f(h*) f(h)

and hence

f(h)

^isrealfor all

f

ⁱⁿ

DA(1). ^Hence, S(A) C_H(A).

Case2. If

A

has noidentity element thenit willhaveanapproximate identity ofnormless thanorequalto one. Also,withtheinvolution definedby

(a, a)* (a*, ()

^for

(a, a)

^E

^A +,

^and

CHARACTERIZATION OF B -ALGEBRAS 485

by Theorem 2.3in

[4], A

⁺ becomesaunital B*- algebracontaining as asub-B*-algebra,

([3], ..s).

Lethbe aself-adjoint element ofA. ^Then

(h, 0)

isself-adjoint and henceHermitian inthe unitalB*-algebra

A +.

Hence hE

H(A).

^Wehavetherefore foranyB*-algebra,

S(A)

C_

H(A).

Supposeconverselythat h

H(A).

^Thenfor

hi

^and

h2

in

S(A),

h

hl + ih2.

^Thisimplies

that

(h2)

⁰

(where (x) sup{IAI A e VA(x)}

^{and iscalled}numerical radiusofx in

A)

and hence

h2

^{0. Thush}

h

^{so that his}self-adjoint. Thatis

H(A)

C_

S(A)

and hencethe theorem.

Remark 2.1 The above theoremshows thatin aB*-algebrathe Hermitianelementsgenerate the whole algebrainthesensethat each elementamaybewritten inthe forma

hi + ih2

^with h and

h2

ⁱⁿ

H(A).

In anarbitrary Banach algebra

A

this is nottrue. Wetherefore consider theset

J(A) H(A) + ill(A).

^Since

H(A)

^{is areal spaceit follows that}

J(A)

is acomplex linear space. If

A

hasnounit elementthenby Theorem 2.3,

[4], J(A)

^xC

J(A +).

^Wedefine

amapa-->a" from

J(A)

into itselfby

(hi

+ ih=)* hi

^{ih2, forall h.,}

h2 H(A).

The linearmapa-->a* isknownastheVidav-involution on

J(A).

Remark 2.2 If

A

has no unit element then it is a simple matterto verify that the Vidav- involution on

J(A+)

^{is an} extension of the Vidav-involution on

J(A).

The space

J(A)

^{is a}

complex Banachspace and a --> a* is a continuous linear involution on

J(A). ^In

^{general, the}

Banach space

J(A)

^{isnot analgebra,and if}

J(A)

^{isanalgebraundersome}conditions, thenthe Vidav-involutionhas the additionalproperty

(ab)*

^{a’b*, forall a, b}

J(A).

3. Characterization.

Vidavhas shownin

[8]

that a unitalBanachalgebra

A

withthe followingconditions:

(V1) ^A H(A) + ill(A),

(V2)

^{for each h}in

H(A)

^thereexistshx,

h2

ⁱⁿ

H(A)

^suchthat

h. +ih2

^h2and

hh2

^h2h,

isaB*-algebrawith Vidav-involutionandanormequivalenttothe originalnorm onA.

Accordingto Palmer

[7],

^thecondition

(V1)

^implies

(V2).

Also Berkson

[1],

^Glickfeld

[6],

and Palmer

[7]

^have shown that if

(V1)

^{issatisfied by the} algebra

A

the equivalent norm by Vidav isequal to the originalnorm onA. Soby these resultswehave the resultthatifAis a unitalBanachalgebra satisfying

(V1)

^then

A

^is B*-algebraunder the Vidav-involution. The following lemmaextends thisresult to the non-unital case.

Lemma3.1Let

A

beaBanachalgebrawith a boundedapproximate identity of norm less than orequalto one. Suppose thateverya in

A

has theforma

hl +

ih, forallh,

h2

ⁱⁿ

H(A).

Thenwith theVidav-involution,

A

is aB*-algebra.

Proof. FromRemark 2.1 wehave that

J(A +) J(A)

C. Since

J(A) A (by

the hypothesis) we have

J(A +) A +.

Therefore

A

⁺ is a unital B*-algebra under the Vidav- involution. Furthermore,

A

is a closed and self adjoint subalgebra of

A +,

and is therefore a B*-algebra underthe Vidav-involution.

486 A.K. GAUR

Finally,combiningthe results ofTheorem2.1 andLemma3.1 wehavethe following:

Theorem 3.2 Let

A

bea Banach algebrawitha bounded approximate identity ofnormless thanorequaltoone. Then

A

is aB*-algebraundersome involutionif andonlyif each element a ofAcan be written in theforma

hi + ih2

^where

hi

^and

h2

are HermitianelementsofA.

4. Acknowledgement.

The author expresses his appreciation to the referee for his or her valuable suggestions whichimprovedthe clarityofthis presentation.

References

[1]

E. Berkson,

"Some

characterizationsofC*-algebras",Illinois

J.

Math., 10,

(1966),

1-8.

[2] H.F.

Bohnenblust and

S.

Karlin, "Geometrical properties of the unit sphere of Banach algebras", Ann.

of

^Math,62,

^(1955),

^217-229.

[3]

^{J. Dixmier, "LesC-algdbreset}leurs reprentations", Gauthier Villars, 1964.

[4]

^{A.K. Gaur and T. Husain,} Spatial numerical ranges of elements of Banach algebras", Inter’nat. J. Math. and Math. Sci., 12,

(1989),

633-640.

[5]

^A.K. GaurandT. Husain, "Relative numerical

ranges",

Math. Japonica, 36,

(1991),

127- 135.

[6]

^B.W.Glickfeld,

"A

metriccharacterizationof

C(X)

^anditsgeneralizationtoC*-algebras", IllinoisJ.

o.f

Math., 10,

(1966),

547-566.

[71

^{T.W. Palmer,} "Characterization of C’-algebras", ^Bull.

Amer.

Math.

Soc.,

74,

(1968),

538-540.

[8]

I. Vidav, "Eine metrische kennzeichnung der selbstad jungiertenoperatoren’, Math.

Z.,

66,

(1956),

121-128.