PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 128, Number 5, Pages 1419–1425 S 0002-9939(99)05139-4 Article electronically published on October 6, 1999

WEAK AMENABILITY OF SEGAL ALGEBRAS

H. G. DALES AND S. S. PANDEY

(Communicated by Christopher D. Sogge)

Abstract. Let G be a locally compact abelian group, and let p ∈ [1, ∞). We show that the Segal algebra Sp(G) is always weakly amenable, but that it is amenable only if G is discrete.

1. Introduction Let A be an algebra, and let E be an A-bimodule. Then a linear map D : A → E is a derivation if D(ab)=a · Db + Da · b (a, b ∈ A) . For example, let x ∈ E,andset

δx(a)=a · x − x · a (a ∈ A) .

Then δx is a derivation; these derivations are inner derivations. Now let A be a , and let E be a Banach A-bimodule. Then the space of continuous derivations from A into E is denoted by Z1(A, E), and the subspace consisting of the inner derivations is N 1(A, E); the first (Banach) cohomology group of A with coefficients in E is  H1(A, E)=Z1(A, E) N 1(A, E) . (For the general theory of the Banach cohomology groups Hn(A, E), where n ∈ N, see [3] and [7].) Let E0 be the dual of E.ThenE0 is also a Banach A-bimodule for the operations defined by hx, a · λi = hx · a, λi , hx, λ · ai = ha · x, λi (a ∈ A, x ∈ E, λ ∈ E0) . The Banach algebra A is amenable if H1(A, E0)={0} for each Banach A-bimodule E; this important concept was introduced by Johnson in [9], where it is proved that the group algebra L1(G) of a locally compact group G is amenable if and only if G is an amenable group. (See also [7, VII, §2.5].) In particular, L1(G) is amenable for each locally compact abelian (LCA) group G. Amenable Banach algebras have certain rather strong properties. For example, each closed ideal I of finite codimension in an A has a bounded ([7, VII, 2.31]), and so I = I2.

Received by the editors March 10, 1998 and, in revised form, July 3, 1998. 1991 Mathematics Subject Classification. Primary 46J10. The second author acknowledges with thanks the support of the Royal Society-INSA exchange program which enabled him to visit the University of Leeds to work with the first author. He is also thankful to the Department of Pure Mathematics at Leeds for hospitality.

c 2000 American Mathematical Society 1419

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1420 H. G. DALES AND S. S. PANDEY

Let A be a Banach algebra. A particular example of a Banach A-bimodule is A itself; now A0 is the dual module of A. The algebra A is weakly amenable if H1(A, A0)={0}. Of course, every amenable Banach algebra is weakly amenable; however the class of weakly amenable Banach algebras is considerably larger than that of amenable Banach algebras. For example, the group algebra L1(G)isweakly amenable for each locally compact group G ; a short proof of this result is given in [4]. A commutative, semisimple Banach algebra, regarded as being defined on its character space ΦA,istermedaBanach function algebra. Examples of weakly amenable, but not amenable, Banach function algebras are given in [1], where it is noted that a commutative Banach algebra is weakly amenable if and only if H1(A, E)={0} for each Banach A-module E. 1 k·k Let G be an LCA group, and let (L (G), 1) be the group algebra on G,so that L1(G) is a Banach algebra for the convolution product (f,g) 7→ f?g,where Z (f?g)(y)= f(x)g(y − x)dx (y ∈ G), G 1 for f, g ∈ L (G). The dual group of G is denoted by Γ or ΓG,andwewritedx and dγ for Haar measures on G and Γ, respectively. The Fourier transform is denoted by F : f 7→ fb,where Z fb(γ)= f(x) h−x, γi dx (γ ∈ Γ) , G and F identifies L1(G) with the Banach function algebra A(Γ) on Γ ([3], [8], [13], [15]). There has been some study of certain subalgebras of the group algebras L1(G); these are the Segal algebras ([13, Chapter 6, §2.1], [14]). Indeed, a subalgebra S of L1(G)isaSegal algebra if the following conditions are satisfied: 1 k·k (i) S is dense in (L (G), 1); (ii) S is translation-invariant (i.e., τxf ∈ S for each f ∈ S and x ∈ G,where (τxf)(y)=f(y − x)(y ∈ G)); k·k (iii) S is a Banach algebra with respect to a S,and k k k k ∈ ∈ τxf S = f S (f A, x G); 7→ → k·k (iv) the map x τxf, G (S, S), is continuous. It is noted in [13, Chap. 6, §2.3] that the subalgebra of S consisting of functions b k·k f such that supp f is compact is dense in (S, S). Particular examples of Segal algebras are the algebras Sp(G), which we now define. Definition 1.1. Let G be an LCA group, and let p ∈ [1, ∞). Then 1 b p Sp(G)={f ∈ L (G):f ∈ L (Γ)} ,

and

||| ||| k k b ∈ f p = f 1 + f (f Sp(G)) . p ||| · ||| The algebras (Sp(G), p) are Segal algebras on S, and they may be identified with Banach function algebras on Γ by using the Fourier transform. Our purpose here is to consider when the algebras Sp(G) are amenable, and when they are weakly amenable.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use WEAK AMENABILITY OF SEGAL ALGEBRAS 1421

Basic properties of the algebras Sp(G) are given in [13] and [14]. For example, the character space of Sp(G) is naturally identified with Γ (and indeed it is shown 1 in [2] that the closed ideal theory of Sp(G) coincides with that of L (G)). It is 1 2 clear that Sp(G) ⊂ Sq(G)when1≤ p ≤ q<∞ and that S2(G)=L (G) ∩ L (G). 1 In the case where G is discrete (and Γ is compact), the algebras Sp(G)andL (G) coincide, and so Sp(G) is an amenable Banach algebra. In the case where G is not discrete, it is proved in [11] that

2 (1) Sp(G) ⊂ Sq(G) ( Sp(G)(p>1) ,

2 where q =max{1,p− 1},andthatS1(G) ( S1(G). It follows that Sp(G)doesnot have a bounded approximate identity, and so Sp(G) is not amenable in this case. However, Sp(G) does have an approximate identity. We shall use the following 1 fact. Let f ∈ Sp(G), and take ε>0. Then there exists u ∈ L (G) such that k k b ||| − ||| 2 u 1 = 1, supp u is compact, and f u?f p <ε;inparticular,Sp(G) is dense ||| · ||| in (Sp(G), p). It remains to prove that Sp(G) is always weakly amenable. For this, we shall establish some preliminary results in §2 and conclude the proof in §3.

2. Preliminaries Let G be an LCA group, and let p ∈ [1, ∞). We shall often write Lp for Lp(G) and Sp for Sp(G).

Lemma 2.1. The subalgebra of Sp consisting of functions with compact support is ||| · ||| dense in (Sp, p).

1 2 Proof. The result is immediate in the case where p = 2, for in this case S2 = L ∩L . Now consider the case where p =1.Letf ∈ S1,andtakeε>0. Then there ∈ ||| − ||| ∈ exist f1,f2 S1 with f f1 ?f2 1 <ε.Wehavef1,f2 S2, and so there exist g1,g2 ∈ S2 such that supp gj is compact and ||| − ||| fj gj 2 <ε/m {||| ||| } for j =1, 2, where m =maxj=1,2 fj 1 +1 . Clearly supp(g1 ?g2)iscompact. Set h = f1 ?f2 − g1 ?g2.Thenkhk < 2ε and 1 b b b b h ≤ f1 · (f2 − gb2) + (f1 − gb1) · gb2 1 1 1

≤ b b − b b − b kb k f1 f2 g2 + f1 g1 g2 2 2 2 2 kbk ||| ||| by H¨older’s inequality, and so h 1 < 2ε.Thus h 1 < 4ε, giving the result in this case. The case of general p follows immediately.

Let A and B be algebras. Then the tensor product A ⊗ B is an algebra with respect to a product that satisfies the conditions

(a1 ⊗ a2)(b1 ⊗ b2)=a1b1 ⊗ a2b2 (a1,a2 ∈ A, b1,b2 ∈ B); the algebra A ⊗ B is commutative when A and B are commutative. Now suppose that (A, k·k)and(B,k·k) are Banach algebras. Then A ⊗ B is a normed algebra

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1422 H. G. DALES AND S. S. PANDEY

with respect to the projective norm k·k ,where  π  Xn Xn  k k k kk k ⊗ ∈ N z π =inf aj bj : z = aj bj,n  ; j=1 j=1 ⊗ k·k ⊗b k·k the completion of (A B, π) is the projective tensor product (A B, π). For example, let G and H be locally compact groups. For f ∈ L1(G)and g ∈ L1(H), identify f ⊗ g ∈ L1(G) ⊗ L1(H) with the element of L1(G × H)given by (f ⊗ g)(x, y)=f(x)g(y)(x ∈ G, y ∈ H) . Then the identification extends to an isometric isomorphism of L1(G)⊗bL1(H)with 1 L (G × H). The dual group of G × H is ΓG × ΓH .Wewrite

(k ⊗ `)(γ,δ)=k(γ)`(δ)(γ ∈ ΓG,δ∈ ΓH )

whenever k and ` are functions on ΓG and ΓH , respectively. Clearly we have fb⊗ gb = f[⊗ g for f ∈ L1(G)andg ∈ L1(H). Let p ∈ [1, ∞), and suppose that f ∈ S (G)andg ∈ S (H). Then p p

[⊗ b kbk f g = f g p , p p and so

||| ⊗ ||| k k k k b kbk ≤ ||| ||| ||| ||| f g p = f 1 g 1 + f g p f p g p . p

Thus f ⊗ g ∈ Sp(G × H), and the map

(f,g) 7→ f ⊗ g, Sp(G) × Sp(H) → Sp(G × H) , is continuous and bilinear. It follows that there is a continuous linear map b T : Sp(G) ⊗ Sp(H) → Sp(G × H)

such that T (f ⊗ g)=f ⊗ g for f ∈ Sp(G)andg ∈ Sp(H). Clearly T is a homo- morphism. The image of T is denoted by Ap.

Lemma 2.2. Let p ∈ [1, ∞).ThenAp is dense in Sp(G × H).

Proof. First consider the case where p = 2. The algebra A2 contains χE×F for each rectangle E × F in G × H such that E and F are Borel subsets of G and H, respectively, and the linear span of the collection of these functions is dense in 1 2 S2(G × H)=(L ∩ L )(G × H). Thus the result holds. Now consider the case where p =1.LetF ∈ S1(G × H), and take ε>0. ∈ × ||| − ||| Then there exist F1,F2 S1(G H)with F F1 ?F2 1 <ε. Clearly, we have F1,F2 ∈ S2(G × H), and so there exist G1,G2 ∈ A2 with ||| − ||| Fj Gj 2 <ε/m {||| ||| } ∈ for j =1, 2, where m =maxj=1,2 fj 1 +1 . Essentially as before, G1 ?G2 A2 ||| − ||| × ||| · ||| and F G1 ?G2 1 < 4ε.ThusA2 is dense in (S1(G H), 1). Again, the case of general p follows immediately. The following results are proved in [6, §2]. Let A and B be commutative Banach algebras. (i) Suppose that A is weakly amenable and that there is a continuous homomorphism T : A → B such that T (A)isdenseinB.ThenB is weakly

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use WEAK AMENABILITY OF SEGAL ALGEBRAS 1423

amenable. (ii) Suppose that A and B are weakly amenable. Then A⊗bB is weakly amenable. Using these results, we immediately obtain the following proposition. Proposition 2.3. Let G and H be LCA groups, and let p ∈ [1, ∞). Suppose that Sp(G) and Sp(H) are weakly amenable. Then Sp(G × H) is weakly amenable. 

3. The result We first examine some special cases. Throughout, p ∈ [1, ∞).

Lemma 3.1. The algebra Sp(Z) is amenable. 1 Proof. We have Sp(Z)=` (Z), which is amenable. Let A be a commutative algebra, and let D : A → E be a derivation into an A-module E.ThenDa = 0 for each idempotent a ∈ A.

Lemma 3.2. Let K be a compact group. Then Sp(K) is weakly amenable. Proof. The dual group of K is denoted by Γ. Since Γ is discrete, the linear span of b functions f ∈ Sp(K) such that f = χ{γ} for some γ ∈ ΓisdenseinSp(K). Each such function is an idempotent in Sp(K). Let D : Sp(K) → E be a continuous derivation into a Banach Sp(K)-module E. Then D(f) = 0 for each such idempotent f,andsoD =0.

Lemma 3.3. The algebra Sp(R) is weakly amenable. Proof. Set Π = {ζ = ξ +iη : ξ>0}, the open right-hand half-plane. The Poisson semigroup (P ζ : ζ ∈ Π) is defined by the formula 1 ζ P ζ (t)= · (t ∈ R) . π ζ2 + t2 It is proved in [16, 2.17] that (P ζ : ζ ∈ Π) is an analytic semigroup in L1(R)and that

1+iη | | | |→∞ P 1 = O(log η )as η . Further, it is proved that (FP ζ )(y)=exp(−ζ |y|)(y ∈ R) , and so

(FP ζ )(y) =exp(−ξ |y|)(y ∈ R) . Thus FP ζ ∈ Lp(R)foreachζ ∈ Π, and, for each η ∈ R,wehave Z  ∞ 1/p 1+iη − | | FP p = exp( p y )dy , −∞ ζ a constant independent of η.Thus(P : ζ ∈ Π) is an analytic semigroup in Sp(R) satisfying the growth condition that

1+iη | | | |→∞ P p = O(log η )as η . 0 Now let D : Sp(R) → Sp(R) be a continuous derivation. A theorem of Gal´e [5, Theorem 2.3] asserts the following. Let A be a Banach algebra generated by an analytic semigroup (aζ : ζ ∈ Π). Suppose that

a1+iη = O(|η|ρ)as |η|→∞,

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1424 H. G. DALES AND S. S. PANDEY

where 0 ≤ ρ<1/2. Then A is weakly amenable. By applying this theorem to the ζ closed subalgebra A of Sp(R) generated by the semigroup (P : ζ ∈ Π), we see that D(P ζ )=0 (ζ ∈ Π). We extend this result by using a theorem of White [17, Theorem 2.4]. Let t0 ∈ R. ∈ Z ∈ ζ ζ 7→ ζ → R For n and ζ Π, define an = τnt0 P . Then the function ζ an, Π Sp( ), is analytic, and aζ · aλ = aζ+λ (m, n ∈ Z,ζ,λ∈ Π) . m n m+n 1 1 ∈ Z Further an = P (n ), and so p p  1 1 lim a a− n =0. n→∞ n p n p ζ Thus the elements an satisfy the conditions in White’s theorem, and so, by that theorem, ζ · ζ ζ · ζ ∈ Z ∈ an D(a−n)=P D(P )(n ,ζ Π) . ζ · ζ ∈ In particular, P D(τt0 P )=0 (ζ Π), and so ζ ζ/2 ζ/2 ζ/2 · ζ/2 ∈ D(τt0 P )=D(P ?τt0 P )=P D(τt0 P )=0 (ζ Π) . Let f ∈ S (R). For each ζ ∈ Π, we have p Z ζ ζ f?P = f(t)τtP dt R ζ 1/n in Sp(R), and so D(f?P )=0 (ζ ∈ Π). Finally, f?P → f in Sp(R)asn →∞, and so Df =0. We have proved that D =0,andsoSp(R) is weakly amenable. We are grateful to the referee for a valuable remark about the above proof. Theorem 3.4. Let G be a locally compact abelian group, and let p ∈ [1, ∞).Then ||| · ||| the Segal algebra (Sp(G), p) is weakly amenable.

Proof. Let E be a Banach Sp(G)-module, and let D : Sp(G) → E be a continuous derivation. We claim that D =0. By Lemma 2.1, it suffices to prove that D(f) = 0 whenever f ∈ Sp(G)and supp f is compact. Let f be such a function. By [8, (5.14)], there is an open and closed, compactly generated subgroup, say H,ofG such that supp f ⊂ H.Wemay regard Sp(H) as a closed subalgebra of Sp(G)withf ∈ Sp(H). By the structure theorem for compactly generated LCA groups ([8, (9.8)]), the compactly generated group H is topologically isomorphic to a group of the form Rm × Zn × K for some m, n ∈ Z+ and some compact abelian group K. It follows from Lemmas 3.1, 3.2, and 3.3, and Proposition 2.3 that the algebra Sp(H)is weakly amenable, and so D | Sp(H) = 0. In particular, D(f) = 0, as required. References

[1] W.G.Bade,P.C.CurtisandH.G.Dales,Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. London Math. Soc., (3) 55 (1987), 359–377. MR 88f:46098 [2] J. T. Burnham, Closed ideals in subalgebras of Banach algebras I,Proc.AmericanMath. Soc., 32 (1972), 551–555. MR 45:4146 [3] H. G. Dales, Banach algebras and automatic continuity, Clarendon Press, Oxford, to appear. [4] M. Despi´c and F. Ghahramani, Weak amenability of group algebras of locally compact groups, Canadian Math. Bulletin, 37 (1994), 165–167. MR 95c:43003

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use WEAK AMENABILITY OF SEGAL ALGEBRAS 1425

[5] J. E. Gal´e, Weak amenability of Banach algebras generated by some analytic semigroups, Proc. American Math. Soc., 104 (1988), 546–550. MR 90a:46144 [6] N. Grønbæk, A characterization of weakly amenable Banach algebras, Studia Math., 94 (1989), 149–162. MR 92a:46055 [7] A. Ya. Helemskii, The homology of Banach and topological algebras, Kluwer Academic Pub- lishers, Dordrecht, 1989. MR 92d:46178 [8] E. Hewitt and K. A. Ross, Abstract harmonic analysis, Vol. I, Springer-Verlag, Berlin, 1963. MR 28:158 [9] B. E. Johnson, Cohomology in Banach algebras, Memoir American Math. Soc, 127 (1972). MR 51:11130 [10] R. Larsen, T. S. Liu and J. K. Wang, On functions with Fourier transforms in Lp,Michigan Math. J, 11 (1964), 369–378. MR 30:412 [11] J. C. Martin and L. Y. H. Yap, The algebra of functions with Fourier transforms in Lp,Proc. American Math. Soc, 24 (1970), 217–219. MR 40:646 [12] S. Poornima, Multipliers of Sobolev spaces, J. , 45 (1982), 1–28. MR 83c:46033 [13] H. Reiter, Classical harmonic analysis and locally compact groups, Oxford University Press, 1968. MR 46:5933 [14] H. Reiter, L1-algebra and Segal algebras, Springer-Verlag, 1971. MR 55:13158 [15] W. Rudin, Fourier analysis on groups, J. Wiley, New York, 1962. MR 27:2808 [16] A. M. Sinclair, Continuous semigroups in Banach algebras, London Math. Soc. Lecture Note Series, 63, Cambridge University Press, 1982. MR 84b:46053 [17] M. C. White, Strong Wedderburn decompositions of Banach algebras containing analytic semigroups, J. London Math. Soc. (2), 49 (1994), 331–342. MR 95d:46056

Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England E-mail address: [email protected] Department of Mathematics, R. D. University, Jabalpur, India E-mail address: [email protected]

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use