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Publicationes Mathematicae Debrecen (00333883) (1-2)
Let A and B be commutative and semisimple Banach algebras. Suppose that ∥ · ∥γ is an algebra cross-norm on A B such that ∥ · ∥γ ≥ ∥ · ∥e, and AbγB is a semisimple Banach algebra. In this paper, we verify the BED property for AbγB. In fact, we show that if AbγB is of BED, then both A and B are so, whenever either A or B is unital. We also show that if B (resp., A) is unital and Ab ⊆ CBSE0 (∆(A)) (resp., Bb ⊆ CBSE0 (∆(B))), then A\bγB ⊆ CBSE0 (∆(AbγB)). We also establish that if B (resp., A) is finite dimensional, then AbγB is of BED if and only if A (resp., B) is of BED. © 2024 Institute of Mathematics, University of Debrecen. All rights reserved.
Banach Journal of Mathematical Analysis (26622033) (3)
Let (X, d) be a compact metric space and A be a commutative and semisimple Banach algebra. Some of our recent works are related to the several BSE concepts of the vector-valued Lipschitz algebra Lip(X,A). In this paper as the main purpose, we verify the BED property for Lip(X,A), which is actually different from the BSE feature. We first prove as an elementary result that Lip(X,A) is regular if and only if A is so. Then we prove that A is a BED algebra, whenever Lip(X,A) is so. Afterwards, we verify the converse of this statement. Indeed, we prove that if A is a BED algebra then CBSE0(Δ(Lip(X,A)))⊆Lip(X,A)^ and LipX⊗A^⊆CBSE0(Δ(Lip(X,A))). It follows that if LipX⊗A is dense in Lip(X,A) then Lip(X,A) is a BED algebra, provided that A is so. Moreover, we conclude that the necessary and sufficient condition for the unital and in particular finite dimensional Banach algebra A, to be a BED algebra is that Lip(X,A) is a BED algebra. Finally, regarding to some known results which disapproves the BSE property for lipα(X,A)(0<α<1), we show that for any commutative and semisimple Banach algebra A with A0≠∅, lipα(X,A) fails to be a BED algebra, as well. © Tusi Mathematical Research Group (TMRG) 2024.
Acta Mathematica Scientia (02529602) (5)
In this paper, X is a locally compact Hausdorff space and A is a Banach algebra. First, we study some basic features of C0(X, A) related to BSE concept, which are gotten from A. In particular, we prove that if C0(X, A) has the BSE property then A has so. We also establish the converse of this result, whenever X is discrete and A has the BSE-norm property. Furthermore, we prove the same result for the BSE property of type I. Finally, we prove that C0 (X, A) has the BSE-norm property if and only if A has so. © Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences 2024.
Rocky Mountain Journal of Mathematics (00357596) (1)
Let A be a separable Banach algebra, G be a locally compact group and 1 < p < ∞. We first provide a necessary and sufficient condition for which Lp(G, A) is a Banach algebra, under convolution product. Then we characterize the character space of Lp(G, A), in the case where A is commutative and G is abelian. Moreover, we investigate the BSE-property for Lp(G, A) and prove that Lp(G, A) is a BSE-algebra if and only if A is a BSE-algebra and G is finite. Finally, we study the BSE-norm property for Lp(G, A) and show that if Lp(G, A) is a BSE-norm algebra then A is so. We prove the converse of this statement for the case where G is finite and A is a unital BSE-algebra. © Rocky Mountain Mathematics Consortium.
Bulletin of the Malaysian Mathematical Sciences Society (01266705) (4)
Let H1 and H2 be two separable Hilbert spaces, K1∈ B(H1) and K2∈ B(H2) . Based on some previous results about tensor product of frames, in this paper we generalize them to tensor product of K-frames. We provide equivalent conditions for that the tensor product of two K1 -frame and K2 -frame is a K1⊗ K2 -frame. Moreover, we investigate whenever the tensor product of two Bessel sequences F and G in H1 and H2 , respectively, is a K1⊗ K2 -dual frame for F⊗ G . © 2023, The Author(s), under exclusive licence to Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia.
Journal of Mathematical Analysis and Applications (0022247X) (1)
Let X be a non-empty set, A be a commutative and semisimple Banach algebra and 1≤p<∞. In this paper, we first investigate the regularity and Tauberian property for the Banach algebra ℓp(X,A) and show that ℓp(X,A) is regular (resp. Tauberian) if and only if A is regular (resp. Tauberian). Then as the main result, we study the BED property for ℓp(X,A). Indeed, we establish that ℓp(X,A) is a BED algebra if and only if A is so. © 2022 Elsevier Inc.
Acta Mathematica Hungarica (02365294) (2)
Let (A, ‖ · ‖ A) and (B, ‖ · ‖ B) be commutative and semisimple Banach algebras. In this paper, we verify the BED property for the direct sum of A and B, denoted by A⊕ B. In particular, we show that A⊕ B is a BED algebra if and only if A and B are so. © 2022, Akadémiai Kiadó, Budapest, Hungary.
Rocky Mountain Journal of Mathematics (00357596) (4)
Let H be a separable Hilbert space. Recently, the concept of K-g-frame was introduced as a special generalization of g-Bessel sequences. In this paper, we point out some gaps in the proof of some existent results concerning K-g-frame. We present examples to indicate that these results are not necessarily valid. Then we remove the gaps and provide some desired conclusions. In this respect, we deal with Schur–Horn problem, which characterizes sequences {k fnk2}∞n=1, for all frames { fn}∞n=1 with the same frame operator. We introduce the concept of synthesis related frames. Finally, as the main result, we investigate around Schur–Horn problem, for the case where H is finite dimensional. In fact, we prove that two frames have the same frame operator if and only if they are synthesis related. c Rocky Mountain Mathematics Consortium
Mediterranean Journal of Mathematics (16605446) (5)
Let H be a separable Hilbert space. It is known that the finite sum of Bessel sequences in H is still a Bessel sequence. But the finite sum of generalized notions of frames does not necessarily remain stable in its initial form. In this paper, for a prescribed Bessel sequence F={fn}n=1∞, we introduce and study KF, the set consisting of all operators K∈ B(H) , such that {fn}n=1∞ is a K-frame. We show that KF is a right ideal of B(H). We indicate by an example that KF is not necessarily a left ideal. Moreover, we provide some sufficient conditions for the finite sum of K-frames to be a K-frame. We also use some examples to compare our results with existing ones. These examples demonstrate that our achievements do not depend on the available results. Furthermore, we study the same subject for K-g-frames and controlled frames and get some similar significant results. © 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
Communications on Pure and Applied Analysis (15340392) (3)
Let (K; d) be a compact metric space, A be a commutative semisimple Banach algebra and 0 < α ≤ 1. The overall purpose of the present paper is to demonstrate that all BSE concepts of Lipα(K;A) are inherited from A and vice versa. Recently, the authors proved in the case that A is unital, Lipα(K;A) is a BSE-algebra if and only if A is so. In this paper, we generalize this result for an arbitrary commutative semisimple Banach algebra A. Furthermore, we investigate the BSE-norm property for Lipα(K;A) and prove that Lipα (K;A) belongs to the class of BSE-norm algebras if and only if A is owned by this class. Moreover, we prove that for any natural number n with n ≥ 2, if all continuous bounded functions on Δ(Lipα (K;A)) are n-BSE-functions, then K is finite. As a result, we obtain that Lipα (K;A) is a BSE-algebra of type I if and only if A is a BSE-algebra of type I and K is finite. Furthermore, in according to a result of Kaniuth and Ulger, which disapproves the BSE-property for lipαK, we show that for any commutative semisimple Banach algebra A, lipα (K;A) fails to be a BSE-algebra, as well. Finally, we concentrate on the classical Lipschitz algebra LipαX, for an arbitrary metric space (not necessarily compact) (X; d) and α > 0, when LipαX separates the points of X. In particular, we show that LipαX is a BSE-algebra, as well as a BSE-norm algebra. © 2021 American Institute of Mathematical Sciences. All rights reserved.
Canadian Mathematical Bulletin (00084395) (3)
Let A and B be commutative and semisimple Banach algebras and let > (B). In this paper,we prove thatA ? Bis a type I-BSE algebra if and only ifAe andBare so.As amain application of this result, we prove that A B is isomorphic with a C*-algebra if and only if Ae and B are isomorphic with C*-algebras. Moreover, we derive related results for the case where A is unital. © Canadian Mathematical Society 2020.
Ukrainian Mathematical Journal (00415995) (7)
The notion of essential amenability of Banach algebras has been defined and investigated. We introduce this concept for Fr´echet algebras. Then numerous well-known results concerning the essential amenability of Banach algebras are generalized for Fréchet algebras. Moreover, related results for the Segal–Fréchet algebras are also provided. As the main result, it is proved that if (A,pl) is an amenable Fréchet algebra with a uniformly bounded approximate identity, then every symmetric Segal–Fréchet algebra in (A,pl) is essentially amenable. © 2020, Springer Science+Business Media, LLC, part of Springer Nature.
Mathematica Slovaca (01399918) (1)
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Mediterranean Journal of Mathematics (16605446) (3)
Let X be a non-empty set, A be a commutative Banach algebra, and 1 ≤ p< ∞. In this paper, we establish some basic properties of ℓp(X, A) , inherited from A. In particular, we characterize the Gelfand space of ℓp(X, A) , denoted by Δ (ℓp(X, A) ). Mainly, we investigate the BSE property of the Banach algebra ℓp(X, A). In fact, we prove that ℓp(X, A) is a BSE algebra if and only if X is finite and A is a BSE algebra. Furthermore, in the case that A is unital, we show that for any natural number n, all continuous bounded functions on Δ (ℓp(X, A) ) are n-BSE functions. However, through an example, we indicate that there is some continuous bounded function on Δ (ℓp(X, A) ) which is not BSE. Finally, we prove that if ℓ1(X, A) is a BSE-norm algebra, then A is so. We also prove the converse of this statement, whenever A is a supremum norm algebra. © 2020, Springer Nature Switzerland AG.
Archivum Mathematicum (00448753) (4)
Let (X, d) be a metric space and α > 0. We study homological properties and different types of amenability of Lipschitz algebras Lipα X and their second duals. Precisely, we first provide some basic properties of Lipschitz algebras, which are important for metric geometry to know how metric properties are reflected in simple properties of Lipschitz functions. Then we show that all of these properties are equivalent to either uniform discreteness or finiteness of X. Finally, some results concerning the character space and Arens regularity of Lipschitz algebras are provided. © 2019, Masarykova Universita. All rights reserved.
Journal of Mathematical Analysis and Applications (0022247X) (1)
Let (K,d) be a compact metric space, 0<α≤1 and LipαK the space of the Lipschitz functions on K. It is known that the Banach algebra LipαK is a BSE-algebra. In this paper, for a commutative unital semisimple Banach algebra A, we prove that the Banach algebra Lipα(K,A) of the A-valued Lipschitz functions is a BSE-algebra if and only if A is so. © 2019 Elsevier Inc.
Periodica Mathematica Hungarica (00315303) (2)
We introduce and study strict, uniform, and compact-open locally convex topologies on an algebra B, by the fundamental system of seminorms of a locally convex subalgebra (A, pα). Moreover, we investigate when B is a locally convex algebra with respect to these topologies. Furthermore, we generalize an essential result related to derivations, from Banach to the Fréchet case. Finally we provide a useful example in this field. © 2017, Akadémiai Kiadó, Budapest, Hungary.
Analysis Mathematica (01333852) (4)
In this paper, we study the concept of a Segal Fréchet algebra and investigate and generalize many known results about abstract Segal algebras, for Segal Fréchet algebras. Moreover, we characterize closed ideals of Segal Fréchet algebras, and show that the ideal theorem is also valid for Fréchet algebras. © 2017, Akadémiai Kiadó, Budapest, Hungary.
Forum Mathematicum (09337741) (6)
Right φ-Amenability and right character amenability have been introduced for Banach algebras. Here, these concepts will be generalized for Fréchet algebras. Then some of the previous available results about right φ-Amenability and right character amenability for the case of Banach algebras will be verified for Fréchet algebras. Related results about Segal Fréchet algebras are provided. © 2018 Walter de Gruyter GmbH, Berlin/Boston 2018.
Canadian Mathematical Bulletin (00084395) (4)
Let (X, d) bea metric space and let J ⊆ [0, ∞) be nonempty. We study the structure of the arbitrary intersections of Lipschitz algebras and define a special Banach subalgebra of ∩y∈j Lipy X, denoted by ILipj X. Mainly, we investigate the C-character amenability of ILipj X, in particular Lipschitz algebras. We address a gap in the proof of a recent result in this field. Then we remove this gap and obtain a necessary and sufficient condition for C-character amenability of ILipj X, specially Lipschitz algebras, under an additional assumption. © Canadian Mathematical Society 2017.
Annals of Functional Analysis (20088752) (1)
Right φ-contractibility and right character contractibility of Ba- nach algebras have been introduced and investigated. Here, we introduce and generalize these concepts for Fréchet algebras. We then verify available results about right φ-contractibility and right character contractibility of Banach alge- bras for Fréchet algebras. Moreover, we provide related results about Segal- Fréchet algebras. © 2010 Mathematics Subject Classification.
UPB Scientific Bulletin, Series A: Applied Mathematics and Physics (12237027) (3)
Let (X, μ) be a measure space. For p, q ϵ (0,∞] and arbitrary subsets P,Q of (0,∞], we introduce and characterize some intersections of Lorentz spaces, denoted by ILp,Q(X, μ), ILJ,q(X, μ) and ILJ,Q(X, μ).
Bulletin of the Australian Mathematical Society (00049727) (2)
Let be a Banach algebra homomorphism from a Banach algebra to a Banach algebra with . Recently, Bhatt and Dabhi ['Arens regularity and amenability of Lau product of Banach algebras defined by a Banach algebra morphism', Bull. Aust. Math. Soc. 87 (2013), 195-206] showed that cyclic amenability of is stable with respect to , for the case where is commutative. In this note, we address a gap in the proof of this stability result and extend it to an arbitrary Banach algebra . © 2015 Australian Mathematical Publishing Association Inc.
Bulletin of the Australian Mathematical Society (00049727) (1)
Let φ be a homomorphism from a Banach algebra B to a Banach algebra A. We define a multiplication on the Cartesian product space A× B and obtain a new Banach algebra A×φB. We show that biprojectivity as well as biflatness of A×φB are stable with respect to φ. © 2014 Australian Mathematical Publishing Association Inc..
Proceedings of the Romanian Academy Series A - Mathematics Physics Technical Sciences Information Science (14549069) (2)
Let G be a locally compact group, ω be a weight function on G and 1 < p < ∞ . We show that projectivity of Lp (G,ω) as a Banach left L1 (G) -module is table with respect to ω . In fact we prove that Lp (G,ω) is a projective L1 (G) -module if and only if G is compact.
Periodica Mathematica Hungarica (00315303) (2)
In this paper, we recall the concept of Segal Fréchet algebra in a Fréchet algebra (A, pℓ) and show that in some cases, every continuous linear left multiplier on (A, p ℓ) is a continuous linear left multiplier of any Segal Fréchet algebra (B, q m) in (A, p ℓ). As the main result, we prove that if A is a commutative Fréchet Q-algebra with an approximate identity, A is semisimple if and only if B is semisimple. © 2015 Akadémiai Kiadó, Budapest, Hungary.
UPB Scientific Bulletin, Series A: Applied Mathematics and Physics (12237027) (4)
Let A be a Fréchet algebra. We introduce and study the notion of weak amenability of A. In fact we show that some results in the field of weak amenability of Banach algebras can be generalized for Fréchet algebras. For example we prove that if A is weakly amenable then A is essential. Moreover if I is a quasinormable closed ideal in the Fréchet algebra A such that A I and I are weakly amenable, then A is weakly amenable, as well.
Bulletin of the Iranian Mathematical Society (10186301) (6)
We present a characterization of Arens regular semi-group algebras ℓ1(S), for a large class of semigroups. Mainly, we show that if the set of idempotents of an inverse semigroup S is finite, then ℓ1(S) is Arens regular if and only if S is finite. © 2014 Iranian Mathematical Society.
Banach Journal of Mathematical Analysis (26622033) (2)
We investigate generalized amenability, contractibility, biprojectivity and biflatness properties of various classes of abstract Segal algebras with respect to the Banach algebra A. Moreover, we verify some of the previous available results about Segal algebras, for abstract Segal algebras.
UPB Scientific Bulletin, Series A: Applied Mathematics and Physics (12237027) (3)
Let G be a locally compact group and 1 ≤ p < ∞. This paper is mainly concerned with introducing the concept of pointwise Lt p functions and studying the structure of the space L tpp (G), consisting of all these functions. Furthermore, some algebraic and topological properties of Ltpp (G) as a Banach algebra under pointwise multiplication are investigated.
Proceedings of the Indian Academy of Sciences: Mathematical Sciences (02534142) (4)
Let A be a Banach algebra. It is obtained a necessary and sufficient condition for the complete continuity and also weak complete continuity of symmetric abstract Segal algebras with respect to A, under the condition of the existence of an approximate identity for B, bounded in A. In addition, a necessary condition for the weak complete continuity of A is given. Moreover, the applications of these results about some group algebras on locally compact groups are obtained. © Indian Academy of Sciences.
Mathematica Slovaca (13372211) (2)
We have recently shown that, for 2 < p < ∞, a locally compact group G is compact if and only if the convolution multiplication f * g exists for all f, g ∈ Lp(G). Here, we study the existence of f * g for all f, g ∈ Lp(G) in the case where 0 < p ≤ 2. Also, for 0 < p < ∞, we offer some necessary and sufficient conditions for Lp(G) * Lp(G) to be contained in certain function spaces on G. © 2013 Versita Warsaw and Springer-Verlag Wien.
Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica (18440835) (3)
Let G be a locally compact group, 1 < p < ∞ and let ω be a weight function on G. Recently, we introduced the Lebesgue weighted Lp-algebra L1ωp (G). Here, we establish necessary and sufficient con- ditions for L1ω,p (G) to be Φ-contractible, pseudo-contractible or con- tractible. Moreover we give some similar results about Lp(G, ω).
Abstract and Applied Analysis (16870409)
Let G be a locally compact group Ω an arbitrary family of the weight functions on G, and 1≤p<∞. The locally convex space ILp(G,Ω) as a subspace of ∩ω∈ΩLp(G,ω) is defined. Also, some sufficient conditions for that space to be a Banach space are provided. Furthermore, for an arbitrary subset J of [1,∞) and a positive submultiplicative weight function ω on G, Banach subspace ILJ(G,ω) of ∩p∈JLp(G,ω) is introduced. Then some algebraic properties of ILJ(G,ω), as a Banach algebra under convolution product, are investigated. © 2013 F. Abtahi et al.
Proceedings of the Romanian Academy Series A - Mathematics Physics Technical Sciences Information Science (14549069) (2)
Let G be a locally compact group, ω be a weight function on G and 1 < p < ∞. In the present note, it is proved that Lp (G,ω) can be considered as a Segal algebra or an abstract Segal algebra with respect to L1 (G), just when G is compact.
Bulletin of the Australian Mathematical Society (17551633) (3)
For a locally compact group G and an arbitrary subset J of [1,∞], we introduce ILJ(G) as a subspace of ∩ pεJL p(G) with some norm to make it a Banach space. Then, for some special choice of J, we investigate some topological and algebraic properties of ILJ(G) as a Banach algebra under a convolution product. © 2012 Australian Mathematical Publishing Association Inc.
Mathematica Slovaca (13372211) (3)
Let A{fraktur} be a normed algebra with identity, Ω be a locally compact Hausdorf space and λ be a positive Radon measure on Ω with supp(λ) = Ω. In this paper, we establish a necessary and sufficient condition for L 1(Ω, A{fraktur}) to be an algebra with pointwise multiplication. Under this condition, we then characterize compact and weakly compact left multipliers on L 1(Ω, A{fraktur}). © 2012 Versita Warsaw and Springer-Verlag Wien.
Proceedings of the Romanian Academy Series A - Mathematics Physics Technical Sciences Information Science (14549069) (2)
Let G be a locally compact group and 2 < p < ∞. We have recently considered the property that convolutions of functions in the L p -space of G exist, and have shown that this is equivalent to compactness of G. Here, we study this property on the weighted Lp -space of G; as the main result, we prove that G is σ -compact if convolutions of functions in the weighted Lp -space of G exist. Copyright © Academia Românǎ 2006.
Bulletin of the Belgian Mathematical Society - Simon Stevin (13701444) (2)
Let G be a locally compact group, to be a weight function on G and 0 < p < 1. We investigate the space Lp (G, w)*Lp(G, w) as a subset of some special spaces such as Lp(G, w), L ∞(G, 1,/w̃) and C0(G, 1/w̃). As the main result we show that all of the mentioned inclusions are equivalent to the discreteness of G.
Acta Mathematica Hungarica (15882632) (4)
Let G be a locally compact group, ω a weight function on G, and 1
Periodica Mathematica Hungarica (15882829) (1)
Let G be a locally compact group, ω be a weight function on G and 1 < p < ∞. Here, we give a sufficient condition for that the weighted Lp-space Lp(G, ω) is a Banach algebra. Also, we get some necessary conditions on G and the weight function ω for Lp(G, ω) to be a Banach algebra. As a consequence, we show that if G is abelian and Lp(G, ω) is a Banach algebra, then G is σ-compact. © Akadémiai Kiadó, Budapest.
Publicationes Mathematicae Debrecen (00333883) (3-4)
Let G be a locally compact group and 1 < p < ∞. The L p-conjecture asserts that LP(G ) is closed under the convolution if and only if G is compact. For 2 < p < ∞, we have recently shown that f * g exists and belongs to L∞(G) for all f, g ε LP(G) if and only if G is compact. Here, we consider the weighted case of this result for a discrete group G and a weight function ω on G; we prove that f * g exists and belongs to l∞ (G, 1/ωÃ) for all f, g ε lp(G,ω) if and only if lp(G,ω) Ç lq(G, 1/ωÃ), the dual of lP(G, ωÃ).
Archiv der Mathematik (0003889X) (3)
Let G be a locally compact group. For 1 < p < ∞, it is well-known that f*g exists and belongs to L p (G) for all f, g ∈ L p (G) if and only if G is compact. Here, for 2 < p < ∞, we show that f*g exists for all f, g ∈ L p (G) if and only if G is compact. We also show that this result does not remain true for 1 < p ≤ 2. © 2007 Birkhäuser Verlag Basel/Switzerland.
