Mahmood Alaghmandan

Approximate amenability of Segal algebras

In this paper we first show that for a locally compact amenable group

Approximate amenability of segal algebras II

G

ZL-amenability constants of finite groups with two character degrees

, every proper abstract Segal algebra of the Fourier algebra on

Character Density in Central Subalgebras of Compact Quantum Groups

G

ZL-amenability and characters for the restricted direct products of finite groups

is not approximately amenable; consequently, every proper Segal algebra on a locally compact abelian group is not approximately amenable. Then using the hypergroup generated by the dual of a compact group, it is shown that all proper Segal algebras of a class of compact groups including the

Weighted hypergroups and some questions in abstract harmonic analysis

2\times 2

Mapping ideals of quantum group multipliers

special unitary group, SU(2), are not approximately amenable.

Seminormed

We prove that every proper Segal algebra of a SIN group is not approximately amenable. Keyword: Segal algebras; approximate amenability; SIN groups; Commutative Banach algebras. AMS codes: 46H20, 43A20. Gourdeau in [5] showed that a Banach algebra A is amenable if and only if every bounded derivation D : A → X for any Banach A-bimodule X can be approximated by a net of inner derivations. A weaker version of this notion is approximate amenability of Banach algebras that is, a Banach algebra A is approximately amenable if and only if every bounded derivation D : A → X∗ for every Banach dual A-bimodule X∗ can be approximated by a net of inner derivations. The concept of approximate amenability first was introduced and studied in [4]. Similar to amenability, different algebras were studied for their approximate amenability property including Segal algebras (for the definition of Segal algebras and their basic properties, look at [9].) In [3], Dales and Loy studied some specific Segal algebras on the commutative groups T and R. They proved that those Segal algebras are not approximately amenable; consequently, they suggested that the same should be true for every proper Segal algebra on these groups. We call a Segal algebra proper if it not equal to the group algebra. Subsequently, Choi and Ghahramani, [2], proved that this conjecture is true by showing that proper Segal algebras on Td and Rd are not approximately amenable (for any dimension d). To do so, they developed a criterion for “ruling out approximate amenability” of Banach algebras. Ghahramani, in the Banach algebra 2011 conference, conjectured that every proper Segal algebra of a locally compact group cannot be approximately amenable. In [1], the author applied the criterion developed in [2] to show that in fact, every proper Segal algebra of a locally compact abelian group is not approximately amenable. Also, applying the hypergroup structure on the dual of compact groups, it was proved that for some classes of compact groups, including SU(2), every proper Segal algebra is not approximately amenable. In this short manuscript we prove that this conjecture is actually true for every SIN group. Recall that a locally compact group G is called a SIN group if there exists a topological basis of 1 ar X iv :1 40 4. 68 54 v1 [ m at h. FA ] 2 8 A pr 2 01 4 2 Mahmood Alaghmandan conjugate invariant neighbourhoods of the identity element of the group G. This class of locally compact groups includes abelian, compact, and discrete groups. Theorem 1 Every proper Segal algebra of a SIN group is not approximately amenable. We prove this theorem, for a generalized version of Segal algebras, called abstract Segal algebras. A Banach algebra (B, ‖·‖B) is an abstract Segal algebra of a Banach algebra (A, ‖·‖A) if B is a dense left ideal in A, there exists C > 0 such that ‖b‖A ≤ C‖b‖B (for each b ∈ B), and there exists M > 0 such that ‖ab‖B ≤M‖a‖A‖b‖B for all a, b ∈ B. We call B a proper abstract Segal algebra of A if B 6= A. It is clear that every (proper) Segal algebra of a locally compact group G is a (proper) abstract Segal algebra of L1(G). Let A be a commutative Banach algebra and let ∆(A) denote the Gelfand spectrum of A and for each a ∈ A, â is the Gelfand transform of a. For definitions related to the Gelfand spectrum of commutative Banach algebras, we refer to [6]. We denote the set of all elements a ∈ A such that supp(â) is compact by Ac. A semisimple commutative Banach algebra A is called a Tauberian algebra when Ac is dense in A. For a Banach algebra A and a constant D > 0, A has a D-bounded approximate identity if there is a net (eα)α ⊆ A such that for every a ∈ A, ‖aeα − a‖A → 0, ‖eαa − a‖A → 0, and supα ‖eα‖A ≤ D. Note that if A is a unital commutative Banach algebra with the unit e ∈ A, then u is constantly one on ∆(A). The following lemma shows that the existence of a bounded approximate identity approximately plays a similar role for a regular Tauberian algebra. Lemma 2 Let A be a regular commutative Tauberian Banach algebra. Then A has a D-bounded approximate identity if and only if for each compact set K ⊆ ∆(A) and > 0, there is some aK, ∈ A such that ‖aK, ‖A ≤ D and âK, |K ≡ 1. Proof. Suppose that (eα)α is a bounded approximate identity of A such that ‖eα‖A ≤ D for some D > 0. For each K ⊆ ∆(A), let bK ∈ Ac such that bK |K ≡ 1 and IK be the ideal {b ∈ A : b̂(K) = {0}}. Therefore, for each b ∈ A, bbK − bK ∈ IK . Considering the quotient norm of A/IK , one gets ‖bK + IK‖A/IK = limα ‖bKeα + IK‖A/IK = limα ‖eα + IK‖A/IK ≤ D. So, there is some b ∈ Ac ∩ IK such that ‖bK + b‖A < D + . Note that for aK := (bK + b), aK |K ≡ 1 and aK ∈ Ac. Conversely, for each > 0 and K ⊆ ∆(A) compact, let aK ∈ A such that âK |K ≡ 1 and ‖aK‖A ≤ D(1 + ). Define eK, := (1 + )aK . It is not hard to show that (eK, )K, forms an approximate identity of the Tauberian algebra A which is ‖ · ‖A-bounded by D where → 0 and K → ∆(A). Let B be an abstract Segal algebra with respect to a Banach algebra A and A has a ‖ · ‖Abounded approximate identity. Then, by an approximation argument, one can show that A has a ‖ · ‖A-bounded approximate identity which lies in B. If A is a Tauberian algebra, the ‖ · ‖Abounded approximate identity of A may belong to Ac ∩ B. The density condition and relation Approximate amenability of Segal algebras II 3 of the norms implies that a proper abstract Segal algebra never has a bounded approximate identity. For a Banach algebra A, let ZA denote the center of A which is the commutative subalgebra of A consisting of all elements a ∈ A such that ab = ba for every b ∈ A. The following proposition proves the non-approximate amenability of Segal algebras with notable centres. Proposition 3 Let B be a proper abstract Segal algebra of a Banach algebra A which has a central bounded approximate identity and ZB is dense in ZA. If ZA is a regular Tauberian algebra, then B is not approximately amenable. Proof. To prove that such an abstract Segal algebra is not approximately amenable, we apply the criterion developed in [2]. To do so, we should construct a sequence (an)n∈N in B such that is ‖ · ‖A-bounded, ‖ · ‖B-unbounded, and satisfying anan+1 = an+1an = an for every n ∈ N. Fist note that ZB is an abstract Segal algebra of ZA. For a fixed > 0 and K0 ⊆ ∆(ZA), for each compact set K such that K0 ⊆ K ⊆ ∆(ZA), there is some aK ∈ ZAc such that âK |K ≡ 1 and ‖aK‖A ≤ D + , by Lemma 2. Define the ‖ · ‖A-bounded net (aK)K0⊆K⊆∆(ZA) as above directed by inclusion over compact sets K. Therefore aK1aK2 = aK1 if supp(âK1) ⊆ K2. We claim that (aK)K0⊆K⊆ΩZA is ‖ · ‖B-unbounded. Note that ZB is a Tauberian algebra. Therefore, A has a ‖ · ‖A-bounded approximate identity (eα) ⊆ ZAc ∩ ZB. So, for each α, for K = supp(eα), eαaK = eα. Hence, ‖eα‖B = lim K0⊆K→∆(ZA) ‖eαaK‖B ≤ lim sup K0⊆K ‖eα‖A‖aK‖B. Therefore, if (aK)K0⊆K is ‖ · ‖B-bounded, (eα)α is a ‖ · ‖B-bounded approximate identity of A which violates the properness of B. To generate a sequence which satisfies the desired conditions mentioned before, fix a nonempty compact set K0 ⊆ ∆(ZA). By our claim, we inductively construct a sequence of compact sets K0 ⊂ K1 ⊂ · · · in ∆(ZA) such that aKnaKn−1 = aKn−1 and ‖B‖aKn ≥ n for all n ∈ N. Then B is not approximately amenable. Now we can prove the main theorem of the paper. Proof of Theorem 1. Note that for every SIN group G, L1(G) has a central bounded approximate identity. Moreover, for each Segal algebra S1(G), ZS1(G) is dense in ZL1(G), [7, Theorem 2]. On the other hand, [8, Theorem 1.8] implies that ZL1(G) is a semisimple regular commutative Tauberian algebra. So Proposition 3 can be applied to finish the proof. Question. Applying some results about structure of locally compact groups, Kotzmann, [7], showed that for every Segal algebra S1(G), ZS1(G) is dense in ZL1(G). The group structure in his proof is essential. It seems that there is not any immediate argument to generalize this proof for a wider class of abstract Segal algebras. It would be of interest if one can generalize 4 Mahmood Alaghmandan this result to abstract Segal algebras. In other words, is there an abstract Segal algebra whose centre is not dense in its ancestor? Question. Is every proper Segal algebra of a locally compact group not approximately amenable? Acknowledgements. This research was supported by a Ph.D. Dean’s Scholarship at University of Saskatchewan and a Postdoctoral Fellowship from the Fields Institute and University of Waterloo. These supports are gratefully acknowledged. The author also would like to express his deep gratitude to Yemon Choi and Ebrahim Samei for many constructive discussions.

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We calculate the exact amenability constant of the centre of

-subalgebras of

\ell^1(G)