THE STRUCTURE OF A SUBCLASS OF AMENABLE BANACH ALGEBRAS

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PII. S0161171204401069 http://ijmms.hindawi.com

THE STRUCTURE OF A SUBCLASS OF AMENABLE BANACH ALGEBRAS

R. EL HARTI Received 8 January 2004

We give sufficient conditions that allow contractible (resp., reflexive amenable) Banach algebras to be finite-dimensional and semisimple algebras. Moreover, we show that any contractible (resp., reflexive amenable) Banach algebra in which every maximal left ideal has a Banach space complement is indeed a direct sum of finitely many full matrix algebras.

Finally, we characterize Hermitian^∗-algebras that are contractible.

2000 Mathematics Subject Classiﬁcation: 46H25, 18G55, 46M20.

1. Introduction. The purpose of this note is to establish the structure of some class of amenable Banach algebras. Let Ꮽ be a Banach algebra over the complex field C. We define a Banach leftᏭ^-moduleᐄto be a Banach space which is also a unital left Ꮽ-module such that the linear mapᏭ×ᐄ→ᐄ,(a,x)→ax, is continuous. Right modules are defined analogously. A BanachᏭ-bimodule is a Banach space with a structural Ꮽ-bimodule such that the linear mapᏭ×ᐄ×A→ᐄ^,(a×x×b)→axb, is jointly continuous, whereᏭ×ᐄ×Ꮽcarries the Cartesian product topology. A submoduleᐅof a Banachleft, right, bi-Ꮽ-moduleᐄis a closed subspace ofᐄwith the structural Banach left, right, or bi-Ꮽ-module. A Banach leftᏭ-module morphismθ:ᐄ→ᐅis a continuous linear map between two left BanachᏭ-modules such thatθ(ax)=aθ(x)for all a∈Ꮽ and allx∈ᐄ. A Banach rightᏭ-module morphism and a BanachᏭ-bimodule morphism are defined analogously. For each Banachleft, bi-moduleᐄonᏭ, the dual ᐄ^∗ is naturally a Banachleft, bi- Ꮽ-bimodule with the module actions defined by aT (x)=T (xa), aT (x)=T (xa), andT a(x)=T (ax), for alla∈Ꮽ, T ∈ᐄ^∗, and x∈ᐄ, whereT (x)denotes the evaluation ofT atx. Ifᐄ,ᐅ, andᐆare Banachleft, or bi-ᐄ-modules andθ:ᐄ→ᐅ^,β:ᐅ→ᐆâreleft, bi-module morphisms, then the sequence

Σ: 0 →ᐄ →ᐅ →ᐆ →0 (1.1)

is exact ifθis one-to-one,β=ᐆ, andθ=kerβ. The exact sequenceΣis admissible ifβ has a continuous right inverse, equivalently, kerβhas a Banach space complement inᐅ. The admissible exact sequence splits if the right inverse ofβis Banachleft, bi-module, equivalently, kerβis a Banach space complement inᐅwhich is anᏭ-submodule.

A derivation fromᏭ into a Banach Ꮽ-bimodule ᐄis a linear operator D:Ꮽ→ᐄ which satisﬁesD(ab)=D(a)b+aD(b), for alla,b∈Ꮽ. Recall that for anyx∈ᐄ^{, the} mappingδx:Ꮽ→ᐄdeﬁned byδx(a)=ax−xa, a∈Ꮽ, is a continuous derivation,

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called an inner derivation. A Banach algebraᏭis said to be contractible if for every BanachᏭ-bimoduleᐄ, each continuous derivation fromᏭintoᐄis inner. We say that Ꮽis amenable whenever every continuous derivation fromᏭintoᐄ^∗is inner for each Banach Ꮽ^-bimodule ᐄ. Obviously, every contractible Banach algebra is an amenable Banach algebra and the converse is true in the finite-dimension case. It is well known that a finite-dimensional algebra is semisimple if and only if it is isomorphic to a finite Cartesian product of a family of full matrix algebras. UsingTheorem 2.1, it is easy to check that a finite Cartesian product of a family of full matrix algebras is contractible.

The purpose of this note is to contribute to the study of the following questions, raised, respectively, in [2], [3, page 817], and [5, page 212].

Question1.1. Is every contractible Banach algebra semisimple?

Question1.2. Is every reﬂexive amenable Banach algebra ﬁnite-dimensional and semisimple?

Question1.3. Is every contractible Banach algebra ﬁnite-dimensional?

Recall that a Banach algebra is called a reflexive Banach algebra if it is reflexive as a Banach space. In this note, we will present two situations in which a contractible Banach algebra is finite-dimensional. First, we will give a partial answer to the above questions, where we assume that each maximal left ideal is complemented as a Banach space. This result improves [5, Proposition IV.4.3] for contractible Banach algebras and [3, Corollary 2.3] for reflexive amenable Banach algebras, where the authors suppose only that all of their primitive ideals have finite codimensions. Second, we will show that a Hermitian Banach^∗-algebra is contractible if and only if it is a finite-dimensional semisimple algebra.

2. Preliminaries. In this section, we recall some facts about the structure of contractible and amenable Banach algebras. Let Ꮽ be a Banach algebra over the complex fieldCand letᏭ^∗∗ be the bidual ofᏭwith the usual multiplication defined by ψ·φ(f )=ψ(f )φ(f )for allψ,φ∈Ꮽ^∗∗andf∈Ꮽ^∗. Consider onᏭ^∗∗the BanachᏭ- bimodule structure defined byaT=η(a)T,T a=T η(a)withη:Ꮽ→Ꮽ^∗∗the canonical map. Notice that if a Banach algebraᏭhas a bounded approximate identity, then its bidualᏭ^∗∗has an identity. It is a fact that a contractible Banach algebra has an identity and an amenable Banach algebra admits bounded right, left, bilateral approximate identities. Of course, a reflexive amenable Banach algebra must be unital. We denote the identity element ofᏭby 1 and we writeᏭ⊗ˆᏭfor the completed projective tensorial product (see [4]). The Banach spaceᏭ⊗^ˆᏭis a BanachᏭ-bimodule if we define

a(b⊗c)=ab⊗c, (b⊗c)a=b⊗ca, a,b,c∈Ꮽ. (2.1)

For a unital Banach algebraᏭ, a diagonal of Ꮽis an element d∈Ꮽ⊗^ˆᏭ such that ad=da, for alla∈Ꮽ^{, and}π(d)=1, whereπ:Ꮽ⊗^ˆᏭ→Ꮽis the canonical BanachᏭ^- bimodule morphism. For such a Banach algebraᏭ, a virtual diagonal ofᏭis an element

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d∈(Ꮽ⊗^ˆᏭ)^∗∗such that

ad=da, ∀a∈Ꮽ, π^∗∗(d)=1, (2.2)

whereπ^∗∗:(Ꮽ⊗^ˆᏭ)^∗∗→Ꮽ^∗∗is the bidual BanachᏭ-module morphism ofπ. In the following theorems, we present characterizations of contractible (resp., amenable) Banach algebras. We recall, respectively, [1, Theorem 6.1] and [6, Theorem 1.3].

Theorem2.1. LetᏭbe a Banach algebra. The following are equivalent:

(1) Ꮽis contractible;

(2) Ꮽhas a diagonal.

Theorem2.2. LetᏭbe a Banach algebra. The following are equivalent:

(1) Ꮽis amenable;

(2) Ꮽhas a virtual diagonal.

We choose as a basis of the algebraMn(C)of alln×ncomplex matrices the set of el- ementary matriceseij. Considerd=n

i,jδijeij⊗eji∈Mn(C)⊗Mn(C). ThenMd=dM, for allM∈Mn(C), andπ(d)=1, whereπ:Mn(C)⊗Mn(C)→Mn(C)is the canonical morphism. It follows thatMn(C)is contractible.

Next, the following propositions hold.

Proposition2.3. LetᏭ^{be a}contractible, amenableBanach algebra. Then, ifθ: Ꮽ→Ꮾis a continuous homomorphism fromᏭinto another Banach algebraᏮwith dense range, thenᏮiscontractible, amenable. In particular, ifᏵis a closed two-sided ideal of acontractible, amenableBanach algebraᏭ^{, then}Ꮽ/Ᏽ^iscontractible, amenabletoo.

Proof. Assume thatᏭis contractible. Letᐄbe a BanachᏮ-bimodule. Consider on ᐄthe structure ofᏭ-bimodule deﬁned bya·x=θ(a)xandx·a=xθ(x). Sinceθ is continuous,ᐄis a BanachᏭ-bimodule. Now, letD:Ꮾ→ᐄbe a continuous derivation.

It is easy to see thatD◦θis a continuous derivation fromᏭto the BanachᏭ^-bimodule ᐄ, and thus it is inner. Therefore, there existsx∈ᐄsuch thatD(θ(a))=a·x−x·a= θ(a)x−xθ(a)for alla∈Ꮽ. Sinceθ(Ꮽ)is dense inᏮ, we haveD(b)=bx−xbfor all b∈Ꮾ. It follows thatDis inner andᏮis contractible. IfᏭis amenable, we will consider a continuous derivationD:Ꮾ→ᐄ^∗ fromᏮto the dual of the bimoduleᐄand we use the same way to prove thatᏮis amenable.

Proposition2.4[1, Theorems 2.3 and 2.5]. LetᏭbe an amenable Banach algebra and let

Σ: 0 →ᐄ^∗ →ᐅ →ᐆ →0 (2.3)

be an admissible short exact sequence of Banach (left, right, or bi-) modules withᐄ^∗^a dual ofᐄ. ThenΣsplits.

Proposition2.5[1, Theorem 6.1]. LetᏭbe a contractible Banach algebra and let

Σ: 0 →ᐄ →ᐅ →ᐆ →0 (2.4)

be an admissible short exact sequence of Banach (left, right, or bi-) modules. ThenΣsplits.

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Remark 2.6. Notice that for each closed two-sided idealᏵ of a reﬂexive Banach algebra,Ᏽand the quotientᏭ/Ᏽare reﬂexive Banach algebras too.

Proposition2.7. LetᏭbe a contractible or reflexive amenable Banach algebra and assume thatᏵis a closedleft, two-sidedideal ofᏭwhich has a Banach space comple- ment. Then there exists a closedleft, two-sidedideal᏶^ofᏭ^{such that}

Ꮽ=Ᏽ+᏶. (2.5)

Proof. LetᏭbe an amenable Banach algebra and letᏵbe a closedleft, two-sided ideal ofᏵwhich has a Banach space complement. Then the short exact sequenceΣ: 0→Ᏽ→Ꮽ→Ꮽ/Ᏽ→0 is admissible. IfᏭis reﬂexive, then the spaceᏵwill be the same, and so it will be the dual of the Banachleft, bi-Ꮽ-moduleᏵ^∗. ByProposition 2.4,Σ splits andᏵhas a Banach space complement which is aleft, two-sidedideal. WhenᏭ is contractible, byProposition 2.5, we have the result.

3. Main results

Theorem3.1. LetᏭbe a contractible or reflexive amenable Banach algebra. Assume that each maximal left ideal ofᏭis complemented as a Banach space inᏭ. Then there aren1,n2,...,nk∈Nsuch that

ᏭMn1(C)⊕Mn2(C)⊕···⊕Mn_k(C). (3.1) Proof. BySection 2, the algebraᏭhas an identity 1_Ꮽ. Let(ᏹi)i∈I be the family of all maximal left ideals. Sinceᏹiis complemented as a Banach space for eachi, there exists a left ideal᏶isuch thatᏭ=ᏹi⊕᏶i. Notice that

Rad(Ꮽ)=

i

ᏹi (3.2)

is the Jacobson radical ofᏭ^and

i

᏶i⊆Soc(Ꮽ), (3.3)

where Soc(Ꮽ)is the socle of the algebraᏭ, that is, it is the sum of all minimal left ideals ofᏭand it coincides with the sum of all minimal right ideals ofᏭ. Recall that every minimal left ideal of Ꮽ is of the formᏭe, where e is a minimal idempotent, that is,e²=e≠0 andeᏭe=Ce. On the other hand, for each ﬁnite family of minimal idempotents(ek)k∈K, we have

Ꮽ=

k∈K

Ꮽek k∈K

Ꮽ1_Ꮽ−ek

. (3.4)

It follows from (3.3) and (3.4) that Soc(Ꮽ)is dense inᏭ/Rad(Ꮽ). This shows that Ꮽ/Rad(Ꮽ)is ﬁnite-dimensional. Therefore

Ꮽ=Rad(Ꮽ)

Soc(Ꮽ). (3.5)

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If Rad(Ꮽ)≠{0}, this would mean that Rad(Ꮽ)has an identity, which is impossible.

So,Ꮽ=Soc(Ꮽ), and then it is a ﬁnite direct sum of certain full matrix algebras.

Corollary3.2. Every commutativecontractible, reflexive amenableBanach alge- braᏭis finite-dimensional and semisimple.

Corollary3.3. LetᏭbe a contractible or reflexive amenable Banach algebra such that every irreducible representation of Ꮽ is finite-dimensional. Then Ꮽ ^{is finite-} dimensional and semisimple.

Proof. It is easy to check that every primitive ideal of a Banach algebra is ﬁnite- codimensional if and only if each of its maximal left ideals is ﬁnite-codimensional. So, the corollary follows.

It should be emphasized that the following result appears in [9] or [5, Corollary in page 212].

Corollary 3.4. Every contractible, reflexive amenable C^∗-algebra Ꮽ is finite- dimensional and semisimple.

Proof. Suppose thatᏭis a contractible or reﬂexive amenableC^∗-algebra. Letᏹbe a maximal left ideal. By [7, Theorems 5.3.5 and 5.2.4], the spaceᏭ/ᏹis a Hilbert space.

It follows that the short exact sequence

Σ: 0 →ᏹ →Ꮽ →Ꮽ/ᏹ →0 (3.6)

is admissible, and thusᏹhas a Banach space complement. ByTheorem 3.1,Ꮽ^{is iso-} morphic to a ﬁnite direct sum of full matrix algebras.

Remark3.5. Recall that a simple algebra is an algebra which has no proper ideals other than the zero ideal. To show that everycontractible, reflexive amenable Ba- nach algebra is finite-dimensional and semisimple, it suffices to prove that everycontractible, reflexive amenablesimple contractible Banach algebra is finite-dimensional.

Indeed, letᏭbe a contractible Banach algebra. Letᏼbe a primitive ideal ofᏭ. Then the algebraᏭ/ᏼ^{is a}contractible, reflexive amenableBanach algebra. PutᏮ=Ꮽ/ᏼând consider some maximal two-sided idealᏹofᏮ. SinceᏮ/ᏹis acontractible, reflexive amenablesimple Banach algebra, it is finite-dimensional. There exists then a closed two-sided ideal᏶^{such that}Ꮾ=ᏹ⊕᏶. Recall that in a primitive algebra, every nonzero ideal is essential, that is, it has a nonzero intersection with every nonzero ideal of the algebra. It follow thatᏹ=0, and soᏮis finite-dimensional. UsingCorollary 3.2,Ꮽmust be a finite-dimensional and semisimple algebra. This completes the proof.

Proposition3.6. LetᏭ^{be a}contractible, reflexive amenablesimple contractible Banach algebra having a maximal left ideal complemented as a Banach space. ThenᏭ is finite-dimensional.

Proof. IfᏭis an infinite-dimensional simple algebra, then Soc(Ꮽ)=0. Moreover, ifᏭîscontractible, reflexive amenablewith a maximal left ideal complemented as a Banach space, thenᏭhas a nontrivial minimal left ideal. This is a contradiction.

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Now, assume thatᏭis a unital Banach^∗-algebra which admits at least one stateτ. Then there exists a^∗-representationπτofᏭon a Hilbert spaceHτ, with a cyclic vector ζ of norm 1 inHτ such thatτ(a)= πτ(a)ζ,ζ, for all a∈Ꮽ,·,·being the inner product inHτ.

Theorem3.7. A Hermitian Banach^∗-algebraᏭis contractible if and only if there aren1,n2,...,nk∈Nsuch that (3.1) holds.

Proof. It suﬃces to show the “only if” part. Suppose that a Hermitian Banach al- gebraᏭ is contractible. Let T(Ꮽ) be the set of all states ofᏭand let R^∗(Ꮽ)be the

∗-radical ofᏭ, that is, the intersection of the kernels of all^∗-representations ofᏭon Hilbert spaces. SinceᏭis Hermitian and has an identity, T(Ꮽ)≠∅, and soR^∗(Ꮽ)≠Ꮽ^. Putπ =

τ∈T(Ꮽ)πτ and H=

τ∈T(Ꮽ)Hτ. Then π is a ^∗-representation ofᏭ on H. Consider

π(a)= sup

τ∈T(Ꮽ)

πτ(a). (3.7)

Then · is aC^∗-norm onπ(A). LetᏮdenote the closure of(π(Ꮽ), · ). Moreover, π:Ꮽ→Ꮾis a continuous mapping into aC^∗-algebraᏮsuch that ker(π)=R^∗(Ꮽ). AsᏭ is contractible,Ꮾis also contractible. UsingCorollary 3.4, the algebraᏮhas to be finite- dimensional. Notice thatᏭ/R^∗(Ꮽ)is isometric with the^∗-subalgebraπ(Ꮽ)ofᏮ^{. Thus,} it follows thatᏭ/R^∗(Ꮽ) is finite-dimensional. SinceR^∗(Ꮽ) is a finite-codimensional closed two-sided^∗-ideal, there exists a closed two-sided ideal᏷such that

Ꮽ=R^∗(Ꮽ)⊕᏷. (3.8)

Next, note thatπ(a)²=sup{τ(a^∗a), τ∈T(Ꮽ)} ≥ |a^∗a|σ, where|a|σis the spectral radius of a∈Ꮽ. By Pták [8], we obtainπ(a)²≥ |a|²σ. So, if a∈R^∗(Ꮽ), then

|a|σ=0. Therefore, every element ofR^∗(Ꮽ)is quasinilpotent. Notice that in general Rad(Ꮽ)⊆R^∗(Ꮽ). SinceR^∗(Ꮽ)is a closed two-sided^∗-ideal, we haveR^∗(Ꮽ)=Rad(Ꮽ), and soᏭis ﬁnite-dimensional and semisimple.

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R. El Harti: Faculty of Sciences and Techniques (FST), University Hassan I-Settat, BP 577, 2600 Settat, Morocco

E-mail address:[email protected]

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