Banach algebras with trivial cohomology

Fi s Ad anced Cou se in Ope a o Theo y and Complex Analysis, Uni e si y o Se ille, June 2004 BANACH ALGEBRAS WITH TRIVIAL COHOMOLOGY RACHID EL HARTI Abs ac . A plausible conjec u e is ha Banach algeb as wi h i ial cohomology ha e o be semisimple and ini e dimensional. In his pape , we show ha a He mi ian Banach ∗-algeb a has i ial cohomology i and only i i is ∗-isomo phic o a ini e di ec sum o ull ma ix algeb as. 1. In oduc ion Le (A,k·k,∗) be a Banach algeb a Awi h an iden i y and some in olu ion. Ais called a He mi ian Banach ∗-algeb a i o e e y elemen ain A,a∗ahas eal spec um. I is said o be a C∗-algeb a i i s no m k·k is a C∗-no m, i.e, k·k sa is ies: ka∗ak=kak2 o all a∈ A. Ob iously, a C∗-algeb a is a He mi ian Banach ∗-algeb a. A s a e τis a linea o m on Asuch ha τ(1) = kτk= 1. A∗- ep esen a ion o Aon a p e-Hilbe space Xis a ∗-homomo phism o A in o he algeb a B(X) o bounded ope a o on X. Gi en a Banach algeb a, A, we de ine a Banach le A-module X o be a Banach space which is a s uc u e le module Asuch ha he linea map (a, x)∈ A × X 7−→ ax ∈ X is con inuous. Righ modules a e de ined analogously. A Banach A-bimodule is a Banach space wi h a s uc u e bimodule o e Asuch ha he linea map (a, x, b)∈ A × X × A 7−→ axb ∈ X is con inuous. A Banach A-submodule o a gi en Banach A-module Xis a he same ime a closed subspace o Xand an algeb aic A-submodule o X. A Banach le A-module mo phism θ:X → Y is a con inuous linea map Re ised Decembe 15, 2004. This is an imp o ed e sion o a p e ious pape by he same au ho , [2]. 79 80 R. EL HARTI be ween wo le Banach A-modules such ha θ(ax) = aθ(x) o all a∈ A and all x∈ X. Banach igh A-module mo phisms and Banach A-bimodule mo phisms a e de ined analogously. Fo each Banach A-bimodule X, he dual X∗is na u ally a Banach bimodule o e Awi h he module ac ions de ined by aT(x) = T(xa) and Ta(x) = T(ax), o all a∈ A,T∈ X∗, and x∈ X; whe e T(x) deno es he e alua ion o Ta x. A de i a ion om Ain o a Banach A-bimodule Xis a linea ope a o D: A→Xwhich sa is ies D(ab) = D(a)b+aD(b), ∀a, b ∈ A. No ice ha o any x∈ X, he mapping δx:A → X de ined by δx(a) = ax −xa, a ∈ A, is a con inuous de i a ion called an inne de i a ion. An n-linea bounded ope a o T:A× n) · · · ×A→ X is called a n-cochain. The se o n-cochains o ms a Banach space deno ed by Cn(A,X). The s anda d cohomological complex is 0→ X δ0 −→ C1(A,X)δ1 −→ · · · δn−1 −→ Cn(A,X)δn −→ Cn+1(A,X)δn+1 −→ · · · , whe e δnis de ined as ollows o 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 ke δnand im δn−1a e deno ed by Zn(A,X) and Bn(A,X) e- spec i ely, and hei elemen s a e called n-dimensional cocycles and n-dimensional cobounda ies espec i ely. The quo ien Zn(A,X)/Bn(A,X) is called he n-dimensional cohomology g oup o Awi h coe icien s in Xand deno ed by Hn(A,X) (see [3]). No e ha Hn(A,X) = 0 means ha e e y bounded n-linea map Twi h δnT= 0 is o he o m δn−1S o some bounded (n−1)-linea map S. No e also ha Z1(A,X) is he space o all de i a ions om A o Xand ha B1(A,X) is he space o all inne de i a ions. Thus, H1(A,X) = 0 means ha Z1(A,X) = B1(A,X) and consequen ly all de i a ions om A o Xa e inne . Fixed a Banach algeb a A, he s a emen H1(A,X) = 0 o e e y Banach A-bimodule Ximply he same s a emen o all n≥1. De ini ion 1.1. We say ha a Banach algeb a Ahas i ial cohomology i H1(A,X) anishes o each Banach A-bimodule X. Ob iously, a ini e-dimensional algeb a Ahas i ial cohomology i and only i i is semisimple (i.e, i s Jacobson adical is i ial) and so, i is a ini e di ec sum o ull ma ix algeb as. In iew o he di icul ies o ind an in ini e dimensional Banach algeb a wi h i ial cohomology, he ollowing ques ion has been equen ly asked (see [6, p. 180]): BANACH ALGEBRAS WITH TRIVIAL COHOMOLOGY 81 Ques ion. Is e e y Banach algeb a wi h i ial cohomology a ini e di ec sum o ull ma ix algeb as? In he pape men ioned abo e, J.L. Taylo gi es a pa ial answe o his ques ion by p o ing ha Banach algeb as wi h i ial cohomology and he bounded app oxima ion p ope y a e ini e di ec sum o ull ma ix algeb as. No e ha no e e y He mi ian Banach ∗-algeb a o C∗-algeb a does sa is y he bounded app oxima ion p ope y, so i is impo an o know wha happens whene e his class o algeb as ha e i ial cohomology. The ollowing heo em, which is he main esul o his pape , sol es he abo e ques ion o he class o He mi ian Banach ∗-algeb as. Theo em 1.2. Le Abe a He mi ian Banach ∗-algeb a. Then, Ahas i ial cohomology i and only i he e a e n1, n2, . . . , nk∈Nsuch ha A∼ =Mn1(C)⊕Mn2(C)⊕ · · · ⊕ Mnk(C) whe e Mni(C)is he algeb a o complex ni×ni-ma ices. 2. P elimina ies Gi en X,Yand ZBanach hle , bi-i A-modules and θ:X → Y,β:Y → Z hle , bi-imodule mo phisms, he sequence Σ : 0 → X → Y → Z → 0 is a sho exac sequence i θis one- o-one, im β=Z, and im θ= ke β. A sho exac sequence Σ is called admissible i βhas a con inuous igh in e se o , equi alen ly, i ke βhas a Banach space complemen in Y. An admissible sho exac sequence spli s i he igh in e se o βis a Banach hle , bi-imodule mo phism o , equi alen ly, i ke βhas a Banach space complemen in Ywhich is an A-submodule. P oposi ion 2.1. Le Aand Bbe Banach algeb as and θ:A −→ B be a con inuous homomo phism wi h dense ange. I Ahas i ial cohomology, hen so has B. In pa icula , i Iis a closed wo-sided ideal o a Banach algeb a A wi h i ial cohomology, hen so is A/I. P oo . Assume ha Ahas i ial cohomology. Le Xbe a Banach B-bimodule. Conside on X, he s uc u e o A-bimodule de ined by ax =θ(a)xand xa = xθ(a). Since θis con inuous, Xis a Banach A-bimodule. Now, le D:B → X be a con inuous de i a ion. I is easy o see ha D◦θis a con inuous de i a ion om A o he Banach A-bimodule Xand, hus, i is inne . The e o e, he e exis s x∈ X such ha D(θ(a)) = ax −xa =θ(a)x−xθ(a) o all a∈ A. Since θ(A) is dense in B, we ha e D(b) = bx −xb o all b∈ B. I ollows ha Dis inne and hus, Bhas i ial cohomology. ¤ No ice ha a Banach algeb a Awi h i ial cohomology is uni al and e e y admissible sho exac sequence o hle , bi-iBanach modules on Aspli s ([1, Theo em 6.1]). This is he eason why he ollowing p oposi ion holds: 82 R. EL HARTI P oposi ion 2.2. Le Abe a Banach algeb a wi h i ial cohomology and I be a closed hle , wo-sidediideal o Awhich has a Banach space complemen . Then he e exis s a closed hle , wo-sidediideal Jo Asuch ha A=I ⊕ J . P oo . Le Abe a Banach algeb a and le Ibe a closed hle , wo-sidediideal o Awhich has a Banach space complemen . Then he sho exac sequence Σ : 0 → I → A → A/I → 0 is admissible. Mo eo e , i Ahas i ial cohomology, hen Σ spli s and Ihas a Banach space complemen which is a hle , wo-sidediideal. ¤ Theo em 2.3. Le Abe a Banach algeb a wi h i ial cohomology. Assume ha each maximal le ideal o Ais complemen ed as a Banach space in A. Then, he e a e n1, n2, . . . , nk∈Nsuch ha A∼ =Mn1(C)⊕Mn2(C)⊕ · · · ⊕ Mnk(C). P oo . As s a ed abo e, he algeb a Ahas an iden i y 1A. Le (Mi)i∈Ibe a amily o all maximal le ideals o A. By hypo esis each Miis complemen ed as a Banach space. By P oposi ion 2.2, he e exis s a le ideal Jisuch ha A=Mi⊕ Ji. No ice ha by de ini ion Rad(A) = i Mi is he Jacobson adical o Aand (1) M i∈I Ji⊆Soc(A), whe e Soc(A) is he socle o he algeb a A, i.e, i is he sum o all minimal le ideals o Aand i coincides wi h he sum o all minimal igh ideals o A. Recall ha e e y minimal le ideal o Ais o he o m Aewhe e eis a minimal idempo en , i.e, e2=e6= 0 and eAe=Ce. On he o he hand, o each ini e amily o minimal idempo en s (ek)k∈K, we ha e (2) A=M k∈K AekM k∈K A(1A−ek). I ollows om (1) and (2) ha Soc(A) is dense in A/Rad(A). This shows ha A/Rad(A) is ini e-dimensional. The e o e A= Rad(A)⊕Soc(A). I Rad(A)6={0}, his would mean ha Rad(A) has an iden i y and his is impossible. The e o e, A= Soc(A) and hen i is a ini e di ec sum o ce ain ull ma ix algeb as. ¤ BANACH ALGEBRAS WITH TRIVIAL COHOMOLOGY 83 3. P oo o Theo em 1.2 I su ices o show he only i pa . Fi s , suppose now ha Ais a C∗-algeb a. Le Mbe a maximal le ideal. By [4, Theo em 5.3.5 and Theo em 5.2.4], he space A/Mis a Hilbe space. I ollows ha he sho exac sequence Σ : 0 → M → A → A/M → 0 is admissible and hus Mhas a Banach space complemen . By Theo em 2.3, Ais ∗-isomo phic o a ini e di ec sum o ull ma ix algeb as. Conside now he gene al case whe e Ais a He mi ian Banach ∗-algeb a wi h i ial cohomology. Le T(A) be he se o all s a es o Aand le R∗(A) be he ∗- adical o A, i.e., he in e sec ion o he ke nels o all ∗- ep esen a ions o Aon Hilbe spaces. Since Ais He mi ian and has an iden i y, T(A)6=∅ and so R∗(A)6=A. Le τ∈T(A) and Iτ={a∈ A :τ(b∗a) = 0 o all b∈ A}. As shown, τis posi i e, i.e τ(a∗a)≥0 o all a∈ A. I is easy o check ha Xτ=A/Iτis a p e-Hilbe space wi h espec o he induced inne p oduc : ha+Iτ, b +Iτi=τ(b∗a). Fo each a∈ A de ine a linea ope a o θτ(a) on Xτby θτ(a)(b+Iτ) = ab +Iτ. Since Iτis a le ideal, θτ(a) is well de ined. I is easy o p o e ha a→θτ(a) is ∗- ep esen a ion o Aon Xτ. Le Hτbe he Hilbe space comple ion o A/Iτand le πτ(a) deno e he unique ex ension o θτ(a) o a bounded linea ope a o on Hτ. Hence a→πτ(a) is a ∗- ep esen a ion o Aon Hτ. Le π=Lτ∈T(A)πτand H=Lτ∈T(A)Hτ. Then πis a ∗- ep esen a ion o Aon H. Conside kπ(a)k= sup τ∈T(A) kπτ(a)k. Then k·k is a C∗-no m on π(A). Mo eo e , le Bdeno e he closu e o (π(A),k· k), hen π:A→Bis a con inuous mapping in o he C∗-algeb a Bsuch ha ke (π) = R∗(A). The Banach algeb a Ahas i ial cohomology. By P oposi ion 2.1, he C∗-algeb a Bhas also i ial cohomology and hus, i has o be ini e dimensional. No ice ha A/R∗(A) is ∗-isome ic o he ∗- subalgeb a π(A) o B. Thus, i ollows ha A/R∗(A) is ini e-dimensional. Since R∗(A) is a ini e-codimensional closed wo-sided ∗-ideal, he e exis s a closed wo-sided ideal Ksuch ha A=R∗(A)⊕ K. Nex , no e ha kπ(a)k2= sup{τ(a∗a), τ ∈T(A)} ≥ |a∗a|σwhe e |a|σis he spec al adius o a∈ A. By P ´ak [5], we ob ain kπ(a)k2≥ |a|2 σ. So, i a∈R∗(A), hen |a|σ= 0. The e o e e e y elemen o R∗(A) is quasinilpo en . No ice ha in gene al Rad(A)⊆R∗(A). Since R∗(A) is a closed wo-sided ∗- ideal, we ha e R∗(A) = Rad(A) and so, Ais ini e-dimensional and semisimple. 84 R. EL HARTI Re e ences [1] Cu is, C. and Loy, R. J. The s uc u e o amenable Banach algeb as, J. London Ma h. Soc., II Se . 40 (1989), 89-104. [2] R. EL Ha i The s uc u e o a subclass o amenable Banach algeb as, In . JMMS., Vol 55 (2004) 2963-2969. [3] Johnson, B. E. Cohomology in Banach algeb as, Memoi s o Ame . Ma h. Soc, ol. 127 (1972). [4] Mu phy, G. J. C∗-algeb as and Ope a o Theo y, Academic p ess, (1990). [5] P ´ak, V On he spec al adius in Banach algeb as wi h in olu ion, Bull. London. Ma h. Soc. 2(1970), 327-334. [6] Taylo , J. L. Homology and cohomology o opological algeb as, Ad . Ma h. 9(1972) 137-182. Rachid El Ha i, Uni e si y Hassan I, FST de Se a , BP 577, 2600 Se a , Mo- occo E-mail add ess:[email p o ec ed]