Amenability and Orlicz Figa-Talamanca Herz Algebras

In this paper, we characterize the amenablity of locally compact groups in terms of the properties of the Orlicz Figa-Talamanca Herz algebras.


Introduction
Let G be a locally compact group and let A p (G) be the Figà-Talamanca Herz algebra introduced by Herz [10]. The following theorem on the characterization of amenability in terms of the A p (G) algebras is well known. (e) Each homomorphism from A p (G) with finite dimensional range is continuous. [2] Amenability and Orlicz Figà-Talamanca Herz algebras 219 In [16], we have introduced and studied the L Φ -versions of the Figà-Talamanca Herz algebras. Here L Φ denotes the Orlicz space corresponding to the Young function Φ. The space A Φ (G) is defined as the space of all continuous functions u, where u is of the form where f n ∈ L Φ (G), g n ∈ L Ψ (G), (Φ, Ψ) is a pair of complementary Young functions satisfying the Δ 2 -condition, and ∞ n=1 N Φ ( f n ) g n ψ < ∞. This paper has the modest aim of proving the above said equivalent statements in the context of A Φ (G) algebras. We shall begin with some preliminaries that are needed in the sequel.
Then Ψ is also a Young function and is termed as the complementary function to Φ. Further, the pair (Φ, Ψ) is called a complementary pair of Young functions.
Let G be a locally compact group with a left Haar measure dx. We say that a Young function Φ satisfies the Δ 2 -condition, denoted Φ ∈ Δ 2 , if there exists a constant K > 0 and x 0 > 0 such that Φ(2x) ≤ KΦ(x) whenever x ≥ x 0 if G is compact and the same inequality holds with x 0 = 0 if G is noncompact. A Young function Φ is called an N-function if it is continuous and satisfies Φ(x) = 0 if and only if x = 0 and lim x→0 Φ(x)/x = 0, lim x→∞ Φ(x)/x = ∞. If (Φ, Ψ) is a complementary Young pair such that both Φ and Ψ satisfy the Δ 2 -condition, then Φ and Ψ are N-functions.
The Orlicz space, denoted L Φ (G), is a complex vector space consisting of measurable functions, defined as The above norm is called the Luxemburg norm or Gauge norm. If (Φ, Ψ) is a complementary Young pair, then there is a norm on L Φ (G), equivalent to the Luxemburg norm, given by, This norm is called the Orlicz norm. Let C c (G) denote the space of all continuous functions on G with compact support. If a Young function Φ satisfies the Δ 2 -condition, then C c (G) is dense in L Φ (G). Further, if the complementary function Ψ is such that Ψ is continuous and Ψ(x) = 0 if and only if x = 0, then the dual of (L Φ (G), N Φ (·)) is isometrically isomorphic to (L Ψ (G), · Ψ ). In particular, if both Φ and Ψ satisfy the Δ 2 -condition, then L Φ (G) is reflexive.
We say that a Young function Φ satisfies the Milnes-Akimovič condition (in short, MA condition) if for each > 0 there exists c > 1 and an We would like to mention that the above condition will be used again and again in many of the theorems because of the following result due to Rao [ Since the MA condition will be used again and again, in the form of the Theorem 2.1, we would like to give an example of a Young function which is also an N-function and satisfies the MA condition. Consider the Young function Φ(x) = e x − x − 1, 0 ≤ x ∈ R. Then Φ is an N-function satisfying the MA condition [21, page 270]. We would also like to give an example of a Young function which is also an N-function but does not satisfy the MA condition. Consider the Young function Φ(x) = (1 + x) log(1 + x) − x, 0 ≤ x ∈ R. Then Φ is an N-function which does not satisfy the MA condition [21, page 270].
For more details on Orlicz spaces, we refer the readers to [20]. Let Φ and Ψ be a pair of complementary Young functions satisfying the Δ 2 condition. Let The space A Φ (G) equipped with the above norm and with the pointwise addition and multiplication becomes a commutative Banach algebra [16,Theorem 3.4]. In fact, A Φ (G) is a commutative, regular and semisimple Banach algebra with spectrum homeomorphic to G [16,Corollary 3.8]. This Banach algebra A Φ (G) is called the Orlicz Figà-Talamanca Herz algebra.
Let B(L Φ (G)) be the linear space of all bounded linear operators on L Φ (G) equipped with the operator norm. For a bounded complex Radon measure μ on G and f ∈ and it is proved in [16,Theorem 3.5], that for a locally compact group G, the dual of A Φ (G) is isometrically isomorphic to PM Ψ (G).
Let A be a regular, semisimple, commutative Banach algebra with the Gelfand structure space Δ(A). For a closed ideal I of A, the zero set of I, denoted by Z(I), is a closed subset of Δ(A) defined as For a closed subset E ⊂ Δ(A), we define the following ideals in A: By [16,Theorem 3.6] singletons are sets of spectral synthesis for A Φ (G). Further, every closed subgroup is a set of local synthesis for A Φ (G).
The closed set E is a Ditkin set if for every u ∈ I A (E), there exists a sequence {u n } ⊂ j A (E) such that u.u n converges in norm to u; if the condition holds for every compactly supported u ∈ I A (E) then E is called a local Ditkin set. If the sequence can be chosen in such a way that it is bounded and is the same for all u ∈ I A (E), then we say that E is a strong Ditkin set. Note that every Ditkin set is a set of spectral synthesis. The Banach algebra A is called a strong Ditkin algebra if all the singletons and the empty set are strong Ditkin sets.
For more on spectral synthesis see [14,22]. Throughout this paper, G will denote a locally compact group and (Φ, Ψ) will denote a complementary pair of Young functions satisfying the Δ 2 -condition.

Amenability and bounded approximate identities
We begin this section with the main result of this paper on the characterization of amenable groups in terms of the existence of bounded approximate identities in A Φ (G). PROOF. Suppose that G is amenable. Let K be a compact subset of G and let > 0. It follows from Leptin's condition [18,Definition 7 Consider the set Λ = {(K, ) : K is a compact subset of G and > 0} directed as follows: We now proceed further to prove the converse. Suppose that A Φ (G) posseses an approximate identity {u α } α∈Λ bounded by c, for some c > 0. For a positive function ψ ∈ C c (G), using [16,Theorem 3.5], it can be shown as in [18,Theorem 10.4], that ψ 1 = T Ψ CV Ψ (G) . We now show that a similar equality holds if we replace ψ by a positive measure having compact support. Let μ ∈ M(G) be a positive measure having compact support. Choose f 0 ∈ C c (G) such that f 0 is positive,f 0 = f 0 and f 0 1 = 1. Note that, for every f ∈ C c (G) with N Φ ( f ) ≤ 1, we have f 0 * f is also positive and has compact support. Further, As T μ CV Ψ (G) ≤ μ M(G) for all μ ∈ M(G), we have T μ CV Ψ (G) = μ M(G) for all positive μ having compact support. Thus G is amenable, thanks to Theorem 2.1.
REMARK. Note that the condition that Φ satisfies the MA condition in the above theorem is needed only while proving the converse, that is, while invoking Theorem 2.1. As mentioned in [19], the assumption that Φ satisfies the MA condition is needed only to avoid the Riesz-convexity theorem. Note that the proof of the above theorem for the A p (G)-algebras uses the Riesz-convexity theorem. Although an extended Riesz-convexity theorem for Orlicz spaces is available, it cannot be used here.
We now begin to prove some corollaries. In the first corollary, we characterize amenability in terms of certain weak*-closed A Φ (G)-submodules of PM Ψ (G).
Then the space B Φ (G) when equipped with the operator norm becomes a commutative Banach algebra.

COROLLARY 3.3. Let G be an amenable group and let Φ satisfy the MA condition.
Then the two norms · A Φ (G) and · B Φ (G) are equivalent.

PROOF. By definition of B Φ (G), it is clear that, for any u ∈
For this inequality, the assumption on the group to be amenable is not needed.
For the other inequality, note that, since G is amenable, by Theorem 3.1, A Φ (G) possesses a bounded approximate identity {u α } such that u α A Φ (G) ≤ 2 ∀ α. Thus, for any u ∈ A Φ (G), we have, Hence the result.
One of the classical results of Reiter states that every closed subgroup of a locally compact abelian group is a set of spectral synthesis for the Fourier algebra A(G). This result is known as the subgroup lemma [22]. This result was generalized to locally compact groups by Takesaki and Tatsuuma [23]. For 1 < p < ∞, Herz generalized the subgroup lemma to A p (G) algebras under the assumption that G is amenable. For other generalizations see [2]. Our next corollary is the subgroup lemma for spectral synthesis. The proof of this is an immediate consequence of Theorem 3.1 and [16, Theorem 3.6]. COROLLARY 3.4. Let G be an amenable group and let Φ satisfy the MA condition. Then every closed subgroup is a set of spectral synthesis for A Φ (G).

Ideals with bounded approximate identities
In this section, our aim is to characterize amenable groups in terms of Ditkin sets. We shall begin this section by introducing some notations. Let A, B ⊂ G be closed sets of G. Let Our first result is an analogue of [8,Proposition 3.4]. This theorem proves the existence of bounded approximate identities with some properties, in certain closed ideals.

THEOREM 4.1. Let G be an amenable locally compact group and let E be a closed subset of G. If E is a set of synthesis for A Φ (G) and s Φ (E) < ∞, then the ideal I(E) has
a bounded approximate identity {u α } α∈Λ such that the following holds: PROOF. Since G is amenable, it follows from the proof of Theorem 3.1, that A Φ (G) possesses an approximate identity {u K, } F (E)×R + such that . This {v K, } (K, )∈F (E)×R + will satisfy the requirements of the theorem. PROOF. Let C be a compact subset of G such that C ∩ K = ∅. Choose an open neighbourhood U of e such that U is symmetric, relatively compact and Further, note that u is 1 on K and 0 on C, that is, u ∈ S (K, C). Hence the result.
As an immediate consequence we have the following corollary. Here is the characterization of amenable groups in terms of the Ditkin sets.

Weak factorization and cofinite ideals
In this section, we characterize amenable groups in terms of weak factorization and cofinite ideals.
For A p (G) algebras, the following theorem was proved by Losert [15].
, which implies that φ n 1 ≤ c T φ n CV Ψ (G) and hence it follows that φ 1 ≤ T φ CV Ψ (G) . Now proceeding as in the proof of the converse of Theorem 3.1, one can show that G is amenable.
Before we proceed to our next characterization, here are some preparatory lemmas. PROOF. (a) ⇒ (b). Let G be amenable and let I be a cofinite ideal in A Φ (G). By [5,Theorem 2.3], it is enough to show that every closed cofinite ideal is idempotent. Therefore, let us assume that I is a closed cofinite ideal in A Φ (G). Since I is cofinite, the zero set Z(I) is finite and hence, by Lemma 5.2, Z(I) is a set of spectral synthesis. Thus, it follows that I = I(Z(I)). Further, by Theorem 4.1, I(Z(I)) has a bounded approximate identity and hence it follows from Cohen's factorization theorem that I is idempotent.

Derivations and splittings
In this section, we characterize amenable groups in terms of continuous derivations. Next we study algebraic splittings and strong splittings of the extensions of the algebra A Φ (G) in the spirit of [17].
We begin this section by showing the existence of a discontinuous derivation. The proof of this Lemma follows from Lemma 5.3 and [5, page 402]. LEMMA 6.1. Let G be a nonamenable group and let Φ satisfy the MA condition. Then there exists a discontinuous derivation of A Φ (G) into a finite dimensional commutative Banach A Φ (G)-bimodule. LEMMA 6.2. Let G be an amenable group and let I be a closed ideal in A Φ (G) of infinite codimension. Then there exist sequences {u n }, {v n } in A Φ (G) such that u n v 1 · · · v n−1 I but u n v 1 · · · v n−1 u 1 v n ∈ I for all n ≥ 2. Here is the characterization of amenable groups in terms of continuous derivations. THEOREM 6.3. Let G be a locally compact group and let Φ satisfy the MA condition. Then the following are equivalent: PROOF. (a) ⇒ (b) is trivial and (b) ⇒ (c) follows from Lemma 6.1. We shall now prove (c) ⇒ (a). In order to prove this, it is enough to verify the conditions of [13, Theorem 2] for a closed cofinite ideal of A Φ (G). But this follows from Theorem 5.4 and Lemma 6.2.
Before we proceed to study algebraic and strong splittings, here are some notations. Let x, y ∈ G ∪ {0}. For z ∈ C and u ∈ A Φ (G), define u.z and z.u as follows: u.z : = u(x)z for a nonzero x and u.z = 0 otherwise; z.u : = u(y)z for a nonzero y and z.u = 0 otherwise. Note that the above left and right action turns C into an A Φ (G)-bimodule. In order to emphasize the role of x and y, we shall denote this bimodule as C x,y .
A It is shown in [16,Corollary 6.2] that the set of all topologically invariant means on PM Ψ (G) is nonempty. As a result, by following the arguments given for [17, Lemma 3.1], we have the following Lemma for A Φ (G). LEMMA 6.4. If x ∈ G, then there exists an A Φ (G)-bimodule homomorphism Θ x : For algebraic and strong splittings of extensions of Banach algebras, we shall refer to [4].
As a consequence of the above Lemma along with [16, Theorem 3.6] and [16, Corollary 3.8], we have the following Lemma, whose proof is similar to [17,Lemma 3.4]. LEMMA 6.5. Let X be a finite-dimensional Banach A Φ (G)-bimodule. Suppose that X is also essential as a left module. Then every singular extension of A Φ (G) by X splits strongly. COROLLARY 6.6. Let X be a finite-dimensional A Φ (G)-bimodule. Then X is isomorphic to n ⊕ i=1 C x i ,y i , for some n ∈ N and x i , y i ∈ G ∪ {0}. THEOREM 6.7. Let Φ satisfy the MA condition. If G is amenable, then all finite-dimensional extensions of A Φ (G) split strongly.
PROOF. Let X be a finite-dimensional A Φ (G)-module. As G is amenable, by Theorem 3.1, it follows that X is an essential A Φ (G)-module. Thus, by Lemma 6.5, every singular extension of A Φ (G) by X splits strongly. Now the conclusion follows from [4, Corollary 1.9.8].