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Amenability and essential amenability of certain
convolution Banach algebras on compact
hypergroups
M. Lashkarizadeh Bami H. Samea
Abstract In this paper we investigate amenability, and essential amenability of the convolution Banach algebra A(K) for a compact hypergroup K together with their applications to convolution Banach algebras Lp(K) ( 1 < p ≤ ∞), and C(K).
Introduction
In [2] F. Ghahramani and R. J. Loy proved that for a compact group G, the con- volution Banach algebras (Lp^ (G), ∗) (1 < p < ∞) are essentially amenable. Their given proof heavily depends on the amenability of the group algebra L^1 (G) (see Theorem 7.1 and Corollary 7.1(1),(2) of [2]). In the present paper by a quite dif- ferent technique we generalize this result to compact hypergroups. Note that we do not know whether L^1 (K) is amenable for a compact hypergroup K. Vrem ([9]) gave a definition of A(K) for a compact hypergroup K and proved that A(K) is a Banach algebra with convolution product. This Banach algebra plays a key role throughout the paper. The organization of this paper is as follows. The preliminaries and notations are given in section 1. In section 2 we state and prove a basic result on essential amenability of general Banach algebras that is needed for the rest of the paper. In the main theorem of this section (Theorem 2.1) we introduce a class of essentially
Received by the editors August 2007 - In revised form in November 2007. Communicated by A. Valette. Key words and phrases : Hypergroup, Fourier space, Banach algebra, Amenability, Essential amenability.
Bull. Belg. Math. Soc. Simon Stevin 16 (2009), 145–
146 M. Lashkarizadeh Bami – H. Samea
amenable Banach algebras. In section 3, through a different technique, we gen- eralize Lemma 28.1 of [4] from compact groups to compact hypergroups. Indeed we prove that for an infinite compact hypergroup K, K̂ is infinite. As an appli- cation we prove that L^1 (K) for a locally compact hypergroup K is contractible if and only if K is finite. Furthermore, we prove that the convolution Banach al- gebra A(K) on a compact hypergroup K is essentially amenable. Moreover this Banach algebra is amenable if and only if K is finite. In section 4 we prove that for a compact hypergroup K, the convolution Banach algebras Lp^ (K) ( 1 < p ≤ ∞), and C(K) are essentially amenable. Also, we prove such Banach algebras are amenable if and only if K is finite.
1 Preliminaries
For a Banach algebra A, an A-bimodule will always refer to a Banach A-bimodule X, that is a Banach space which is algebraically an A-bimodule, and for which there is a constant CX ≥ 0 such that ‖a.x‖, ‖x.a‖ ≤ CX ‖a‖‖x‖ (a ∈ A, x ∈ X). A bounded linear map D : A → X is called an X-derivation, if for each a, b ∈ A, D(ab) = D(a).b + a.D(b). For every x ∈ X, we define adAx by adAx (a) = a.x − x.a (a ∈ A). It is easily seen that adAx is a derivation. Derivations of this form are called inner derivations. Let A be a Banach algebra and X be a Banach A-bimodule. Then the Banach space X∗^ with the dual module multiplications given by
( f a)(x) = f (ax), (a f )(x) = f (xa) (a ∈ A, f ∈ X∗, x ∈ X),
defines a Banach A-bimodule called the dual Banach A-bimodule X∗. A Banach algebra A is called amenable if for each Banach A-bimodule X, every continuous derivation from A into X∗^ is inner. An A-bimodule X is neo-unital if X = A.X.A. Recall from [2], a Banach algebra A is called essentially amenable if for any neo-unital A-bimodule X, every continuous derivation D : A → X∗^ is inner (See also [6]). A Banach algebra A is called contractible if for each Banach A-bimodule X, every continuous derivation D : A → X is inner. Throughout this paper K is a (measured) locally compact hypergroup with involution x 7 → x¯ and the identity e as defined by Jewett ([5]). By the term mea- sured we mean that K admits a left Haar measure ω K. Let M(K) be the space of all bounded regular Borel measures on K. For 1 ≤ p ≤ ∞, let Lp(K) = Lp(K, ω K ). For x, y ∈ K we define x ∗ y as the set supp( ε x ∗ ε y ). For Borel functions f and g, at least one of which is σ -finite, we define the convolution f ∗ g on K by ( f ∗ g)(x) =
K f^ (x^ ∗^ y)g(^ y¯)d ω K^ (y) (x^ ∈^ K), where^ f^ (x^ ∗^ y) =^
K f d( ε x^ ∗^ ε y^ ). Let K be a compact hypergroup. By Theorem 1.3.28 of [1], K admits a left Haar measure. Throughout the present paper we use the normalized Haar measure ω K on the compact hypergroup K (i.e. ω K (K) = 1). If π ∈ K̂ (where K̂ is the set of equivalence classes of continuous irreducible representations of K, c.f. [1], 11. of [5], and [9]), then by Theorem 2.2 of [9], π is finite dimensional. Furthermore by the proof of Theorem 2.2 of [9], there exists a constant c π such that for each ξ ∈ H π with ‖ ξ ‖ = (^1) ∫
K
|〈 π (x) ξ , ξ 〉|^2 d ω K (x) = c π.
148 M. Lashkarizadeh Bami – H. Samea
and so by induction X = Am.X.Am. Therefore
X = Am.X.Am^ ⊆ B.X.B ⊆ X,
and so X is a neo-unital Banach B-bimodule. Since
‖D(b)‖ ≤ ‖D‖‖b‖A ≤ C‖D‖‖b‖B (b ∈ B),
so the mapping D : B → X∗, b 7 → D(b),
is a continuous derivation. By essential amenability of B, D is inner. Hence there exists ξ ∈ X∗^ such that D = adB ξ. Define D˜ = D − adA ξ. Clearly D˜ ∈ Z(A, X∗^ )
and D˜(B) = { 0 }. Let a ∈ A and x ∈ X. Since X = Am.X.Am, so there exists b ∈ Am^ and y ∈ X such that x = b.y. Now, since b, ab ∈ Am, we have
D^ ˜(a)(x) = D˜(a)(b.y) =
D˜(a).b
(y) =
D˜(ab) − a. D˜(b)
(y) = 0.
Hence D˜ = 0 and so D = adA ξ. Therefore A is essentially amenable.
Corollary 2.2. Let A be a Banach algebra and B be a closed subalgebra of A containing A.A. If B is essentially amenable, then so is A.
3 Amenability and essential amenability of the convolution Ba-
nach algebra A(K) on the compact hypergroup K
Before starting the first result of this section, we note that by Lemma 28.1 of [4] a compact group G is finite if and only if Ĝ is finite. In the following lemma, by a different technique, we generalize this result to compact hypergroups.
Lemma 3.1. A compact hypergroup K is finite if and only if K is finite.̂
Proof. If K is finite, then clearly K̂ is finite. Conversely, if K̂ is finite, then E 1 ( K̂ ) is
finite-dimensional. Since T̂ (K) = E 00 ( K̂ ), so T(K) is finite-dimensional. By The- orem 2.13 of [9], T(K) is uniformly dense in C(K). Since each finite dimensional subspace of a Banach space is closed, it follows that T(K) = C(K). Therefore C(K) is finite-dimensional. Now, by the comment on page 57 of [7] K is finite.
As an application of the above lemma, we have the following result.
Convolution Banach algebras on compact hypergroups 149
Proposition 3.2. Let K be a locally compact hypergroup. Then L^1 (K) is contractible if and only if K is finite.
Proof. If K is a finite hypergroup, then L^1 (K) = ℓ^1 (K) = C(K) = A(K), and K̂ is finite. Hence
ℓ^1 (K) ∼= Â (K) = E 1 ( K̂ ) ∼= ℓ∞^ − ⊕ π ∈ K̂ M d π ( C ),
and so by Exercise 4.1.3 of [8], L^1 (K) is contractible. Suppose L^1 (K) is contractible. By Examples C.1.2(c) and 3.1.12(b) of [8], the Banach space L^1 (K) has the approximation property (c.f. Definition C.1.1(i) of [8]). Now, by Theorem 4.1.5 of [8], L^1 (K) is finite-dimensional. Since A(K) ⊆ L^1 (K), so A(K) is finite-dimensional. Hence by Lemma 4.1, K is finite.
The following theorem is adapted from Theorem 2.3 of [6].
Theorem 3.3. If K is compact, then the convolution Banach algebra A(K) is essentially amenable. Moreover the convolution Banach algebra A(K) is amenable if and only if K is finite.
Proof. By Proposition 4.2 of [9], the mapping f 7 → f̂ is an isometric algebra iso- morphism from the convolution Banach algebra A(K) onto E 1 ( K̂ ). Clearly E 0 ( K̂ ) = c 0 − ⊕ π ∈ K̂ B(H π ), where B(H π ) is equipped with the norm
‖.‖ ϕ ∞. By Remark D.42 of [3] and Example 2.3.16 of [8], for each π ∈ K̂ , the Banach algebra B(H π ) with the norm ‖.‖ ϕ ∞ is 1-amenable. So by Corollary 2.3. of [8], E 0 ( K̂ ) is amenable. For each finite subset F of K̂ define EF by
(EF) π =
idH π for π ∈ F 0 otherwise,
where idH π is the identity operator of B(H π ). It is easy to show that (EF )F is an approximate identity for both Banach algebras E 0 ( K̂ ) and E 1 ( K̂ ). By Theorems 28.32(ii,iii) of [3] and 7.1 of [2], E 1 ( K̂ ) is essentially amenable. So (A(K), ∗) is essentially amenable. If (E α ) α is an approximate identity for E 1 ( K̂ ), then for each finite subset F of I
Card(F) ≤ (^) ∑ π ∈F
k π ‖idH π ‖ ϕ 1
= ‖EF‖ 1 = lim α ‖EF E α ‖ 1
= lim α
(EF) π (E α ) π
π
1 ≤ lim inf α ‖E α ‖ 1.
So for infinite set I, lim α ‖E α ‖ 1 = ∞, and hence E 1 ( K̂ ) does not have a bounded approximate identity. Therefore by Proposition 2.2.1 of [8], this Banach algebra is not amenable. Now, E 1 ( K̂ ) is amenable if and only if K̂ is finite. By Lemma 3.1, K̂ is finite if and only if K is finite.
Convolution Banach algebras on compact hypergroups 151
Lemma 4.2. Let K be a compact hypergroup. Then the following statements are valid:
(i) For 1 < p < 2 , Lp(K) ∗ Lp(K) ⊆ L
p 2 −p (^) (K). (ii) For 2 ≤ p ≤ ∞, Lp(K) ∗ Lp^ (K) ⊆ L^2 (K) ∗ L^2 (K) = A(K). (iii) C(K) ∗ C(K) ⊆ L^2 (K) ∗ L^2 (K) = A(K). (iv) For f ∈ A(K), and 1 ≤ p ≤ ∞, ‖ f ‖p ≤ ‖ f ‖∞ ≤ ‖ f ‖ ϕ 1.
Proof. (i): Since for each x, y ∈ K, ε x ∗ ε y is a probability measure, so for f ∈ Lp(K)
| f (x ∗ y)|p^ =
∫
K
f d( ε x ∗ ε y)
p ≤
K
| f |d( ε x ∗ ε y )
)p
∫
K
| f |p^ d( ε x ∗ ε y ) = | f |p^ (x ∗ y),
and hence
K |^ f^ (x^ ∗^ y)|
p (^) dy ≤ ‖ f ‖pp. Now, an exact method as the proof of Theorem
20.18 of [3], proves (i). (ii),(iii): Since K is compact, so for each 2 ≤ p ≤ ∞, C(K) ⊆ Lp^ (K) ⊆ L^2 (K), and by Theorem 4.9 of [9], A(K) = L^2 (K) ∗ L^2 (K). Hence (ii),(iii) are valid. (iv): Since ω K(K) = 1 and A(K) ⊆ Lp^ (K) for 1 ≤ p ≤ ∞, so ‖ f ‖p ≤ ‖ f ‖∞ for every f ∈ A(K). By Proposition 4.2 of [9], ‖ f ‖∞ ≤ ‖ f ‖ ϕ 1 ( f ∈ A(K)).
Theorem 4.3. Let K be a compact hypergroup and A be any of Banach spaces (Lp^ (K), ∗) ( 1 < p ≤ ∞), and (C(K), ∗). Then A is essentially amenable. Moreover A is amenable if and only if K is finite.
Proof. Let 1 < p < 2 and A = Lp^ (K). There is m ∈ N such that
1 +
2 m+^1 − 1
≤ p < 1 +
2 m^ − 1
So (^2) m− (^1) −( 2 pm− (^1) − 1 )p < 2 ≤ (^2) m (^) −( 2 pm− 1 )p. Now, by induction and using Lemma 4.2(i)
we have
A^2
m ⊆ L
p 2 m−( 2 m− 1 )p (^) (K) ⊆ L^2 (K).
Hence A^2
m+ 1 = A^2
m ∗ A^2
m ⊆ L^2 (K) ∗ L^2 (K) = A(K).
If A is any of Banach spaces (Lp^ (K), ∗) ( 2 ≤ p ≤ ∞), and (C(K), ∗), then by Lemma 3.4(ii),(iii)
A^2 = A ∗ A ⊆ L^2 (K) ∗ L^2 (K) = A(K). Let B = (A(K), ∗). By Lemma 4.2(iv), ‖ f ‖A ≤ ‖ f ‖B ( f ∈ B). Since by Theorem 3.3 B is essentially amenable, from Theorem 2.1 it follows that A is essentially amenable. If A is amenable, then by Proposition 2.2.1 of [8] and Cohn’s Factorization Theorem A ∗ A = A. Hence for each m ∈ N , Am^ = A. But in the first paragraph we proved that there exists m ∈ N such that Am^ ⊆ A(K). Therefore A ⊆ A(K). Clearly A(K) ⊆ A. Hence A(K) = A, and so
A(K) ∗ A(K) = A ∗ A = A = A(K).
Now, from Lemma 4.1, K is finite. Conversely, if K is finite, then A is an essentially amenable Banach algebra with the identity ε e. Hence by Proposition 2.1.5 of [8], A is amenable.
152 M. Lashkarizadeh Bami – H. Samea
Acknowledgements. The authors would like to express their deep gratitude to the referees for their careful reading of the earlier version of the manuscript and several insightful comments. The first author also wishes to thank both The Center of Excellence for Mathematics and The Research Affairs (Research Project No. 850709) of the University of Isfahan for their financial supports. The second author wishes to thank the University of Bu-Ali Sina for moral support.
References
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[3] E. Hewitt and K. A. Ross; Abstract harmonic analysis, Vol. I, 2nd ed., Springer, Berlin, 1979.
[4] E. Hewitt and K. A. Ross; Abstract Harmonic Analysis, Vol. II, Springer Verlag, Berlin, 1970.
[5] R. I. Jewett; Spaces with an abstract convolution of measures, Adv. Math. 18 (1975), 1-110.
[6] M. Lashkarizadeh Bami and H. Samea; Amenability and essential amenability of certain Banach algebras, to appear in Studia. Math. Hungarica.
[7] G. J. Murphy; C∗-algebras and operator theory, Academic Press, San Diego, Calif, 1990.
[8] V. Runde; Lectures on amenability, Lecture Notes in Mathematics, Vol. 1774 , Springer, Berlin, 2002.
[9] R. C. Vrem; Harmonic analysis on compact hypergroups, Pacific J. Math. 85 (1979), 239-251.
Department of Mathematics, University of Isfahan, Isfahan, Iran E-mail : lashkari@sci.ui.ac.ir
Department of Mathematics, Bu-Ali Sina University, Hamedan, Iran. E-mail : h.samea@basu.ac.ir
