Dedicated to Professor Jamshid Moori on the occasion of his 70th birthday NOTE ON CHARACTER AMENABILITY IN BANACH ALGEBRAS O.T. MEWOMO Communicated by Henri Moscovici Wegiveasurveyofresultsandproblemsconcerningthenotionofcharacterame- nability in Banach algebras. We also provide different characterizations of this notion of amenability and the relationship that exists between this notion and some important properties of the algebras. Results and problems are surveyed over general Banach algebras and Banach algebras in different classes. AMS 2010 Subject Classification: Primary 46H20; Secondary 46H10. Key words: character, Banach algebra, Beurling algebra, semigroup algebra, group algebra, measure algebra, Segal algebra, left character ame- nable, right character amenable, character amenable. 1. INTRODUCTION This article is a survey of the character amenability results around the general Banach algebras and Banach algebras in different classes which are known to the author. The notion of amenability of groups was first defined for discrete locally compact groups by Von Neumann [49]. This was later generalized to arbitrary locally compact groups by M. Day [14]. B.E. Johnson in[26]whiletryingtostudytherelationshipbetweenthegroupamenabilityofa locallycompactgroupGandthegroupalgebraL1(G)cameupwiththenotion of amenability for Banach algebras. He proved that a locally compact group G is amenable as a group if and only if the group algebra L1(G) is amenable as a Banach algebra. This result of Johnson laid the groundwork for amenability in Banach algebras. Ever since this groundwork, the notion of amenability has become a major issue in Banach algebra theory and in harmonic analysis. For details on amenability in Banach algebras see [36]. After the pioneering work of Johnson in [26], several modifications of the original notion of amenability in Banach algebras are introduced. One of the most important modifications was introduces by A.T. Lau [31] where he introduced the notion of left amenability for a class of F-algebras. This MATH. REPORTS 19(69), 3 (2017), 293–312 294 O.T. Mewomo 2 waslattergeneralizedbyE.KaniuthinjointpaperswithA.T.LauandJ.Pym [29,30] where they introduced the notion of ϕ-amenability of Banach algebras. Recently, M.S. Monfared [44], gave an extension of these notions, where he introduced the notion of character amenability. The notion of character ame- nability as defined in [44] is stronger than left amenability of Lau and also modifies the original definition by Johnson in the sense that it requires conti- nuous derivations from A into dual Banach A-bimodules to be inner, but only those modules are concerned where either of the left or right module action is defined by characters on A. As such character amenability is weaker than the classical amenability introduced by Johnson in [26]. Severalauthorshavestudiedthenotionofcharacteramenabilityfordiffe- rentclassesofBanachalgebras,mostnotablyareAlaghmandan,Nasr-Isfahami andNemati[1],Dashti,Nasr-IsfahamiandRenani[12],EssmailiandFilali[18], Hu, Monfared and Traynor [25], Kaniuth, Lau and Pym [29], Mewomo and Okelo [39], Mewomo, Maepa and Uwala [42], Mewomo and Meapa [40,41], Mewomo and Ogunsola [38], see also [43] and Monfared [44]. The purpose of this note is to give an overview of what has been done so far on character amenability in general Banach algebras and Banach algebras in different classes and raise some problems of interest. 2. PRELIMINARIES First, werecallsomestandardnotions; forfurtherdetails, see[8]and[36]. A locally compact group G is amenable if it possesses a translation in- variant mean. That is, if there exists a linear functional µ : L∞(G) → C, satisfying µ(1) = (cid:107)µ(cid:107) = 1 and µ(δ ∗f) = µ(f) (x ∈ G,f ∈ L∞(G)). x Let A be an algebra. Let X be an A-bimodule. A derivation from A to X is a linear map D : A → X such that D(ab) = Da · b+a · Db (a,b ∈ A). For example, δ : a → a · x−x · a is a derivation; derivations of this form are x the inner derivations. Let A be a Banach algebra, and let X be an A-bimodule. Then X is a Banach A-bimodule if X is a Banach space and if there is a constant k > 0 such that (cid:107)a · x(cid:107) ≤ k(cid:107)a(cid:107)(cid:107)x(cid:107), (cid:107)x · a(cid:107) ≤ k(cid:107)a(cid:107)(cid:107)x(cid:107) (a ∈ A, x ∈ X). Forexample,AitselfisBanachA-bimodule,andX(cid:48),thedualspaceofaBanach A-bimodule X, is a Banach A-bimodule with respect to the module operations 3 Note on character amenability in Banach algebras 295 defined by (cid:104)x, a · λ(cid:105) = (cid:104)x · a, λ(cid:105), (cid:104)x, λ · a(cid:105) = (cid:104)a · x, λ(cid:105) (x ∈ X) for a ∈ A and λ ∈ X(cid:48); we say that X(cid:48) is the dual module of X. Let A be a Banach algebra, and let X be a Banach A-bimodule. Then Z1(A,X) is the space of all continuous derivations from A into X, N 1(A,X) is the space of all inner derivations from A into X, and the first cohomology group of A with coefficients in X is the quotient space H1(A,X) = Z1(A,X)/N 1(A,X). The Banach algebra A is amenable if H1(A,X(cid:48)) = {0} for each Banach A- bimodule X. For example, the group algebra L1(G) of a locally compact group GisamenableifandonlyifGisamenable[26]. Also, aC∗-algebraisamenable if and only if it is nuclear [7,22]. Let A be a Banach algebra. Then the projective tensor product A⊗ˆA is a Banach A-bimodule where the multiplication is specified by a·(b⊗c) = ab⊗c and (b⊗c)·a = b⊗ca (a,b,c ∈ A.) Definition 2.1. Let A be a Banach algebra. 1. A bounded approximate diagonal for A is a bounded net (m ) in A⊗ˆA α such that a·m −m ·a → 0 and aπ (m ) → a (a ∈ A). α α A α 2. A virtual diagonal for A is an element M ∈ (A⊗ˆA)(cid:48)(cid:48) such that a·M = M ·a and π(cid:48)(cid:48)(M)a = a (a ∈ A). A Johnson [27] gave the following characterization of amenability: Theorem 2.1. Let A be a Banach algebra. Then the following are equi- valent: 1. A is amenable 2. A has a bounded approximate diagonal 3. A has a virtual diagonal Definition 2.2. A Banach algebra A is an F-algebra if it is the unique predual of a C∗-algebra B and the identity element e of B is a multiplicative linear functional on A. Example 2.3. The following are examples of F-algebras: 1. The group algebra L1(G) for any locally compact group G. 2. The semigroup algebra (cid:96)1(S). 296 O.T. Mewomo 4 3. TheFourieralgebraA(G)foranylocallycompactgroupGwithpointwise multiplication. Let A be an F-algebra associated with a C∗-algebra B and let P (A(cid:48)(cid:48)) = 1 {µ ∈ B(cid:48) : µ ≥ 0,(cid:104)e,µ(cid:105) = 1}. P (A(cid:48)(cid:48)) ⊂ A(cid:48)(cid:48) = B(cid:48). An element m ∈ P (A(cid:48)(cid:48)) is 1 1 called a left invariant mean on A(cid:48) if m(f ·a) = (cid:104)a,e(cid:105)m(f) (a ∈ A,f ∈ A(cid:48)). We recall from [31] that the F-algebra A is called left amenable if there is a left invariant mean on A(cid:48). Example 2.4. LaugavethefollowingexamplesofleftamenableF-algebras. 1. ThegroupalgebraL1(G)foranylocallycompactgroupGisleftamenable if and only if G is an amenable group. 2. The semigroup algebra (cid:96)1(S) is left amenable if and only if S is a left amenable semigroup. 3. All commutative F-algebras are left amenable. We next introduce the notations and basic concepts on character ame- nability. Let A be a Banach algebra over C and ϕ : A → C be a character on A, that is, an algebra homomorphism from A into C, we let Φ denote the A character space of A. Also, MA denote the class of Banach A- bimodule X for ϕr which the right module action of A on X is given by x·a = ϕ(a)x (a ∈ A,x ∈ X,ϕ ∈ Φ ), and MA denote the class of Banach A- bimodule X for which the A ϕ l leftmoduleactionofAonX isgivenbya·x = ϕ(a)x (a ∈ A,x ∈ X,ϕ ∈ Φ ). A If the right module action of A on X is given by x·a = ϕ(a)x, then it is easy to see that the left module action of A on the dual module X(cid:48) is given by a·f = ϕ(a)f (a ∈ A,f ∈ X(cid:48),ϕ ∈ Φ ). Thus, we note that X ∈ MA (resp. A ϕr X ∈ MA) if and only if X(cid:48) ∈ MA (resp. X(cid:48) ∈ MA ). ϕl ϕl ϕr Let A be a Banach algebra and let ϕ ∈ Φ , we recall the next definitions A from [25] and [44]. Definition 2.5. (i) A is left ϕ-amenable if every continuous derivation D : A → X(cid:48) is inner for every X ∈ MA ; ϕr (ii) A is right ϕ-amenable if every continuous derivation D : A → X(cid:48) is inner for every X ∈ MA; ϕ l (iii) A is left character amenable if it is left ϕ-amenable for every ϕ ∈ Φ ; A (iv) A is right character amenable if it is right ϕ-amenable for every ϕ ∈ Φ ; A (v) A is character amenable if it is both left and right character amenable. 5 Note on character amenability in Banach algebras 297 3. RESULTS OVER GENERAL BANACH ALGEBRAS In this section, we surveyed some general theory and results over gene- ral Banach algebras. These were applied and used in establishing results for Banach algebras in different classes. Thefollowingresultsareusefulhereditary,stabilityandgeneralproperties of character amenability for Banach algebras. Proposition 3.1. Let A be a Banach algebra. Suppose A is character amenable and I is a closed ideal of codimension one in A. 1. Then (i) A has a bounded approximate identity and hence factors (ii) the unitization algebra A(cid:93) is character amenable (iii) I has a bounded approximate identity (iv) I is character amenable. 2. Suppose B is another Banach algebra and τ : A → B is a continuous homomorphism with τ(A) = B. (i) Then B is character amenable (ii) In particular, suppose I is a closed ideal of A. Then A/I is character amenable. Proof. 1. (i) This is [40, Proposition 3.1 (i)] (ii) This is [40, Proposition 3.1 (ii)] (iii) This is [40, Theorem 3.2 (i)] (iv) This is [40, Theorem 3.2 (ii)] 2. (i) This is [42, Proposition 3.3] (ii)Thisis[41,Proposition3.1]. Itfollowsfromthefactthatτ : A → A/I is a continuous homomorphism with dense range. (cid:3) Let A⊗ˆB be the projective tensor product of Banach algebras A and B. For f ∈ A(cid:48),g ∈ B(cid:48), let f ⊗g ∈ (A⊗ˆB)(cid:48) such that (f ⊗g)(a⊗b) = f(a)g(b) (a ∈ A,b ∈ B). Then Φ = {ϕ⊗ψ : ϕ ∈ Φ ,ψ ∈ Φ }. Also, let J be a non-empty set. We A⊗ˆB A B denote by M (A) the set of J ×J matrices (a ) with entries in A such that J ij (cid:88) (cid:107)(a )(cid:107) = (cid:107)a (cid:107) < ∞. ij ij i,j∈J Then M (A) with the usual matrix multiplication is a Banach algebra. The J map (cid:88) τ : M (A) → A⊗ˆM (C) defined by τ((a )) = a ⊗E J J ij ij ij i,j∈J 298 O.T. Mewomo 6 ((a ) ∈ M (A)), is an isometric isomorphism of Banach algebras, where (E ) ij J ij are the matrix units in M (C). Thus, we have the next results from [29] J and [41]. Proposition 3.2. 1. Let A and B be Banach algebras with ϕ ∈ Φ A and ψ ∈ Φ . Then A⊗ˆB is (ϕ ⊗ ψ)-amenable if and only if A is ϕ- B amenable and B is ψ-amenable. In particular, A⊗ˆB is left character amenable if and only if A and B are left character amenable. 2. Let A be a Banach algebra and J a non-empty set. Then M (A) is left J character amenable if and only if A is left character amenable. Proof. 1. This is from [29, Theorem 3.3]. 2. Thisis[41,Corollary3.3]. ItclearlyfollowsfromProposition3.2(i). (cid:3) We recall from [2] that a Banach algebra A is left [right] ϕ-biflat if there exists a bounded linear operator ρ : A → (A⊗ˆA)(cid:48)(cid:48) such that (i) ρ(ab) = ϕ(a)ρ(b) = ρ(a)∗b [ρ(ab) = ϕ(b)ρ(a) = a·ρ(b)] (ii) (π(cid:48)(cid:48) ◦ρ(a))(ϕ) = ϕ(a) (a,b ∈ A,ϕ ∈ Φ ). A A Helemskii in [22] showed that a Banach algebra A is amenable if and only if it is biflat and has a bounded approximate identity. We give the character amenability of this result. The next result is due to [2, Proposition 2.2]. Proposition 3.3. Let A be a Banach algebra with ϕ ∈ Φ . A (i) If A is left ϕ-amenable, then A is left ϕ-biflat; (ii If A is left ϕ-biflat and has a bounded approximate identity, then A is left ϕ-amenable. Theabovedefinitionofϕ-biflatnessin[2]wasgeneralizedin[41]asfollows: We say that a Banach algebra A is (i) left [right] character biflat if it is left [right] ϕ-biflat for every ϕ ∈ Φ ; A (ii) character biflat if it is both left and right character biflat. Theorem 3.4. Let A be a Banach algebra. Then the following are equi- valent: (i) A is character amenable (ii) A is character biflat and has a bounded approximate identity. Proof. Thisis[41,Theorem3.6]. ThisclearlyfollowsfromProposition3.3 (i) and (ii) and the above definition. (cid:3) For a commutative and reflexive Banach algebra, we have the next result due to [25]. 7 Note on character amenability in Banach algebras 299 Theorem 3.5. Let A be a character amenable, reflexive, commutative Banach algebra. Then A ∼= Cn for some n ∈ N. 4. SOME CHARACTERIZATIONS AND RESULTS ON SECOND DUAL ALGEBRAS Inthissection,wesurveyedsomecharacterizationsandresultsoversecond dual of general Banach algebras. These were applied and used in establishing results for Banach algebras in different classes. We start by characterizing in terms of bounded approximate identity and certain topological invariant elements in the second dual A(cid:48)(cid:48) of the Banach algebra A. Definition 4.1. Let ϕ ∈ Φ . Ψ ∈ A(cid:48)(cid:48) is called A 1. ϕ-topologically left invariant if (cid:104)Ψ, a·λ(cid:105) = ϕ(a)(cid:104)Ψ, λ(cid:105) (a ∈ A,λ ∈ A(cid:48)); 2. ϕ-topologically right invariant if (cid:104)Ψ, λ·a(cid:105) = ϕ(a)(cid:104)Ψ, λ(cid:105) (a ∈ A,λ ∈ A(cid:48)); 3. ϕ-topologically invariant if it it is both left and right. Theorem 4.1. The Banach algebra A is left character amenable if and only if the following two conditions hold: 1. A has a bounded approximate identity. 2. For every ϕ ∈ Φ ,∃ ϕ-topologically left invariant element Ψ ∈ A(cid:48)(cid:48) such A that Ψ(ϕ) (cid:54)= 0. Proof. This is [44, Theorem 4.2]. (cid:3) We next give a Johnson-like characterization. Recall from [25] that, for ϕ ∈ Φ , a left (right) ϕ-virtual diagonal for A is an element M in (A⊗ˆA)(cid:48)(cid:48) A such that (i) M ·a = ϕ(a)M (a·M = ϕ(a)M) (a ∈ A); (ii) (cid:104)M,ϕ⊗ϕ(cid:105) = π(cid:48)(cid:48)(M)(ϕ) = 1. Also, a left [right] ϕ-approximate diagonal for A is a net (m ) in (A⊗ˆA), such α that (i) (cid:107)m ·a−ϕ(a)m (cid:107) → 0 [(cid:107)a·m −ϕ(a)m (cid:107) → 0] (a ∈ A); α α α α (ii) (cid:104)ϕ⊗ϕ, m (cid:105) → 1. α Thefollowingimportantcharacterizationwasshownin[25,Theorem2.3]. 300 O.T. Mewomo 8 Theorem 4.2. Let A be a Banach algebra and ϕ ∈ Φ . The following A statements are equivalent: (i) A is left [right] ϕ-amenable (ii)A has a bounded left [right] ϕ-approximate diagonal (iii) A has a left [right] ϕ-virtual diagonal. We also give the following characterizations of ϕ-amenability for Banach algebras. These were shown in [29, Theorem 1.1] and [29, Theorem 1.3]. Theorem 4.3. Let A be a Banach algebra and let ϕ ∈ Φ . The following A statements are equivalent: (i) A is ϕ-amenable. (ii) There exists m ∈ A(cid:48)(cid:48) such that m(ϕ) = 1 and a·m = ϕ(a)m (a ∈ A). (iii) There exists a bounded net (u ) in A such that ϕ(u ) = 1 for all α and α α (cid:107)au −ϕ(a)u (cid:107) → 0 (a ∈ A). α α Let A be a Banach algebra. Then the second dual A(cid:48)(cid:48) of A is a Banach A-bimodule for the maps (a,Φ) → a·Φ and (a,Φ) → Φ·a from A×A(cid:48)(cid:48) to A(cid:48)(cid:48) that extend the product map A×A → A, (a,b) → ab on A. Arens in [3] defined two products, (cid:3) and ♦, on the second dual A(cid:48)(cid:48) of a Banach algebra A; A(cid:48)(cid:48) is a Banach algebra with respect to each of these products, and each algebra contains A as a closed subalgebra. The products are called the first and second Arens products on A(cid:48)(cid:48), respectively. For the general theory of Arens products, see [11,16]. We recall briefly the definitions. For Φ ∈ A(cid:48)(cid:48), we set (cid:104)a, λ·Φ(cid:105) = (cid:104)Φ, a · λ(cid:105), (cid:104)a, Φ·λ(cid:105) = (cid:104)Φ, λ · a(cid:105) (a ∈ A, λ ∈ A(cid:48)), so that λ·Φ,Φ·λ ∈ A(cid:48). Let Φ,Ψ ∈ A(cid:48)(cid:48). Then (cid:104)Φ(cid:3)Ψ, λ(cid:105) = (cid:104)Φ, Ψ · λ(cid:105), (cid:104)Φ♦Ψ, λ(cid:105) = (cid:104)Ψ, λ · Φ(cid:105) (λ ∈ A(cid:48)). Suppose that Φ,Ψ ∈ A(cid:48)(cid:48) and that Φ = lim a and Ψ = lim b for nets (a ) α α β β α and (b ) in A. Then β Φ(cid:3)Ψ = limlima b and Φ♦Ψ = limlima b , α β α β α β β α where all limits are taken in the weak-∗ topology σ(A(cid:48)(cid:48),A(cid:48)) on A(cid:48)(cid:48). The following result is well known, see [11, Theorem 2.17]: Theorem 4.4. Let A be a Banach algebra. Then both (A(cid:48)(cid:48),(cid:3)) and (A(cid:48)(cid:48),♦) are Banach algebras containing A as a closed subalgebra. The next result is [25, Theorem 3.8], which is the character amenability version of the result on amenability of the second dual. Theorem 4.5. Let A be a Banach algebra. If (A(cid:48)(cid:48),(cid:3)) is left [right] cha- racter amenable, then A is left [right] character amenable. 9 Note on character amenability in Banach algebras 301 5. RESULTS OVER BANACH ALGEBRAS IN DIFFERENT CLASSES A fruitful area of research in notions of amenability in Banach algebras has been to describe these notions of amenability in terms of the structures that the algebra sits on. The structures that come to mind are: Banach spaces (algebra of operators), locally compact groups and semigroups (group, mea- sure, Segal, Beurling, Fourier and Fourier-Stieljes algebras), locally compact Hausdorff space (uniform and Lipschitz algebras). In the recent years, various authors have considered character amenability of Banach algebras in terms of these structures that the algebras sit. Inthissection,wesurveyedresultsonthisrelationshipfordifferentclasses of Banach algebras. 5.1. RESULTS ON CONVOLUTION ALGEBRAS In this subsection, we surveyed results on group algebras, measure alge- bras, Fourier algebras and Fourier-Stieltjes algebras. WerecallthatalocallycompactgroupisagroupGwhichisalsoalocally compact Hausdorff space such that the maps G×G → G, (g,h) → gh and G → G, g → g−1 are continuous. Each locally compact group G has a left Haar measure µ. L1(G), consisting of measurable functions f on G with (cid:90) (cid:107)f(cid:107) = |f(t)|dµ(t) < ∞, 1 G becomes a Banach algebra for the product (cid:90) (f (cid:63)g)(t) = f(s)g(s−1t)dµ(s). G This is called the group algebra of G. We also denote by M(G), the measure algebra. We recall that the product µ,v ∈ M(G) is specified by the formula (cid:90) (cid:90) (cid:104)µ∗v,λ(cid:105) = λ(st)dµ(sdv(t) (λ ∈ C (G)), 0 G G so that δ ∗δ = δ (s,t ∈ G). It is standard that M(G) is a unital Banach s t st algebra with the convolution product and it is identified with the dual space of all continuous linear functional on the Banach space C (G) and that L1(G) 0 is a closed ideal in M(G). The map ϕ : µ → µ(G), M(G) → C is a character on M(G), called the augmented character. We denote by A(G), the Fourier algebra on G. it is well known that A(G) ∼= L1(Gˆ) when G is abelian and Gˆ is its dual group. It was shown in [19] that for any locally compact group G, A(G) is a commutative Banach algebra with character space G and in [33] that A(G) has a bounded approximate identity if and only if G is an amenable group. 302 O.T. Mewomo 10 TheFourier-StieltjesalgebraB(G)isthecollectionofallcoefficientfuncti- ons of continuous unitary representations of G. It is a Banach algebra contai- ning A(G) as a closed ideal and B(G) = A(G) if and only if G is compact. For details, see [8,9,24] and [25]. ThecharacteramenabilityofthegroupalgebraL1(G)andFourieralgebra A(G) is equivalent to the amenability of the group G. Thus the character ame- nability of L1(G) and A(G) is completely determined by the amenability of G. Character amenability of the measure algebra M(G) is equivalent to G being discrete and amenable. Also, for group and measure algebras, L1(G), M(G), the amenability and character amenability coincide. This follows from John- son’s classical result in [26] and the following result due to [44, Corollary 2.4, Corollary 2.5]. Theorem 5.1. Let G be a locally compact. 1. Then the following statements are equivalent: (i) L1(G) is left character amenable. (ii) L1(G) is right character amenable. (iii) G is amenable. 2. The following statements are equivalent: (i) M(G) is character amenable. (ii) G is a discrete amenable group. For Fourier algebra, A(G), the amenability and character amenability do not coincide. This follows from [44, Corollary 2.4] stated below and from Leptin [33] that G is amenable if and only if A(G) has a bounded approximate identity. ThesethenimplythatifA(G)isamenablethenGmustbeamenable; the converse to this was shown to be false by Johnson in [28], where he shows that for the compact group G = SO(3), A(G) is not amenable. Theorem 5.2. Let G be a locally compact. Then the following statements are equivalent: (i) A(G) is left character amenable. (ii) A(G) is right character amenable. (iii) G is amenable. Proof. This follows [44, Corollary 2.4] with p = 2. (cid:3) It was shown in [25], that B(G) can be character amenable when G is noncompact. For G compact, B(G) = A(G), and so using Theorem 5.2, we have that B(G) is left or right character amenable if and only if G is amenable. 5.2. RESULTS ON BEURLING AND SEGAL ALGEBRAS In this subsection, we surveyed results on Beurling and Segal algebras.