Segal Algebras Research Articles

Globally well and ill posedness for non-elliptic derivative Schrödinger equations with small rough data

We show that there exists sc>0 such that the cubic (quartic) non-elliptic derivative Schrödinger equations with small data in modulation spaces M2,1s(Rn) for n≥3 (n=2) are globally well-posed if s≥sc, and ill-posed if s<sc. In 2D cubic case, using the Gabor frame, we get some time-global dispersive estimates for the Schrödinger semi-group in anisotropic Lebesgue spaces, which include a time-global maximal function estimate in the space Lx12Lx2,t∞. By resorting to the smooth effect estimate together with the dispersive estimates in anisotropic Lebesgue spaces, we show that the cubic hyperbolic derivative NLS in 2D has a unique global solution if the initial data in Feichtinger–Segal algebra or in weighted Sobolev spaces are sufficiently small.

Bochner-Schoenberg-Eberlein property for abstract Segal algebras

Abstract: Let A be a BSE Banach algebra and B be an essential abstract Segal algebrawith respect toA. In this paper we present a necessary and sufficient condition forB to be a BSEalgebra as well. Furthermore we study BSE property of some certain abstract Segal algebraswhich are not discussed in previous works.Key words: Abstract Segal algebra; BSE algebra; -weak bounded approximate identity.1. Introduction. Let A be a commutativeBanach algebra without order. Denote by ðAÞ andMðAÞ the Gelfand spectrum and the multiplieralgebra of A, respectively. A bounded continuousfunction on ðAÞ is called aBSE-function if thereexists a constant C>0 such that for every finitenumber of ’ 1 ;...;’ n in ðAÞ and the same numberof complex numbers c 1 ;...;c n , the inequalityX nj¼1 c j ð’ j Þ C:X nj¼1 c j ’ j bounded approximate identity. A holds. The BSE-norm of , k k BSE , is defined to bethe infimum of all such C.ThesetofallBSE-functions is denoted by C BSE ð ðAÞÞ. Takahasi andHatori [19] showed that under the norm k:k

APPROXIMATE AMENABILITY OF SEGAL ALGEBRAS

AbstractIn this paper we first show that for a locally compact amenable group $G$, every proper abstract Segal algebra of the Fourier algebra on $G$ 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 $2\times 2$ special unitary group, $\mathrm{SU} (2)$, are not approximately amenable.

Segal algebras, approximate identities and norm irregularity in C0(X,A)

We study three closely related concepts in the context of the Banach algebra $C_0(X,A)$. We show that, to a certain extent, Segal extensions, norm irregularity and the existence of approximate identities in $C_0(X,A)$ can be deduced from the corresponding

On dense ideals of C*-algebras and generalizations of the Gelfand–Naimark Theorem

We develop the theory of Segal algebras of commutative $C^\ast $-algebras, with an emphasis on the functional representation. Our main results extend the Gelfand&#8211;Naimark Theorem. As an application, we describe faithful principal ideals of $C^\ast $-

Formula omitted]-Segal algebras with order unit

We introduce the notion of a (noncommutative) C∗-Segal algebra as a Banach algebra (A,‖⋅‖A) which is a dense ideal in a C∗-algebra (C,‖⋅‖C), where ‖⋅‖A is strictly stronger than ‖⋅‖C on A. Several basic properties are investigated and, with the aid of the theory of multiplier modules, the structure of C∗-Segal algebras with order unit is determined.

ONn-IDEAL AMENABILITY OF CERTAIN BANACH ALGEBRAS

AbstractIn this paper we consider some notions of amenability such as ideal amenability,n-ideal amenability and approximaten-ideal amenability. The first two were introduced and studied by Gordji, Yazdanpanah and Memarbashi. We investigate some properties of certain Banach algebras in each of these classes. Results are also given for Segal algebras on locally compact groups.

CHARACTER AMENABILITY AND CONTRACTIBILITY OF ABSTRACT SEGAL ALGEBRAS

AbstractLet ℬ be an abstract Segal algebra with respect to 𝒜. For a nonzero character ϕ on 𝒜, we study ϕ-amenability, and ϕ-contractibility of 𝒜 and ℬ. We then apply these results to abstract Segal algebras related to locally compact groups.

Biflatness and Pseudo-Amenability of Segal Algebras

AbstractWe investigate generalized amenability and biflatness properties of various (operator) Segal algebras in both the group algebra,L1(G), and the Fourier algebra,A(G), of a locally compact groupG.

Approximate weak amenability of abstract Segal algebras

In this paper the approximate weak amenability of abstract Segal algebras is investigated. Applications to compact groups are given. Also an open problem raised by Ghahramani and Lau is answered negatively.

Approximate amenability of semigroup algebras and Segal algebras

In recent years, there have been several studies of various `approximate' versions of the key notion of amenability, which is defined for all Banach algebras; these studies began with work of Ghahramani and Loy in 2004. The present memoir continues such work: we shall define various notions of approximate amenability, and we shall discuss and extend the known background, which considers the relationships between different versions of approximate amenability. There are a number of open questions on these relationships; these will be considered. In Chapter 1, we shall give all the relevant definitions and a number of basic results, partly surveying existing work; we shall concentrate on the case of Banach function algebras. In Chapter 2, we shall discuss these properties for the semigroup algebra `1(S) of a semigroup S. In the case where S has only finitely many idempotents, `1(S) is approximately amenable if and only if it is amenable. In Chapter 3, we shall consider the class of weighted semigroup algebras of the form `1(N^; !), where ! : Z ! [1; 1) is an arbitrary function. We shall determine necessary and sufficient conditions on ! for these Banach sequence algebras to have each of the various approximate amenability properties that interest us. In this way we shall illuminate the implications between these properties. In Chapter 4, we shall discuss Segal algebras on T and on R. It is a conjecture that every proper Segal algebra on T fails to be approximately amenable; we shall establish this conjecture for a wide class of Segal algebras.

On Approximate Ideal Amenability in Banach Algebras

The notions of approximate I-weak amenability and approximate ideal amenability in Banach algebras are introduced. General theory is developed for these notions, and in particular, we show that the approximate ideal amenability of the Segal algebra S(G) in L(G) implies that G, and hence L(G) is amenable. Mathematics Subject Classification 2000: 46H20, 46H10, 46H25.

APPROXIMATE AMENABILITY OF SCHATTEN CLASSES, LIPSCHITZ ALGEBRAS AND SECOND DUALS OF FOURIER ALGEBRAS

Amenability of any of the algebras described in the title is known to force them to be finite-dimensional. The analogous problems for approximate amenability have been open for some years now. In this article we give a complete solution for the first two classes, using a new criterion for showing that certain Banach algebras without bounded approximate identities cannot be approximately amenable. The method also provides a unified approach to existing non-approximate amenability results, and is applied to the study of certain commutative Segal algebras. Using different techniques, we prove that bounded approximate amenability of the second dual of a Fourier algebra implies that it is finite-dimensional. Some other results for related algebras are obtained.

ESSENTIAL AMENABILITY OF ABSTRACT SEGAL ALGEBRAS

AbstractA number of well-known results of Ghahramani and Loy on the essential amenability of Banach algebras are generalized. It is proved that a symmetric abstract Segal algebra with respect to an amenable Banach algebra is essentially amenable. Applications to locally compact groups are given.

Approximate and pseudo-amenability of various classes of Banach algebras

We continue the investigation of notions of approximate amenability that were introduced in work of the second and third authors together with R.J. Loy. It is shown that every boundedly approximately contractible Banach algebra has a bounded approximate identity, and that the Fourier algebra of the free group on two generators is not operator approximately amenable. Further examples are obtained of ℓ 1 -semigroup algebras which are approximately amenable but not amenable; using these, we show that bounded approximate contractibility need not imply sequential approximate amenability. Results are also given for Segal algebras on locally compact groups, and algebras of p-pseudo-functions on discrete groups.

Segal Algebras on Hermitian Groups

Segal Algebras on Hermitian Groups

Operator space structure on Feichtinger's Segal algebra

We extend the definition, from the class of abelian groups to a general locally compact group G, of Feichtinger's remarkable Segal algebra S 0 ( G ) . In order to obtain functorial properties for non-abelian groups, in particular a tensor product formula, we endow S 0 ( G ) with an operator space structure. With this structure S 0 ( G ) is simultaneously an operator Segal algebra of the Fourier algebra A ( G ) , and of the group algebra L 1 ( G ) . We show that this operator space structure is consistent with the major functorial properties: (i) S 0 ( G ) ⊗ ˆ S 0 ( H ) ≅ S 0 ( G × H ) completely isomorphically (operator projective tensor product), if H is another locally compact group; (ii) the restriction map u ↦ u | H : S 0 ( G ) → S 0 ( H ) is completely surjective, if H is a closed subgroup; and (iii) τ N : S 0 ( G ) → S 0 ( G / N ) is completely surjective, where N is a normal subgroup and τ N u ( s N ) = ∫ N u ( s n ) d n . We also show that S 0 ( G ) is an invariant for G when it is treated simultaneously as a pointwise algebra and a convolutive algebra.

Operator Segal algebras in Fourier algebras

Let $G$ be a locally compact group, $\mathrm{A}(G)$ its Fourier algebra and $\mathrm{L}^1(G)$ the space of Haar integrable functions on $G$. We study the Segal algebra ${\mathrm{S}^1\!\mathrm{A}(G)}= {\mathrm{A}(G)}\cap{\rm L}^1(G)$ in ${\mathrm{A}(G)}$.

On characterizations of the image of the Gelfand transform of commutative Banach algebras

AbstractIn this paper, we propose a new method to characterize in terms of “quasi‐topology” the image of Gelfand transform of commutative Banach algebras. Our studies in this paper were inspired by the work of R. Doss in 1967–1968.Among other things, we consider a generalization of Doss' result ([3, Theorem 1] which characterizes Fourier transforms of group algebras) to the case of general commutative Banach algebras.In the last two sections, we study several concrete Banach algebras to see whether our characterization are valid or not; concrete Banach algebras we consider are ideals and quotient algebras of group algebras, Segal algebras for locally compact abelian groups, Lipsitz algebras,C1([0, 1]),C1(ℝ) and etc. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Stability of Gabor frames with arbitrary sampling points

We study the stability of Gabor frames with arbitrary sampling points in the time-frequency plane, in several aspects. We prove that a Gabor frame generated by a window function in the Segal algebra S0(Rd) remains a frame even if (possibly) all the sampling points undergo an arbitrary perturbation, as long as this is uniformly small. We give explicit stability bounds when the window function is nice enough, showing that the allowed perturbation depends only on the lower frame bound of the original family and some qualitative parameters of the window under consideration. For the perturbation of window functions we show that a Gabor frame generated by any window function with arbitrary sampling points remains a frame when the window function has a small perturbation in S0(Rd) sense. We also study the stability of dual frames, which is useful in practice but has not found much attention in the literature. We give some general results on this topic and explain consequences to Gabor frames.