Banach algebras with trivial cohomology

First Advanced Course in Operator Theory and Complex Analysis, University of Seville, June 2004 BANACH ALGEBRAS WITH TRIVIAL COHOMOLOGY RACHID EL HARTI Abstract. A plausible conjecture is that Banach algebras with trivial cohomology have to be semisimple and finite dimensional. In this paper, we show that a Hermitian Banach ∗-algebra has trivial cohomology if and only if it is ∗-isomorphic to a finite direct sum of full matrix algebras. 1. Introduction Let (A,k·k,∗) be a Banach algebra Awith an identity and some involution. Ais called a Hermitian Banach ∗-algebra if for every element ain A,a∗ahas real spectrum. It is said to be a C∗-algebra if its norm k·k is a C∗-norm, i.e, k·k satisfies: ka∗ak=kak2for all a∈ A. Obviously, a C∗-algebra is a Hermitian Banach ∗-algebra. A state τis a linear form on Asuch that τ(1) = kτk= 1. A∗-representation of Aon a pre-Hilbert space Xis a ∗-homomorphism of A into the algebra B(X) of bounded operator on X. Given a Banach algebra, A, we define a Banach left A-module Xto be a Banach space which is a structure left module Asuch that the linear map (a, x)∈ A × X 7−→ ax ∈ X is continuous. Right modules are defined analogously. A Banach A-bimodule is a Banach space with a structure bimodule over Asuch that the linear map (a, x, b)∈ A × X × A 7−→ axb ∈ X is continuous. A Banach A-submodule of a given Banach A-module Xis at the same time a closed subspace of Xand an algebraic A-submodule of X. A Banach left A-module morphism θ:X → Y is a continuous linear map Revised December 15, 2004. This is an improved version of a previous paper by the same author, [2]. 79 80 R. EL HARTI between two left Banach A-modules such that θ(ax) = aθ(x) for all a∈ A and all x∈ X. Banach right A-module morphisms and Banach A-bimodule morphisms are defined analogously. For each Banach A-bimodule X, the dual X∗is naturally a Banach bimodule over Awith the module actions defined by aT(x) = T(xa) and Ta(x) = T(ax), for all a∈ A,T∈ X∗, and x∈ X; where T(x) denotes the evaluation of Tat x. A derivation from Ainto a Banach A-bimodule Xis a linear operator D: A→Xwhich satisfies D(ab) = D(a)b+aD(b), ∀a, b ∈ A. Notice that for any x∈ X, the mapping δx:A → X defined by δx(a) = ax −xa, a ∈ A, is a continuous derivation called an inner derivation. An n-linear bounded operator T:A× n) · · · ×A→ X is called a n-cochain. The set of n-cochains forms a Banach space denoted by Cn(A,X). The standard cohomological complex is 0→ X δ0 −→ C1(A,X)δ1 −→ · · · δn−1 −→ Cn(A,X)δn −→ Cn+1(A,X)δn+1 −→ · · · , where δnis defined as follows for n≥1: (δnT)(a1, . . . , an+1) =a1T(a2, . . . , an+1) + n X k=1 (−1)kT(a1, . . . , ak−1, ak+1, . . . , an+1) + (−1)n+1T(a1, . . . , an)an+1 and δ0(x)(a) = δx(a) = xa −ax. The spaces ker δnand im δn−1are denoted by Zn(A,X) and Bn(A,X) respectively, and their elements are called n-dimensional cocycles and n-dimensional coboundaries respectively. The quotient Zn(A,X)/Bn(A,X) is called the n-dimensional cohomology group of Awith coefficients in Xand denoted by Hn(A,X) (see [3]). Note that Hn(A,X) = 0 means that every bounded n-linear map Twith δnT= 0 is of the form δn−1Sfor some bounded (n−1)-linear map S. Note also that Z1(A,X) is the space of all derivations from Ato Xand that B1(A,X) is the space of all inner derivations. Thus, H1(A,X) = 0 means that Z1(A,X) = B1(A,X) and consequently all derivations from Ato Xare inner. Fixed a Banach algebra A, the statement H1(A,X) = 0 for every Banach A-bimodule Ximply the same statement for all n≥1. Definition 1.1. We say that a Banach algebra Ahas trivial cohomology if H1(A,X)vanishes for each Banach A-bimodule X. Obviously, a finite-dimensional algebra Ahas trivial cohomology if and only if it is semisimple (i.e, its Jacobson radical is trivial) and so, it is a finite direct sum of full matrix algebras. In view of the difficulties to find an infinite dimensional Banach algebra with trivial cohomology, the following question has been frequently asked (see [6, p. 180]): BANACH ALGEBRAS WITH TRIVIAL COHOMOLOGY 81 Question. Is every Banach algebra with trivial cohomology a finite direct sum of full matrix algebras? In the paper mentioned above, J.L. Taylor gives a partial answer to this question by proving that Banach algebras with trivial cohomology and the bounded approximation property are finite direct sum of full matrix algebras. Note that not every Hermitian Banach ∗-algebra or C∗-algebra does satisfy the bounded approximation property, so it is important to know what happens whenever this class of algebras have trivial cohomology. The following theorem, which is the main result of this paper, solves the above question for the class of Hermitian Banach ∗-algebras. Theorem 1.2. Let Abe a Hermitian Banach ∗-algebra. Then, Ahas trivial cohomology if and only if there are n1, n2, . . . , nk∈Nsuch that A∼ =Mn1(C)⊕Mn2(C)⊕ · · · ⊕ Mnk(C) where Mni(C)is the algebra of complex ni×ni-matrices. 2. Preliminaries Given X,Yand ZBanach hleft, bi-i A-modules and θ:X → Y,β:Y → Z hleft, bi-imodule morphisms, the sequence Σ : 0 → X → Y → Z → 0 is a short exact sequence if θis one-to-one, im β=Z, and im θ= ker β. A short exact sequence Σ is called admissible if βhas a continuous right inverse or, equivalently, if ker βhas a Banach space complement in Y. An admissible short exact sequence splits if the right inverse of βis a Banach hleft, bi-imodule morphism or, equivalently, if kerβhas a Banach space complement in Ywhich is an A-submodule. Proposition 2.1. Let Aand Bbe Banach algebras and θ:A −→ B be a continuous homomorphism with dense range. If Ahas trivial cohomology, then so has B. In particular, if Iis a closed two-sided ideal of a Banach algebra A with trivial cohomology, then so is A/I. Proof. Assume that Ahas trivial cohomology. Let Xbe a Banach B-bimodule. Consider on X, the structure of A-bimodule defined by ax =θ(a)xand xa = xθ(a). Since θis continuous, Xis a Banach A-bimodule. Now, let D:B → X be a continuous derivation. It is easy to see that D◦θis a continuous derivation from Ato the Banach A-bimodule Xand, thus, it is inner. Therefore, there exists x∈ X such that D(θ(a)) = ax −xa =θ(a)x−xθ(a) for all a∈ A. Since θ(A) is dense in B, we have D(b) = bx −xb for all b∈ B. It follows that Dis inner and thus, Bhas trivial cohomology. ¤ Notice that a Banach algebra Awith trivial cohomology is unital and every admissible short exact sequence of hleft, bi-iBanach modules on Asplits ([1, Theorem 6.1]). This is the reason why the following proposition holds: 82 R. EL HARTI Proposition 2.2. Let Abe a Banach algebra with trivial cohomology and I be a closed hleft, two-sidediideal of Awhich has a Banach space complement. Then there exists a closed hleft, two-sidediideal Jof Asuch that A=I ⊕ J . Proof. Let Abe a Banach algebra and let Ibe a closed hleft, two-sidediideal of Awhich has a Banach space complement. Then the short exact sequence Σ : 0 → I → A → A/I → 0 is admissible. Moreover, if Ahas trivial cohomology, then Σ splits and Ihas a Banach space complement which is a hleft, two-sidediideal. ¤ Theorem 2.3. Let Abe a Banach algebra with trivial cohomology. Assume that each maximal left ideal of Ais complemented as a Banach space in A. Then, there are n1, n2, . . . , nk∈Nsuch that A∼ =Mn1(C)⊕Mn2(C)⊕ · · · ⊕ Mnk(C). Proof. As stated above, the algebra Ahas an identity 1A. Let (Mi)i∈Ibe a family of all maximal left ideals of A. By hypotesis each Miis complemented as a Banach space. By Proposition 2.2, there exists a left ideal Jisuch that A=Mi⊕ Ji. Notice that by definition Rad(A) = \ i Mi is the Jacobson radical of Aand (1) M i∈I Ji⊆Soc(A), where Soc(A) is the socle of the algebra A, i.e, it is the sum of all minimal left ideals of Aand it coincides with the sum of all minimal right ideals of A. Recall that every minimal left ideal of Ais of the form Aewhere eis a minimal idempotent, i.e, e2=e6= 0 and eAe=Ce. On the other hand, for each finite family of minimal idempotents (ek)k∈K, we have (2) A=M k∈K AekM \ k∈K A(1A−ek). It follows from (1) and (2) that Soc(A) is dense in A/Rad(A). This shows that A/Rad(A) is finite-dimensional. Therefore A= Rad(A)⊕Soc(A). If Rad(A)6={0}, this would mean that Rad(A) has an identity and this is impossible. Therefore, A= Soc(A) and then it is a finite direct sum of certain full matrix algebras. ¤ BANACH ALGEBRAS WITH TRIVIAL COHOMOLOGY 83 3. Proof of Theorem 1.2 It suffices to show the only if part. First, suppose now that Ais a C∗-algebra. Let Mbe a maximal left ideal. By [4, Theorem 5.3.5 and Theorem 5.2.4], the space A/Mis a Hilbert space. It follows that the short exact sequence Σ : 0 → M → A → A/M → 0 is admissible and thus Mhas a Banach space complement. By Theorem 2.3, Ais ∗-isomorphic to a finite direct sum of full matrix algebras. Consider now the general case where Ais a Hermitian Banach ∗-algebra with trivial cohomology. Let T(A) be the set of all states of Aand let R∗(A) be the ∗-radical of A, i.e., the intersection of the kernels of all ∗-representations of Aon Hilbert spaces. Since Ais Hermitian and has an identity, T(A)6=∅ and so R∗(A)6=A. Let τ∈T(A) and Iτ={a∈ A :τ(b∗a) = 0 for all b∈ A}. As shown, τis positive, i.e τ(a∗a)≥0 for all a∈ A. It is easy to check that Xτ=A/Iτis a pre-Hilbert space with respect to the induced inner product: ha+Iτ, b +Iτi=τ(b∗a). For each a∈ A define a linear operator θτ(a) on Xτby θτ(a)(b+Iτ) = ab +Iτ. Since Iτis a left ideal, θτ(a) is well defined. It is easy to prove that a→θτ(a) is ∗-representation of Aon Xτ. Let Hτbe the Hilbert space completion of A/Iτand let πτ(a) denote the unique extension of θτ(a) to a bounded linear operator on Hτ. Hence a→πτ(a) is a ∗-representation of Aon Hτ. Let π=Lτ∈T(A)πτand H=Lτ∈T(A)Hτ. Then πis a ∗-representation of Aon H. Consider kπ(a)k= sup τ∈T(A) kπτ(a)k. Then k·k is a C∗-norm on π(A). Moreover, let Bdenote the closure of (π(A),k· k), then π:A→Bis a continuous mapping into the C∗-algebra Bsuch that ker(π) = R∗(A). The Banach algebra Ahas trivial cohomology. By Proposition 2.1, the C∗-algebra Bhas also trivial cohomology and thus, it has to be finite dimensional. Notice that A/R∗(A) is ∗-isometric to the ∗- subalgebra π(A) of B. Thus, it follows that A/R∗(A) is finite-dimensional. Since R∗(A) is a finite-codimensional closed two-sided ∗-ideal, there exists a closed two-sided ideal Ksuch that A=R∗(A)⊕ K. Next, note that kπ(a)k2= sup{τ(a∗a), τ ∈T(A)} ≥ |a∗a|σwhere |a|σis the spectral radius of a∈ A. By Pt´ak [5], we obtain kπ(a)k2≥ |a|2 σ. So, if a∈R∗(A), then |a|σ= 0. Therefore every element of R∗(A) is quasinilpotent. Notice that in general Rad(A)⊆R∗(A). Since R∗(A) is a closed two-sided ∗- ideal, we have R∗(A) = Rad(A) and so, Ais finite-dimensional and semisimple. 84 R. EL HARTI References [1] Curtis, C. and Loy, R. J. The structure of amenable Banach algebras, J. London Math. Soc., II Ser. 40 (1989), 89-104. [2] R. EL Harti The structure of a subclass of amenable Banach algebras, Int. JMMS., Vol 55 (2004) 2963-2969. [3] Johnson, B. E. Cohomology in Banach algebras, Memoirs of Amer. Math. Soc, vol. 127 (1972). [4] Murphy, G. J. C∗-algebras and Operator Theory, Academic press, (1990). [5] Pt´ak, V On the spectral radius in Banach algebras with involution, Bull. London. Math. Soc. 2(1970), 327-334. [6] Taylor, J. L. Homology and cohomology for topological algebras, Adv. Math. 9(1972) 137-182. Rachid El Harti, University Hassan I, FST de Settat, BP 577, 2600 Settat, Morocco E-mail address:[email protected]