manuscripta math. 104, 467 – 478 (2001) © Springer-Verlag 2001

Kaidi El Amin · Antonio Morales Campoy· Angel Rodríguez Palacios

A holomorphic characterizationof C∗- and JB∗-algebras

Received: 3 October 2000 / Revised version: 26 January 2001

Abstract. We characterizeC∗-algebras as those complete normed associative complex al-gebras having approximate units bounded by one and whose open unit balls are boundedsymmetric domains. Such a characterization follows from the more general fact, also provedin the paper, that non-commutativeJB∗-algebras coincide with complete normed (possiblynon-associative) complex algebras having approximate units bounded by one and whoseopen unit balls are bounded symmetric domains.

0. Introduction

A bounded domain� in a complex Banach space is called symmetric if for eachx in � there exists an involutive holomorphic mappingϕ : � → � havingx asan isolated fixed point. It is well-known that the open unit balls ofC∗-algebrasare bounded symmetric domains. It is also folklore thatC∗-algebras have approx-imate units bounded by one. In this paper we show that the above two propertiescharacterizeC∗-algebras among complete normed associative complex algebras(Corollary 3.4). The key tools in the proof are the holomorphic characterization ofthe Banach spaces underlying unitalJB∗-algebras, obtained by R. Braun, W. Kaup,and H. Upmeier in [6], and the Vidav–Palmer theorem (both in its original form [3,Theorem 6.9] and in Moore’s reformulation [4, Theorem 31.10]). Actually, apply-ing the non-associative versions of the Vidav–Palmer and Moore’s theorems (see[19] and [13], respectively), we prove that a complete normed complex algebra isa non-commutativeJB∗-algebra if and only if it has an approximate unit boundedby one and its open unit ball is a bounded symmetric domain (Theorem 3.3).

The results just mentioned arise in Corollary 3.4 and Theorem 3.3 under aslightly different formulation, which involvesJB∗-triples instead of bounded sym-metric domains. As a matter of fact, it follows from Kaup’s algebraic character-ization of bounded symmetric domains ([11] and [12]) thatJB∗-triples coincide

K. El Amin, A. M. Campoy: Departamento de Algebra y Análisis Matemático, Universidadde Almería, Facultad de Ciencias Experimentales, 04120 Almería, Spain.e-mail: elamin@ual.es; amorales@ual.es

A. R. Palacios: Departamento de Análisis Matemático, Universidad de Granada, Facultad deCiencias, 18071 Granada, Spain. e-mail: apalacio@ugr.es

Mathematics Subject Classification (2000): 46K70, 46L70

468 K. El Amin et al.

with those complex Banach spaces whose open unit balls are bounded symmetricdomains.

1. Approximate units in non-commutative JB∗-algebras

Following [22, p. 141], we definenon-commutative Jordan algebras as those alge-bras satisfying theJordan identity (xy)x2 = x(yx2) and theflexibility condition(xy)x = x(yx). For an elementx in a non-commutative Jordan algebraA, wedenote byUx the mappingy → x(xy + yx) − x2y from A to A. If A is a non-commutative Jordan algebra, then the algebraA+, consisting of the vector spaceof A and the product

x.y := 1

2(xy + yx),

becomes a (commutative) Jordan algebra, and forx in A the equalityUx = U+x

holds, where

U+x (y) := 2x.(x.y)− x2.y.

By a non-commutative JB∗-algebra we mean a complete normed non-com-mutative Jordan complex algebra (sayA) endowed with a conjugate-linear algebrainvolution ∗ satisfying‖ Ux(x∗) ‖=‖ x ‖3 for everyx in A. We note that (asso-ciative)C∗-algebras are precisely those non-commutativeJB∗-algebras which areassociative. IfA is a non-commutativeJB∗-algebra, thenA+ is a (commutative)JB∗-algebra in a natural way, and therefore the setAsa of all self-adjoint elementsof A (regarded as a closed real subalgebra ofA+) is aJB-algebra [10, Proposi-tion 3.8.2]. In this way, the self-adjoint partAsa of any non-commutativeJB∗-algebraA is endowed with the order induced by the positive cone{x2 : x ∈ Asa}[10, Sect. 3.3].

An approximate unit of a normed algebraA is a net{yλ}λ∈� in A satisfying

lim{xyλ}λ∈� = x and lim{yλx}λ∈� = x

for everyx in A.Every non-commutativeJB∗-algebra has an increasing approximate unit con-

sisting of positive elements with norm≤ 1 [24]. In this section we rediscover theabove result as a consequence of an interesting inequality for non-commutativeJB∗-algebras (see Theorem 1.3 below). Our argument begins with a lemma onC∗-algebras. Before formulating and proving it, let us establish some notation andfacts.

Let A be an algebra. The commutatorxy − yx of elementsx, y of A will bedenoted as usual by[x, y]. For x in A, the symbolsLx andRx will stand for themappingsz → xz andz → zx, respectively, fromA toA.

LetX be a normed space. We denote byBX,�X, andX∗ the closed unit ball,the open unit ball, and the (topological) dual space, respectively, ofX.BL(X) willdenote the normed algebra of all bounded linear operators onX, andIX will standfor the identity operator onX. Now, let us fix a norm-one elementu in X. The

A holomorphic characterization ofC∗- andJB∗-algebras 469

set ofstates of X relative tou, D(X, u), is defined as the non empty, convex, andweak∗-compact subset ofX∗ given by

D(X, u) := {ϕ ∈ BX∗ : ϕ(u) = 1}.For complexX, an elementx in X is said to behermitian relative to u if ϕ(x)belongs toR wheneverϕ is inD(X, u), and the set of all hermitian elements inXis denoted byH(X, u). A bounded linear operatorT on the complex normed spaceX is calledhermitian if T is an hermitian element ofBL(X) relative toIX.

Lemma 1.1. Let A be a C∗-algebra, and let x be in Asa . Then, for every y in A wehave

‖ [x, y] ‖2≤ 16 ‖ y ‖‖ x2.y − x.(x.y) ‖ .Proof. Since the unital hull of everyC∗-algebra is aC∗-algebra [5, Lemma 12.19],we may assume thatA has a unit1. Sincex belongs toAsa , we havex ∈ H(A, 1)[3, Example 5.3]. ThenLx andRx are hermitian operators onA [18, Corollary2.(a)], and henceT := Lx − Rx is also hermitian. By [3, Theorem 10.13.ii], wehave

‖ [x, y] ‖2=‖ T (y) ‖2≤ 4 ‖ y ‖‖ T 2(y) ‖= 4 ‖ y ‖‖ [x, [x, y]] ‖ .But, sinceA is associative, the equality

[x, [x, y]] = 4(x2.y − x.(x.y))

holds. ��Let F be a field containing more than two elements. Following [22, pp. 49-50],

an algebraA overF is calledquadratic if it has a unit1, A �= F1, and, for eachxin A, there are elementst (x) andn(x) in F such thatx2 − t (x)x + n(x)1 = 0.

Lemma 1.2. Let A be a quadratic non-commutative JB∗-algebra, and let x be inAsa . Then, for all y in A we have

‖ [x, y] ‖2≤ 16 ‖ y ‖‖ x2.y − x.(x.y) ‖ .Proof. By [16; Theorems 3.2 and 3.4], there exists a nonzero real Hilbert spaceE,together with a bilinear anticommutative product∧ onE satisfying(u∧ v | w) =(u | v ∧ w) and‖ u ∧ v ‖≤‖ u ‖‖ v ‖ for all u, v,w in E, such that, ifB denotesthe real algebra with vector spaceR1 ⊕ E and product

(α1 + u)(β1 + v) := [αβ − (u | v)]1 + αv + βu+ u ∧ v,then, algebraically viewed,A is equal to the complexification ofB, theJB∗-algebrainvolution ofA is given by

[α1 + u+ i(β1 + v)]∗ := α1 − u− i(β1 − v),

and the norm ofA is determined by

‖ b + ic ‖2:=‖ b ‖2 + ‖ c ‖2 +2[‖ b ‖2‖ c ‖2 −(b | c)2]1/2,

470 K. El Amin et al.

where, in the right hand of the equality, the inner product and norms refer to thenatural hilbertian structure ofB as&2-sum ofR1 andE.

First, takey in Asa , so that, sincex is also inAsa , we havex = α1 + iu andy = β1 + iv, for suitableα, β in R andu, v in E. Then we deduce

x2.y − x.(x.y) = −i(u2.v − u.(u.v)) = i(‖ u ‖2 v − (u | v)u),and hence

‖ x2.y − x.(x.y) ‖2=‖ u ‖2 (‖ u ‖2‖ v ‖2 −(u | v)2),and consequently

‖ x2.y − x.(x.y) ‖≤‖ u ‖2‖ v ‖ .On the other hand, we know (see [19, proof of Proposition 24]) that the inequality

‖ u ∧ v ‖2≤‖ u ‖2‖ v ‖2 −(u | v)2

holds. It follows from the equality[x, y] = 2v ∧ u that

‖ u ‖2‖ [x, y] ‖2 = 4 ‖ u ‖2‖ u ∧ v ‖2≤ 4 ‖ x2.y − x.(x.y) ‖2

≤ 4 ‖ x2.y − x.(x.y) ‖‖ u ‖2‖ v ‖ .Since‖ v ‖≤| β | + ‖ v ‖=‖ β1 + iv ‖=‖ y ‖, we obtain

‖ [x, y] ‖2≤ 4 ‖ y ‖‖ x2.y − x.(x.y) ‖ .Finally, lety be arbitrary inA. Writey = y1 +iy2 with y1, y2 inAsa , and note

that, since theJB∗-involution ofA is isometric [26, Lemma 4], the inequalities‖ y1 ‖≤‖ y ‖ and‖ y2 ‖≤‖ y ‖ are true. Then we have

‖ [x, y] ‖ ≤‖ [x, y1] ‖ + ‖ [x, y2] ‖≤ 2(‖ y1 ‖‖ x2.y1−x.(x.y1) ‖)1/2 + 2(‖ y2 ‖‖ x2.y2−x.(x.y2) ‖)1/2≤ 4(‖ y ‖‖ x2.y − x.(x.y) ‖)1/2. ��

To continue our argument, we need some background on factor representations ofnon-commutativeJB∗-algebras. First of all, we note that, ifA is a non-commutativeJB∗-algebra, and ifλ is a real number with 0≤ λ ≤ 1, then the involutive Banachspace ofA, endowed with the product

(x, y) → λxy + (1 − λ)yx,

becomes a non-commutativeJB∗-algebra (which will be denoted byA(λ)). By anon-commutative JBW ∗-algebra we mean a non-commutativeJB∗-algebra whichis a dual Banach space. Prime non-commutativeJBW ∗-algebras are callednon-commutative JBW ∗-factors. A non-commutativeJBW ∗-factor is said to be ofType I if the closed unit ball of its predual has extreme points (compare [16, Theorem1.11]). If A is a non-commutativeJBW ∗-factor of Type I, and ifA is neitherquadratic nor commutative, then there exist a complex Hilbert spaceH and a realnumberλ with 1

2 < λ ≤ 1 such that, denoting byB theC∗-algebra of all bounded

A holomorphic characterization ofC∗- andJB∗-algebras 471

linear operators onH , we haveA = B(λ) [16, Theorem 2.7]. Now, letA be anon-commutativeJB∗-algebra. Afactor representation of A is aw∗-dense range*-homomorphism fromA into some non-commutativeJBW ∗-factor. The fact thatAhas a faithful family of Type I factor representations [16, Corollary 1.13], togetherwith [15, Proposition 2.1], implies that, forx in A, the equality

‖ x ‖= sup{‖ ϕ(x) ‖ : ϕ a Type I factor representation ofA}holds.

Theorem 1.3. LetA be a non-commutative JB∗-algebra, and let x be inAsa . Then,for all y in A we have

‖ [x, y] ‖2≤ 16 ‖ y ‖‖ x2.y − x.(x.y) ‖ .Proof. First assume thatA is of the formB(λ) for someC∗-algebraB and somereal numberλ with 0 ≤ λ ≤ 1. Denote by◦ the product ofB, and, forz, t inA, put[z, t]◦ := z ◦ t − t ◦ z. Then, for everyy in A, we have[x, y] = (2λ− 1)[x, y]◦,hence‖ [x, y] ‖≤‖ [x, y]◦ ‖, and therefore, since the symmetrized products ofA

andB(λ) coincide, the result follows from Lemma 1.1.Putting together Lemma 1.2 and the above paragraph, we obtain that the result

is true wheneverA is a non-commutativeJBW ∗-factor of Type I.Now, remove any additional assumption on the non-commutativeJB∗-algebra

A. For every Type I factor representationϕ of A we have

‖ ϕ([x, y]) ‖2 =‖ [ϕ(x), ϕ(y)] ‖2

≤ 16 ‖ ϕ(y) ‖‖ (ϕ(x))2.ϕ(y)− ϕ(x).(ϕ(x).ϕ(y)) ‖= 16 ‖ ϕ(y) ‖‖ ϕ(x2.y − x.(x.y)) ‖≤ 16 ‖ y ‖‖ x2.y − x.(x.y) ‖ .

Therefore

‖ [x, y] ‖2= supϕ

‖ ϕ([x, y]) ‖2≤ 16 ‖ y ‖‖ x2.y − x.(x.y) ‖ . ��Corollary 1.4. Every non-commutative JB∗-algebra has an increasing approxi-mate unit consisting of positive elements with norm ≤ 1.

Proof. LetA be a non-commutativeJB∗-algebra. Take an increasing approximateunit {yλ}λ∈� of theJB-algebraAsa satisfying 0≤ yλ and‖ yλ ‖≤ 1 for all λ in�[10, Proposition 3.5.4], and letx be inAsa . Then, by Theorem 1.3, we have

‖ [x, yλ] ‖2≤ 16 ‖ x2.yλ − x.(x.yλ) ‖for all λ in �. But, since lim{x2.yλ − x.(x.yλ)} = 0, we obtain

lim{[x, yλ]}λ∈� = 0,

hence

lim{xyλ}λ∈� = lim{x.yλ + 1

2[x, yλ]}λ∈� = x

and

lim{yλx}λ∈� = lim{x.yλ − 1

2[x, yλ]}λ∈� = x.

Finally, keep in mind thatA = Asa ⊕ iAsa . ��

472 K. El Amin et al.

2. Complete normed algebras whose biduals are non-commutativeJB∗-algebras

From now on, the bidualA∗∗ of any normed algebraA will be regarded withoutnotice as a normed algebra relative to the Arens product [1]. The aim of this sectionis to show that, ifA is a complete normed complex algebra, and ifA∗∗ is a non-commutativeJB∗-algebra for some involution, thenA is a self-adjoint subset ofA∗∗, and thereforeA is a non-commutativeJB∗-algebra, too. In the proof, wewill apply several times the fact, first noticed in [26], that theJB∗-involution ofa unital non-commutativeJB∗-algebraA is determined by the Banach space ofA and the unit1. Indeed, forx in such an algebraA, the assertionsx = x∗ andx ∈ H(A, 1) are equivalent. As a consequence, if a unital normed complex algebrais a non-commutativeJB∗-algebra for some involution, then such an involution isunique. In the proof of the following lemma, for a subsetS of a complex vectorspace, the symbol LinR(S) will stand for the real linear hull ofS.

Lemma 2.1. Let A be a unital complete normed complex algebra such that A∗∗is a non-commutative JB∗-algebra for some involution ∗. Then A is a ∗-invariantsubset of A∗∗, and hence a non-commutative JB∗-algebra.

Proof. First note that the unit1 of A is a unit forA∗∗, and therefore we have‖ 1 ‖= 1. Letf be in LinRD(A, 1) ∩ iLinRD(A, 1). Then there areg, h, r, s inD(A, 1) andα, β, γ, δ in R

+0 satisfying

f = αg − βh = i(γ r − δs).

Denoting byj the canonical injection ofA∗ into A∗∗∗, j (g), j (h), j (r), j (s) areelements ofD(A∗∗, 1), and the equalities

j (f ) = αj (g)− βj (h) = i(γj (r)− δj (s))

hold. Thereforej (f ) belongs to LinRD(A∗∗, 1)∩iLinRD(A∗∗, 1). SinceA∗∗ is a

non-commutativeJB∗-algebra, we can apply [13, Corollary 4.3] to conclude thatj (f ) = 0, and thereforef = 0.Now, we have LinRD(A, 1)∩iLinRD(A, 1) = 0,and hence, again by [13, Corollary 4.3],A is a non-commutativeJB∗-algebrarelative to the involution� given by(x + iy)� = x − iy for x, y in H(A, 1).SinceH(A, 1) = H(A∗∗, 1) ∩ A and(x + iy)∗ = x − iy for x, y in H(A∗∗, 1),it follows thatA is a∗-invariant subset ofA∗∗. ��

The proof of the next lemma involves the concept of aJC∗-algebra.JC∗-algebras are defined as thoseJB∗-algebras which can be seen as norm-closedself-adjoint subalgebras ofA+ for someC∗-algebraA.

Lemma 2.2. Let A be a non-commutative JB∗-algebra, and e an idempotent in Asuch that ‖ e ‖= 1. Then e∗ = e.

Proof. We may assume thatA is commutative. Then the closed subalgebra ofA

generated by{e, e∗} is aJC∗-algebra [25, Corollaries 1.4 and 1.2]. Thereforee canbe regarded as a norm-one idempotent in aC∗-algebra, so that, byC∗-folklore (seefor instance [17, p. 27]), we havee∗ = e, as required. ��

A holomorphic characterization ofC∗- andJB∗-algebras 473

The next lemma is surely of folklore type.

Lemma 2.3. LetB be a normed algebra, andA a subalgebra of B. Then the bipolarA00 of A in B∗∗ is a subalgebra of B∗∗, and the natural identification of A00 withA∗∗ as Banach spaces becomes an algebra isomorphism.

Proof. We recall that, for any normed algebraC, the Arens product ofC∗∗ isthe unique norm-continuous bilinear mappingf : C∗∗ × C∗∗ → C∗∗ extendingthe product ofC and such thatf (. , d) andf (c, .) areσ(C∗∗, C∗)-to−σ(C∗∗, C∗)continuous wheneverd andc are inC∗∗ andC, respectively. Letx, y be inA00.Thenthere are nets{xλ} and{yµ} inA such thatx = w∗− lim{xλ} andy = w∗− lim{yµ}(herew∗ meansσ(B∗∗, B∗)). Therefore we have that

xy = w∗ − limλ(w∗ − lim

µ{xλyµ})

lies inA00. Moreover, since the natural identification

5 : A00 → A∗∗

is (σ (B∗∗, B∗)|A00)-to-σ(A∗∗, A∗) continuous and becomes the identity mappingonA, we deduce that5(xy) = 5(x)5(y). ��

From now on, it is convenient to keep in mind that every non-commutativeJBW ∗-algebra has a unit [15, Corollary 3.3].

Theorem 2.4. LetAbe a complete normed complex algebra such thatA∗∗, endowedwith a suitable involution ∗, is a non-commutative JB∗-algebra. Then A is a ∗-invariant subset of A∗∗, and hence a non-commutative JB∗-algebra.

Proof. In view of Lemma 2.1, we may assume thatA does not have a unit. However,sinceA∗∗ is a non-commutativeJBW ∗-algebra,A∗∗ certainly has a unit1. PutB := A∗∗ (actually, we can takeB equal to any non-commutativeJB∗-subalgebraofA∗∗ containing1 andA), and consider the closed subalgebraC := C1+A of B.Taking bipolars inB∗∗, we obtainC00 = C1 +A00. From Lemma 2.3 and the factthatA∗∗ is a non-commutativeJBW ∗-algebra we deduce in a first instance thatA00

has a norm-one unit element (saye). SinceB∗∗ is a non-commutativeJB∗-algebrafor some involution (say� ) [15, Theorem 1.7], ande is an idempotent inB∗∗, itfollows from Lemma 2.2 thate is � -invariant. Then, keeping in mind [14, p. 188],the set

D := {x ∈ B∗∗ : xe = ex = x}is a non-commutativeJB∗-subalgebra ofB∗∗ with unit e, so that, forx in D,the assertionsx = x� and x ∈ H(D, e) are equivalent. On the other hand,the assumption thatA∗∗ is a non-commutativeJB∗-algebra for some involution,together with Lemma 2.3, provides us with an involution� on A00 such that(A00,�) is a non-commutativeJB∗-algebra. SinceA00 is contained inD, andH(A00, e) = H(D, e) ∩ A00, it follows that, forx in A00, the assertionsx = x�and x = x� are equivalent. Therefore the involution� on A00 is nothing butthe restriction of� to A00, and henceA00 is a �-invariant subset ofB∗∗. Then

474 K. El Amin et al.

C00 = C1 + A00 is a non-commutativeJB∗-subalgebra ofB∗∗, and hence, byLemma 2.3 (withC replacingA),C∗∗ is a non-commutativeJB∗-algebra for someinvolution (say�).

In what follows, the algebraB is not needed any longer. SinceC has a unit,Lemma 2.1 assures thatC is a �-invariant subset ofC∗∗. SinceC = C1 + A,A is also�-invariant [15, Corollary 1.11], and thereforeA is a non-commutativeJB∗-algebra for the involution�.

The involution� on A extends to an involution onA∗∗ convertingA∗∗ intoa non-commutativeJB∗-algebra [15, Theorem 1.7], and this extended involutionmust coincide with the given involution∗ on A∗∗ due to the uniqueness of theJB∗-involution on unital non-commutativeJB∗-algebras. ��

3. The main result

In this section we prove the main result in the paper. Namely, a normed complexalgebra is a non-commutativeJB∗-algebra for some involution if and only if it islinearly isometric to aJB∗-triple and has an approximate unit bounded by one.

LetX be a normed space, andu an element inX. Following [2], we say thatuis avertex of the closed unit ball ofX if ‖ u ‖= 1 and the conditionsx ∈ X andϕ(x) = 0 for allϕ inD(X, u) imply x = 0. Now letQ be a subset ofX. Following[6, p. 285], we define thetangent cone to Q at u, Tu(Q), as the set of allx ∈ Xsuch that

x = limxn − u

tn

for some sequence{xn} inQ with lim{xn} = u and some sequence{tn} of positivereal numbers. WhenX is complex, theholomorphic tangent cone toQ at u, Tu(Q),is defined as

Tu(Q) := ∩λ∈C\{0}λTu(Q).

Lemma 3.1. Let X be a nonzero complex normed space, and u an element in X.Then u is a vertex of BX if and only if Tu(�X) = 0.

Proof. Assume thatu is a vertex ofBX. If x = lim xn−utn

for some sequence{xn}in�X with lim{xn} = u and some sequence{tn} of positive real numbers, and ifϕis inD(X, u), then from the equality

ϕ(x) = limϕ(xn)− 1

tn

we deduceRe(ϕ(x)) ≤ 0. It follows that, ifx is actually inTu(�X), then we haveϕ(x) = 0 for everyϕ in D(X, u), and hence, sinceu is a vertex ofBX, x = 0.ThereforeTu(�X) = 0.

Now assume Tu(�X) = 0. We note that a first easy consequence of thiscondition is that‖ u ‖= 1, and hence, by [9, Theorem V.9.5], we have

limα→0+

‖ u+ αx ‖ −1

α= Max{Re(ϕ(x)) : ϕ ∈ D(X, u)}

A holomorphic characterization ofC∗- andJB∗-algebras 475

for everyx in X. Let x be inX such thatϕ(x) = 0 wheneverϕ is inD(X, u). Forevery positive real numberα, put

xα := u+ αx

‖ u+ αx ‖ +α2 ∈ �X.

From the equality

limα→0+

‖ u+ αx ‖ −1

α= 0

we deduce

limα→0+

xα − u

α= x.

In this way,x belongs toTu(�X). For everyλ in C \ {0}, we can apply the aboveargument toλx instead ofx, so thatx ∈ Tu(�X) = 0. ��

The above proof actually shows that, for an arbitrary norm-one elementu of acomplex normed spaceX, we have

Tu(�X) = {x ∈ X : ϕ(x) = 0 ∀ϕ ∈ D(X, u)}.A JB∗-triple is a complex Banach spaceJ with a continuous triple product

{...} : J × J × J → J which is linear and symmetric in the outer variables, andconjugate-linear in the middle variable, and satisfies:

1. For allx in J , the mappingy → {xxy} from J to J is a hermitian operator onJ and has nonnegative spectrum;

2. Themain identity

{ab{xyz}} = {{abx}yz} − {x{bay}z} + {xy{abz}}holds for alla, b, x, y, z in J ;

3. ‖ {xxx} ‖=‖ x ‖3 for everyx in J .

For a vector spaceE, let L(E) denote the associative algebra of all linearmappings fromE toE, and for a non-commutative Jordan algebraA, let (x, y) →Ux,y be the unique symmetric bilinear mapping fromA × A to L(A) satisfyingUx,x = Ux for everyx in A. It is well-known that every non-commutativeJB∗-algebraA is aJB∗-triple under the triple product{...} defined by{xyz} := Ux,z(y

∗)for all x, y, z in A (see for instance [6] and [27]).

Let X be a normed space. By aproduct onX we mean a continuous bilinearmapping fromX × X into X. Given a productf onX, we can replace the normofX with a suitable positive multiple in such a way that(X, f ) becomes a normedalgebra (sayA). The productf onX is said to beArens regular if the equality(A∗∗)op = (Aop)∗∗ holds, where theopposite algebra Bop of any algebraB (withproduct(x, y) → xy) is defined as the algebra whose vector space is the one ofB

and whose product is(x, y) → yx.

Lemma 3.2. Every product on a JB∗-triple is Arens regular.

476 K. El Amin et al.

Proof. Let J be aJB∗-triple. By [7, Lemma 5], every bounded linear mappingfrom J to J ∗ factors through a Hilbert space, and hence is weakly compact. Then,by [20, Theorem], every product onJ is Arens regular. ��

Now we are ready to conclude the proof of the main result in the paper.

Theorem 3.3. A normed complex algebra A is a non-commutative JB∗-algebra(for some involution) if and only if A is linearly isometric to a JB∗-triple and hasan approximate unit bounded by one.

Proof. The “only if” part follows from Corollary 1.4 and the already commentedfact that non-commutativeJB∗-algebras areJB∗-triples in a natural manner. Toprove the “if” part, letA be a normed complex algebra with an approximate unitbounded by one and such that there exists a linear isometryφ from A onto someJB∗-triple J . SinceA is linearly isometric to aJB∗-triple and every product on aJB∗-triple is Arens regular (by Lemma 3.2),A is an Arens regular normed algebra,and therefore the product ofA∗∗ is w∗-continuous in each of its variables. Then,sinceA has an approximate unit bounded by one, a straightforward verificationshows thatA∗∗ has a unit1 with ‖ 1 ‖= 1 (take aw∗-limit point in A∗∗ ofthe approximate unit ofA). By the non-associative version of the Bohnenblust–Karlin theorem [13, p. 617],1 is a vertex ofBA∗∗ . Considering the surjective linearisometryφ∗∗ : A∗∗ → J ∗∗, we obtain thatφ∗∗(1) is a vertex ofBJ ∗∗ , and hence,by Lemma 3.1, we haveTφ∗∗(1)(�J ∗∗) = 0. On the other hand,J ∗∗ is a JB∗-triple [8], and hence�J ∗∗ is homogeneous under biholomorphic automorphisms[23, Corollary 20.13]. It follows from [6, Theorem 4.4] thatJ ∗∗ is the underlyingBanach space of aJB∗-algebra with unitφ∗∗(1). Therefore we have

J ∗∗ = H(J ∗∗, φ∗∗(1))+ iH(J ∗∗, φ∗∗(1)),

and henceA∗∗ = H(A∗∗, 1)+ iH(A∗∗, 1).

Now, by the non-associative Vidav–Palmer theorem [19, Theorem 12],A∗∗ is anon-commutativeJB∗-algebra. Finally, by Theorem 2.4,A is a non-commutativeJB∗-algebra. ��

The condition in the above theorem thatA has an approximate unit boundedby one cannot be omitted. Indeed, takeA equal to the Banach space of an arbitrarynonzeroJB∗-triple, endowed with the zero product. More illuminating examplesof complex normed algebrasA which are linearly isometric toJB∗-triples but arenot non-commutativeJB∗-algebras are listed in what follows.

1. The associative normed algebraA whose normed space is that of an arbitraryJB∗-triple J of dimension≥ 2, and whose product is given byxy := f (x)y,wheref is a norm-one element ofJ ∗ such thatf (e) = 1 for some norm-oneelemente of J . In this casee becomes a norm-one left unit forA.

2. The associative and commutative normed algebraAwhose normed space is thatof an arbitrary nonzeroJB∗-triple J , and whose product is given byxy :=13(f (x)y+f (y)x−f (x)f (y)e), wheref ande are chosen as in the first-listedexample. In this case 3e is a unit forA.

A holomorphic characterization ofC∗- andJB∗-algebras 477

3. The non-commutative Jordan normed algebraA obtained from an arbitrarynonzero non-commutativeJB∗-algebra by replacing its natural productxy withMxy, where 0< M < 1. In this caseA is bicontinuously isomorphic to a non-commutativeJB∗-algebra, and hence has a bounded approximate unit.

4. The associative normed algebraA of all Hilbert-Schmidt operators on a complexHilbert spaceH of dimension≥ 2. If H is finite-dimensional, thenA has a unit.Otherwise,A has an unbounded approximate unit.

The following corollaries follow from the Theorem 3.3.

Corollary 3.4. An associative normed complex algebra A is a C∗-algebra if andonly ifA is linearly isometric to a JB∗-triple and has an approximate unit boundedby one.

Corollary 3.5. A normed complex algebra A is a non-commutative JB∗-algebraif and only if A is linearly isometric to a non-commutative JB∗-algebra and hasan approximate unit bounded by one.

Corollary 3.4 improves [21, Corollary 1.3]. Concerning Corollary 3.5, wenote that linearly isometric non-commutativeJB∗-algebras need not be Jordan-∗-isomorphic [6, Example 5.7].

Acknowledgements. Part of this work was done while the third author was visiting the Uni-versity of Almería. He is grateful to the Department of Algebra and Mathematical Analysisof that University for its hospitality and support.

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