On Biamenability of Banach Algebras 3

Home , Amenable Banach algebra

arXiv:2011.12653v1 [math.FA] 25 Nov 2020 Let derivation. inner an called defining otnosmdl prtosi one iermapping linear bounded a is operations module continuous oinse[ see notion o each For nydrvtosfrom derivations only iial,tehge duals higher the Similarly, aua fashion. natural mapping aahalgebra Banach A eiainfo aahalgebra Banach a from derivation A e od n phrases. and words Key 2010 Let X A ahmtc ujc Classification. Subject Mathematics lersadw td h eainbtenbaeaiiyo aro B of pair a of them. biamenability of between amenability relation and the algebras study we and algebras nnt iesoa ibr space Hilbert dimensional infinite B n ahinteger each and o xml ewl hwta o ahifiiedmninlHletspac Hilbert dimensional infinite each for that show will we example For biamenable. not un unitization is unconditional algebra unconditional general Banach in the a but although of amenable is that algebra show Banach amenable we Also, amenable. mnbeadhgl ocmuaieBnc lerssc as such algebras Banach commu different, noncommutative that very highly show and we to amenable regard, lead as this such they In algebras nach algebras, theories. Banach opposed, of between somewhat similarities biamenability apparent the and despite ity that show we and algebras Abstract. eaBanach a be ( eaBnc ler and algebra Banach a be a.f x H D ial,w nrdc h ocp fbaeaiiyfrapi fBanach of pair a for biamenability of concept the introduce we Finally, hscneti sdfrfidn h hrce pc fsm Banac some of space character the finding for used is concept This NBAEAIIYO AAHALGEBRAS BANACH OF BIAMENABILITY ON ) ∈ 10 : ⊕ and A X ]. B a.f × ( h mapping the H f.a A nti ae,w nrdc h ocp fbaeaiiyo Banach of biamenability of concept the introduce we paper, this In ( ) x (2 → o each for , = ) n A ) A n A aeepyspectrum. empty have X bmdl.Then -bimodule. d ieiain ne ieiain biamenability. biderivation, inner biderivation, scle mnbei o ahBanach each for if amenable called is ≥ f to ( ab scle ieiainif biderivation a called is ( 0, R xa X = ) and B X ) ∗ δ ( a H x ( 1. r ne eiain.Frmr eal bu this about details more For derivations. inner are n d .BAROOTKOOB S. C ∈ f.a , : ) ) ( (2 X a a Introduction edt akbaeaiiy hl hymybe may they while biamenability, lack to tend A a emd noBanach into made be can n ) → ) b H eaBanach a be and n h oueetninBnc algebra Banach extension module the and + rmr 62;Scnay46H25. Secondary 46H20; Primary edt ebaeal,wiete r not are they while biamenable, be to tend ax ( ad 1 A x f − = ) ( X b oaBanach a to ∈ ( ) xa ∗ X f sada Banach dual a is ( , ( ∗ ,b a, ax a A by ( ) ∈ bmdl,abuddbilinear bounded a -bimodule, D ∈ A sadrvto ihrespect with derivation a is A sabuddderivation, bounded a is ) ) A . x -bimodule ∈ d A : X A B A -bimodule ) ( bmdlsi a in -bimodules algebras. h A aieBa- tative . H amenabil- → bmdl,by -bimodule, itization o an for ) X anach X fan of e and uhthat such H ihthe with , X the , 2 S. BAROOTKOOB to both arguments. That is the mappings aD : A → X and Db : A → X are derivations. Where

aD(b)= D(a, b)= Db(a) (a, b ∈ A). We denote the space of such biderivations by BZ1(A, X). Consider the subspace Z(A, X)= {x ∈ X : ax = xa ∀a ∈ A} of X. Then for each x ∈ Z(A, X), the mapping ∆x : A × A → X deﬁned by

∆x(a, b)= x[a, b]= x(ab − ba) (a, b ∈ A) is a basic example of a biderivation and called an inner biderivation. We denote the space of such inner biderivations by BN 1(A, X). For more applications of biderivations, see the survey article [4, Section 3]. Some algebraic aspects of biderivations on certain algebras were investigated by many authors; see for example [3, 7], where the structures of biderivations on triangular algebras and generalized matrix algebras are discussed, and particularly the question of whether biderivations on these algebras are inner, was considered. We define the first bicohomology group BH1(A, X) as follows, BZ1(A, X) BH1(A, X)= . BN 1(A, X) Obviously BH1(A, X) = 0 if and only if every biderivation from A×A to X is an inner biderivation. Now we are motivated to define the concept of biamenability of Banach algebras as follows. A Banach algebra A is biamenable if for each Banach A-bimodule X we have BH1(A, X∗)=0. Although one might expect that biderivations and biamenability must run par- allel to derivations and amenable Banach algebras what is true is that although there are some external similarities between them but they lead to very different, and somewhat opposed, theories. Indeed we show that commutative Banach algebras tend to lack biamenability, while highly noncommutative Banach algebras tend to be biamenable. Thus for instance, the ground algebras C and R are not biamenable (while they are trivially amenable) and B(H), the algebra of all bounded operators on an infinite dimensional Hilbert space H, is biamenable, but not amenable. Moreover, if H is finite dimensional, it turns out that B(H) is amenable, but it fails to be biamenable.

2. biamenable Banach algebras For an example of a biamenable Banach algebra we commence with the next lemmas. The following lemma is similar to Corollary 2.4 of [5], where it is intro- duced for a biderivation D : A × A → A. Lemma 2.1. For each Banach A-bimodule X and each biderivation D : A×A → X, D(a, b)[c,d] = [a, b]D(c,d) (a,b,c,d ∈ A). In Proposition 2.1.3 of [10] it is shown that if A has a bounded approximate identity, and one of module actions is trivial, then the only derivations from A to ON BIAMENABILITY OF BANACH ALGEBRAS 3

X∗ are inner derivations. The following lemma introduces a condition that not only implies the innerness of biderivations but it also forces them to be zero. Lemma 2.2. If a Banach algebra A has a bounded left approximate identity and span{ab−ba : a, b ∈ A} is dense in A, then for every Banach A-bimodule X such that XA =0 we have BZ1(A, X)=0.

Proof. Let (eα) be a left approximate identity of A. Lemma 2.1 says that for each D ∈ BZ1(A, X), [a, b]D(c,d)=0 (a,b,c,d ∈ A). So by density we have aD(b, c) = 0, for each a, b, c ∈ A. Now since XA = 0 we have D(a, b) = limα D(eαa, b) = limα[eαD(a, b)+ D(eα, b)a] = 0. That is BZ1(A, X) = 0. A very similar proof may be applied if A has a right approximate identity and the left module action is trivial. The following lemma introduces a condition that under which some biderivations are inner. This result will lead to a condition implying biamenability of a Banach algebra. We shall see that B(H), for each inﬁnite dimensional Hilbert space H, satisﬁes this condition. Lemma 2.3. If A is unital and A = span{ab − ba : a, b ∈ A}, then for every unital Banach A-bimodule X, every biderivation D : A × A → X is an inner biderivation. Proof. Let D be a biderivation and e be the identity of A. Since A = span{ab−ba : a, b ∈ A}, there exist ai and bi in A such that e = i[ai, bi]. So by Lemma 2.1, for every a, b ∈ A, we have P D(a, b) = D(a, b)e

= D(a, b) i[ai, bi] = i D(a,P b)[ai, bi] = Pi[a, b]D(ai, bi) =P [a, b]λ.

Where λ = i D(ai, bi). Similarly we have D(a, b)= λ[a, b], and so P λ[a, b] = [a, b]λ (a ∈ A, b ∈ B).

Now since A = span{ab − ba : a, b ∈ A}, λ ∈ Z(A, X). So D = ∆λ is an inner biderivation. Lemma 2.4. If a Banach algebra A has a bounded approximate identity and span{ab − ba : a, b ∈ A} is dense in A, then the following statements are equivalent. (i) A is biamenable. (ii) BH1(A, X∗)=0, for every left approximately unital Banach A-bimodule X. 4 S. BAROOTKOOB

(iii) BH1(A, X∗)=0, for every right approximately unital Banach A-bimodule X. (iv) BH1(A, X∗)=0, for every approximately unital Banach A-bimodule X. Proof. We only prove (i) is equivalent to (ii). The equivalence of (i) and (iii) is similar and then the equivalence of (i) and (iv) is obvious. Clearly if A is biamenable then (ii) is true. Now let BH1(A, Y ∗) = 0, for every left approximately unital Banach A-bimodule Y and let X be a Banach A-bimodule. Then Corollary 2.9.26 of [6] implies that X0 = AX is a left approximately unital closed submodule of X. Also A( X )=0 and so ( X )∗A = 0. Therefore Lemma X0 X0 2.2 says that BZ1(A, X⊥)= BZ1(A, ( X )∗) = 0. 0 X0 1 ∗ ∗ Let D ∈ BZ (A, X ) and J : X0 → X be the inclusion mapping. Then J ◦ D ∈ 1 ∗ ∗ ∗ BZ (A, X0 ) and by assumption J ◦ D = ∆φ0 , for some φ0 ∈ Z(A, X0 ). Now the ∗ ∗ ⊥ equation X = X0 ⊕ X0 , which is implied from Theorem 4.9 of [9], shows that ∗ there exists an extension φ of φ0 such that φ ∈ Z(A, X ). 1 ⊥ Define D0 = D − ∆φ. Then D0 ∈ BZ (A, X0 )=0 and so D = ∆φ. A similar result of the previous lemma in the area of amenability is given in Proposition 2.1.5 of [10]. Now combination of the Lemmas 2.3 and 2.4 gives the following theorem that leads to a condition for biamenability of Banach algebras and then we can find some examples of biamenable Banach algebras which are not amenable. Theorem 2.5. Each unital Banach algebra A with A = span{ab − ba : a, b ∈ A}, is biamenable. Corollary 2.6. If A = span{ab − ba : a, b ∈ A} and A has an identity, then the only biderivation D : A × A → A∗ is zero. Proof. Let D : A × A → A∗ be a biderivation. By Theorem 2.5 A is biamenable, so D is inner. That is there exists an f ∈ Z(A, A∗) such that for every a, b ∈ A, D(a, b)= f[a, b]. Now hf, [a, b]i = hfa − af, bi =0. Hence our assumption implies that f =0 and so D = 0. Every bounded bilinear mapping f : X × Y → Z on normed spaces X,Y and Z, has two natural extensions f ∗∗∗ and f t∗∗∗t from X∗∗ × Y ∗∗ to Z∗∗ as follows. We define the adjoint f ∗ : Z∗ × X → Y ∗ of f by hf ∗(z∗, x),yi = hz∗, f(x, y)i, where x ∈ X,y ∈ Y and z∗ ∈ Z∗. We then define f ∗∗ =(f ∗)∗ and f ∗∗∗ =(f ∗∗)∗. Let f t : Y × X −→ Z be the flip map of f which is defined by f t(y, x) = f(x, y) (x ∈ X,y ∈ Y ). If we continue the latter process with f t instead of f, we come to the bounded bilinear mapping f t∗∗∗t : X∗∗ × Y ∗∗ → Z∗∗. Where π is the multiplication of a Banach algebra A, π∗∗∗ and πt∗∗∗t are actually the first and second Arens products, which are denoted by and ♦, respectively. For more detailes see [1] and [2]. Now we give some examples of biamenable Banach algebras. ON BIAMENABILITY OF BANACH ALGEBRAS 5

Example 2.7. According to Lemma 5.8 of [11], since for each infinite dimensional Hilbert space H and every integer n ≥ 0, B(H)(2n) = span{au − ua : a ∈ B(H),u ∈ B(H)(2n)}. So Theorem 2.5 help us to find some biamenable Banach algebras such as the Banach algebra B(H)(2n) and the module extension Banach algebra B(H) ⊕ B(H)(2n) with the product and norm as follows. (a, u)(b, v)=(ab, av + ub), k(a, u)k = kak + kuk, (a ∈ B(H),u ∈ B(H)(2n)). Although B(H) is not amenable in general. Note that since {au − ua : a ∈ B(H),u ∈ B(H)(2n)} is a subset of {uv−vu : u, v ∈ B(H)(2n)} and {[a, v]−[b, u]: a, b ∈ B(H),u,v ∈ B(H)(2n)}. Therefore the commutators span the whole of B(H)(2n) and B(H) ⊕ B(H)(2n). Also similar to last corollary we can show that the only biderivation from B(H)× B(H) to B(H)(2n+1) is zero. A similar method as Lemma 5.7 of [11], can show that for the Banach algebra K(H) of compact operators on H, K(H)= span{ku − uk : k ∈ K(H): u ∈ B(H)}. So similarly B(H) ⊕ K(H) is biamenable. Although Remark 5.10 of [11] says that it is not amenable. Let G be a locally compact group. m ∈ L∞(G)∗ is a mean on L∞(G) if m(1) = kmk = 1. A mean m on L∞(G) is called a left invariant mean if for each ∞ x ∈ G and g ∈ L (G), m(δx ∗ g)= m(g). G is called amenable if there is a left invariant mean on L∞(G). Consider L∞(G) as an L1(G)-bimodule with the left and right module actions 1 ∞ ∞ ∞ 1 ∞ πℓ : L (G) × L (G) → L (G) πr : L (G) × L (G) → L (G), defined by πℓ(f,g) = f ∗ g and πr(g, f)=( G f)g. Then we have the following proposition. R Proposition 2.8. Let G be a locally compact group such that Z(L1(G), L∞(G)∗) containes an element n such that n(1) =06 . Then G is amenable.

∞ |n| Proof. Deﬁne |n| by |n|(φ) = |n(g)|, for each g ∈ L (G). Then m = |n(1)| is a positive element of Z(L1(G), L∞(G)∗) such that m(1) = 1. Therefore by [10, Proposition 1.1.2], m is a mean on L∞(G). Now we have

hm, f ∗ gi = hm, πℓ(f,g)i ∗ = hπℓ (m, f),gi t∗t = hπr (f, m),gi

= hm, πr(g, f)i

= hm, gi f ZG = hm, gi (f ∈ P (G),g ∈ L∞(G)). 6 S. BAROOTKOOB

1 Where P (G)= {f ∈ L (G); kfk1 =1, f ≥ 0}. So [10, Lemma 1.1.7] implies that G is amenable. A big class of Banach algebras are not biamenable, although they may be amenable. For example if there exists a non-zero derivation d : A → A∗∗ on a commutative Banach algebra A such that for some a, b ∈ A, d(a)d(b) =6 0, then A is not biamenable. Since the map D : A × A −→ A∗∗ (a, b) 7→ d(a)d(b) where denotes the first Arens product of A∗∗, defines a biderivation which is not inner. Also if there is a Banach A-bimodule X such that Z(A, X)= {0} and there is a non zero biderivation from A × A into X∗, then A is not biamenable. Since in this case Z(A, X∗)= {0} and so the only inner biderivation D : A × A → X∗ is zero and therefore we have the following proposition. Proposition 2.9. Let σ(A) be the spectrum of A. If σ(A) =6 ∅, then A is not biamenable. Proof. Let A be a Banach algebra such that σ(A) =6 ∅, f be an element in σ(A) and X be a non zero Banach A-bimodule with module actions ax =0, xa = f(a)x (a ∈ A, x ∈ X). Then Z(A, X)= {0}. But for a non-zero element h ∈ X∗ D : A × A → X∗ (a, b) 7→ f(a)(bh − hb) is a non-zero biderivation. Now by applying Proposition 2.9 and Theorem 1.3.3 of [8], we conclude that every unital commutative Banach algebra is not biamenable. For example C, R, C(Ω), for each Hausdorff space Ω and the group algebra M(G), for each locally compact abelian group G are not biamenable. In the next section we extend this result to arbitrary commutative Banach algebras. Also a combination of Example 2.7 and Proposition 2.9 implies the following. Corollary 2.10. For each integer n ≥ 0 and each infinite dimensional Hilbert space H, σ(B(H)(2n))= ∅ and σ(B(H) ⊕ B(H)(2n))= ∅. Example 2.11. Let A be a Banach space and θ be a non zero element of A∗. Then A is a Banach algebra with the multiplication ab = θ(a)b, (a, b ∈ A). Now since θ ∈ σ(A), A with this multiplication is not biamenable.

3. Some properties In this section we study some properties of biamenable Banach algebras and we tend to some another examples of non biamenable Banach algebras. ON BIAMENABILITY OF BANACH ALGEBRAS 7

Theorem 3.1. Let A be a Banach algebra and consider C as a Banach A- bimodule. If there is a nonzero derivation d : A → C, then biamenability of A implies amenability of A. Proof. Let X be a Banach A-bimodule and d′ : A → X∗ be a bounded derivation. Then D : A × A → X∗ (a, b) 7→ d(a)d′(b) is a bounded biderivation and so there is f ∈ Z(A, X∗) such that d(a)d′(b)= D(a, b)= f[a, b] (a, b ∈ A). Therefore for every b ∈ A and for some a ∈ A such that d(a) =6 0 we have ′ d (b)= δ fa (b). − d(a) Example 3.2. Let D = {z ∈ C; |z| ≤ 1} be the unit disc, and A(D) be the disc algebra. We can consider C as an A(D)-bimodule with module actions αf = αf(0) = fα and d : A(D) → C f 7→ f ′(0) is a nonzero derivation. Therefore since A(D) is not amenable so it is not biamenable. We know that every amenable Banach algebra has an approximate identity (See Proposition 2.2.1 of [10]). A similar result is given in the following. Proposition 3.3. If A = span{ab − ba : a, b ∈ A} and A is biamenable, then A has a bounded approximate identity. Proof. If A is biamenable, then for the biderivation D : A × A → A∗∗ (a, b) 7→ [a, b] ∗∗ there is E ∈ Z(A, A ) such that for each a, b ∈ A, E[a, b] = [a, b]. Now let (eα) be a bounded net in A which is w∗-convergent to E. Then we have

limα eα[a, b] = w − limα eα[a, b] = E[a, b] = [a, b] = [a, b]E = w − limα[a, b]eα = limα[a, b]eα, and by assumption A has an approximate identity (eα). Note that the converse of this proposition is not true in general. For example in the sequel we see that every commutative Banach algebra is not biamenable. Although it may be unital or approximately unital. For each integer n ≥ 0 put AA(2n) + A(2n)A = {aa(2n) + b(2n)b : a, b ∈ A, a(2n), b(2n) ∈ A(2n)}. Then we have the following proposition. 8 S. BAROOTKOOB

Proposition 3.4. If A is biamenable, then for each integer n ≥ 0, span(AA(2n) + A(2n)A) is dense in A(2n). Proof. If span(AA(2n) +A(2n)A) is not dense in A(2n), then there exists a non-zero linear functional f ∈ A(2n+1) such that it is zero on span(AA2n + A(2n)A). Now the bilinear mapping D : A × A → A(2n+1) (a, b) 7→ f(a)f(b)f is a biderivation which is not inner. So A is not biamenable, which is a contradiction. Let A be a Banach algebra and n A = span{a1...an : ai ∈ A} (n ∈ N). As a corollary of the latter proposition we have: Corollary 3.5. If A is biamenable then for each n ∈ N, An is dense in A. Proof. By Proposition 3.4 A2 is dense in A. Now by applying the density of A2 in A we can prove that A3 is dense in A and also by an inductive method we can prove that for each n, An is dense in A. For a Banach algebra A, put [A, A]= {[a, b]: a, b ∈ A} and [A, A]A = {[a, b]c : a, b, c ∈ A}. The following proposition gives a big class of non-biamenable Banach algebras. Proposition 3.6. Let A be a biamenable Banach algebra. Then span([A, A] ∪ [A, A]A) is dense in A. Proof. Suppose S = span([A, A] ∪ [A, A]A) is not dense in A. Then there exists a ∗ nonzero element f ∈ A such that f|S = 0. In particular for each a, b, c ∈ A, we have f(ab) = f(ba) and c.f(ab)= c.f(ba). Consider X = f.A as an A-bimodule with module actions (f.a).b = f.ab, and b.(f.a)=0 (a, b ∈ A). Then Z(A, X∗) = {0} and so the only inner biderivation from A × A to X∗ is zero. Now by Corollary 3.5 the bilinear mapping D : A × A → X∗ defined by hD(a, b), f.ci = f(abc), (a, b, c ∈ A) is nonzero. Also for each a,b,c,d ∈ A we have hD(ab, c), f.di = f(abcd) = f(bcda) = hD(b, c), f.dai = hD(b, c), (f.d).a)i = haD(b, c), f.di = haD(b, c)+ D(a, c)b, f.di, ON BIAMENABILITY OF BANACH ALGEBRAS 9 and similarly hD(a, bc), f.di = f(abcd) = f(cdab) = (b.f)(cda) = (b.f)(acd) = f(acdb) = hD(a, c), f.dbi = hbD(a, c)+ D(a, b)c, f.di. So D is a nonzero biderivation and so it is not inner. That is a contradiction. Note that if a biamenable Banach algebra A has a right approximate identity, then [A, A] ⊆ [A, A]A and therefore span([A, A]A) is dense in A. This may be compared with the converse of Proposition 3.3, for each biamenable Banach algebra. Corollary 3.7. Every non zero commutative Banach algebra is not biamenable. This corollary implies that for every locally compact abelian group G, L1(G) is not biamenable but it is amenable by applying Johnson’s theorem and the Example 1.1.5 of [10]. Theorem 3.8. Let A be a Banach algebra, X be a Banach A-bimodule and I be a closed ideal of it such that Z(A, X∗) = Z(I,X∗). Then if BH1(I,X∗) = {0} A 1 ∗ and I is biamenable, then BH (A, X )= {0}. ∗ 1 ∗ Proof. Let D : A × A → X be a biderivation. Then D0 = D|I×I ∈ BZ (I,X ) ∗ and so D0 = ∆E, for some E ∈ Z(A, X ). Put D˜ = D − ∆E. Then D˜(I × I)=0 D A A ∗ D ˜ and so : I × I → X , defined by ((a + I, b + I)) = D((a, b)) is a well defined map. Put X0 = IX + XI. Then

X ∗ ⊥ ∗ ( ) = X0 = {φ ∈ X ; φi =0= iφ, for all i ∈ I}. X0 ⊥ A and so we can consider X0 as an I -bimodule with the module actions (a+I).φ = ⊥ a.φ and φ.(a + I)= φ.a, for each a ∈ A and φ ∈ X0 . On the other hand we have D˜(a, b)ij =(D˜(ai, b) − aD˜(i, b))j = D˜(ai, bj) − bD˜(ai, j) − aD˜(i, bj)+ abD˜(i, j) =0, (i, j ∈ I, a, b ∈ A). Similarly we can show that ijD˜(a, b) = 0. Therefore by density of I2 in I (Propo- D A A ⊥ sition 3.4) we conclude that ( I × I ) ⊆ X0 and then we can coclude that D A A ⊥ A ⊥ ∗ : I × I → X0 is a biderivation. So there is ψ ∈ Z( I ,X0 ) ⊆ Z(A, X ) such ∗ that D − ∆E = D˜ = ∆ψ. Hence D = ∆E+ψ and E + ψ ∈ Z(A, X ). We know that a Banach algebra A is amenable if and only if the unconditional unitization A♭ of A (see [6, Deﬁnition 1.3.3]) is amenable [10, Corollary 2.3.11]. But it is not true for biamenability of Banach algebras. Indeed we show that the unconditional unitization of each Banach algebra is not biamenable. 10 S. BAROOTKOOB

Lemma 3.9. If θ : A → B is a continuous homomorphism of Banach algebras with dense range and A is biamenable, then so is B. Proof. Let X be a Banach B-bimodule. Consider X as an A-bimodule with module actions ax = θ(a)x and xa = xθ(a). Now for each D ∈ BZ1(B,X∗), 1 ∗ D ◦ (θ × θ) ∈ BZ (A, X ) and biamenability of A implies that D ◦ (θ × θ) = ∆φ for some φ ∈ Z(A, X∗). Now by density we conclude that φ ∈ Z(B,X∗) and D = ∆φ . A Corollary 3.10. If A is biamenable then for each closed ideal I of A, I is biamenable. For analogues of the above two results in the area of amenability see Proposition 2.3.1 and Corollary 2.3.2 of [10]. Now we have the following theorem for the unconditional unitization A♭ of a Banach algebra A. Theorem 3.11. For each Banach algebra A, the unconditional unitization A♭ is not biamenable. ♭ C A♭ Proof. If A is biamenable then = A is biamenable by Corollary 3.10 (recall that A is a closed ideal in A♭). A contradiction.

4. biamenability of a pair of Banach algebras Let A and B be Banach algebras and X be an A − B-bimodule that is X is an A-bimodule and B-bimodule and we have a(xb)=(ax)b , b(xa)=(bx)a (a ∈ A, b ∈ B, x ∈ X). The bounded bilinear mapping D : A × B → X is called a biderivation if D is a derivation with respect to both arguments. That is the mappings aD : B → X and Db : A → X where

aD(b)= D(a, b)= Db(a) (a ∈ A, b ∈ B) are derivations. We denote the space of such biderivations by BZ1(A × B,X). Let x ∈ Z(A, X) ∩ Z(B,X), where Z(A, X)= {x ∈ X; ax = xa ∀a ∈ A}.

The map Dx : A × B → X that

Dx(a, b)= x[a, b]=(xa)b − (xb)a (a ∈ A, b ∈ B) is a basic example of biderivations and called an inner biderivation. We denote the space of such inner biderivations by BN 1(A × B,X). Also we deﬁne the ﬁrst bicohomology group BH1(A × B,X) as follows, BZ1(A × B,X) BH1(A × B,X)= . BN 1(A × B,X) Let A and B be Banach algebras. We say that A and B commute with respect to an A−B-bimodule X if for each a ∈ A, b ∈ B and x ∈ X we have a(bx)= b(ax) and (xb)a =(xa)b. ON BIAMENABILITY OF BANACH ALGEBRAS 11

Note that if A and B commute with respect to X then they commute with respect to A − B-bimodule X∗. For example if we consider X as an A − B-bimodule with module actions zero on A, then A and B commute with respect to this A − B-bimodule. Also if A is commutative then A commutes with itself with respect to A. Proposition 4.1. If A and B commute with respect to an A − B-bimodule X and there are nonzero derivations d : A → X and d′ : B → X, then there is a nonzero biderivation from A × B into X.

Proof. Define D : A × B → X by D(a, b)= δd(a)(b)+ δd′(b)(a). Proposition 4.2. For each biderivation D : A × B → X, D(a, b)[c,d] = [a, b]D(c,d) (a, c ∈ A, b, d ∈ B). Proof. Since X is an A − B-bimodule, we have for each a, c ∈ A, b, d ∈ B, a(bD(c,d))+ a(D(c, b)d) + (D(a, b)d)c +(bD(a, d))c = aD(c,bd)+ D(a, bd)c = D(ac, bd) = bD(ac,d)+ D(ac, b)d = b(aD(c,d))+ b(D(a, d)c)+(aD(c, b))d +(D(a, b)c)d. So [a, b]D(c,d)−D(a, b)[c,d]= a(bD(c,d))+b(aD(c,d)+(D(a, b)d)c−(D(a, b)c)d =0. Definition 4.3. We say that the pair (A, B) is biamenable if for each A − B- bimodule X, BH1(A × B,X∗)= {0}. Obviously biamenability of a Banach algebra A is nothing else than some re- finement of biamenability of the pair (A, A) with the same module actions on each argument, Or briefly biamenability of the pair (A, A) implies biamenability of the Banach algebra A. Remark 4.4. If Z(A, X)= {0} or Z(B,X)= {0} then the only inner biderivation D : A × B → X is zero. Therefore if X is an A − B-bimodule such that there is a non zero biderivation from A × B into X∗ and Z(A, X) = {0}, then (A, B) is not biamenable. The following theorem says that amenability of Banach algebras A and B are necessary conditions for biamenability of the pair (A, B). Theorem 4.5. (i) If σ(B) =6 ∅ and (A, B) is biamenable then A is amenable. (ii) If σ(A) =6 ∅ and (A, B) is biamenable then B is amenable. Proof. We only prove (i). Let X be a Banach A-bimodule and d : A → X∗ be a derivation. consider B- bimodule actions bx = 0 and xb = ϕ(b)x, for some ϕ ∈ σ(B). Then it is easy to see that X is a Banach A − B-bimodule. 12 S. BAROOTKOOB

Now D : A×B → X∗ with D(a, b)= ϕ(b)d(a) is a biderivation and so D(a, b)= ∗ ∗ ∗ ∗ D(a,b) x [a, b]= δx∗b(a), for some x ∈ Z(A, X ) ∩ Z(B,X ). so d(a)= = δ x∗b (a). ϕ(b) ϕ(b) for some b ∈ B such that ϕ(b) =06 . i.e. A is amenable.

Note that the converse of this theorem is not true, in general. For example C is not biamenable by Corollary 3.7 and so (C, C) is not biamenable. Also σ(C) =6 ∅, but it is amenable.

5. a condition for preserving biamenability In this section we investigate the relation between biamenability of two pairs (A, B) and (E, F ) of Banach algebras, where there exist homomorphisms T : E → A and S : F → B. Let A, B, E and F be Banach algebras and T : E → A and S : F → B be bounded homomorphism of Banach algebras. If X is a Banach A − B-bimodule, then we may consider X as a Banach E − F -bimodule with module actions e.x = T (e)x, x.e = xT (e), f.x = S(f)x and x.f = xS(f), which x ∈ X, e ∈ E and f ∈ F . Theorem 5.1. If D : A×B → X is a biderivation, then D◦(T ×S): E ×F → X is a biderivation. Also if D is inner so is D ◦ (T × S). The converse is true if T and S are onto.

Proof. It is easy to check that for biderivation D, D ◦ (T × S) is a biderivation. Also if D is inner, then there is x ∈ Z(A, X) ∩ Z(B,X) such that

D(a, b)=(xa)b − (xb)a (a ∈ A, b ∈ B).

Therefore

D◦(T ×S)(e, f)=(xT (e))S(f)−(xS(f))T (e)=(x.e).f−(x.f).e (e ∈ E, f ∈ F ).

Similarly we can show that x ∈ Z(E,X) ∩ Z(F,X). For the next part note that if T and S are onto, then for each a ∈ A and b ∈ B, D(a, b)= D ◦ (T × S)(e, f) for some e ∈ E and f ∈ F . So for each a, a′ ∈ A and b ∈ B there are e, e′ ∈ E and f ∈ F such that D(aa′, b) = D ◦ (T × S)(ee′, f) = D ◦ (T × S)(e, f).e′ + e.D ◦ (T × S)(e′, f) = D(a, b)T (e′)+ T (e)D(a′, b) = D(a, b)a′ + aD(a′, b).

Similarly we have for each a ∈ A and b, b′ ∈ B, D(a, bb′) = D(a, b)b′ + bD(a, b′). By a similar argument we may prove that if T and S are onto and D ◦ (T × S) is inner so is D. Corollary 5.2. Consider A, B, E and F as above. If T and S are onto and (E, F ) is biamenable, then so is (A, B). ON BIAMENABILITY OF BANACH ALGEBRAS 13

References [1] A. Arens, Operations induced in function classes, Monatsh. Math., 55 (1951), 1–19. [2] A. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc., 2 (1951), 839–848. [3] D. Benkoviˇc, Biderivations of triangular algebras, Linear Algebra Appl. 431 (2009), 1587– 1602. [4] M. Breˇsar, Commuting maps: A survey, Taiwanese J. Math. 8 (2004), 361–397. [5] M. Breˇsar, On generalized biderivations and related maps, J. Algebra 172 (1995) 764–786. [6] H.G. Dales, Banach algebras and automatic continuity, volume 24 of London Mathematical Society Monographs. New Series. The Clarendon Press Oxford University Press, New York, 2000. Oxford Science Publications. [7] Y. Du and Y. Wang, Biderivations of generalized matrix algebras, Linear Algebra. Appl. 438 (2013), 4483–4499. [8] J. R. Murphy, C∗-Algebras and operator theory, Academic Press, Inc., (1990). [9] W. Rudin, Functional Analysis, Mc Graw-Hill, New York Inc., (1973). [10] V. Runde, Lectures On Amenability, Lecture Notes In Mathematics, 1774, Springer (2002). [11] Y. Zhang, Weak amenability of module extensions of Banach algebras, Trans. Amer. Math. Soc. 354 (10) (2002), 4131–4151.

Faculty of Basic Sciences, University of Bojnord, P.O. Box 1339, Bojnord, Iran Email address: [email protected]