scholarly journals Tensor products of commutative Banach algebras

1982 ◽  
Vol 5 (3) ◽  
pp. 503-512
Author(s):  
U. B. Tewari ◽  
M. Dutta ◽  
Shobha Madan
Keyword(s):  
Abelian Group ◽  
Banach Algebra ◽  
Group Algebra ◽  
Compact Abelian Group ◽  
Banach Algebras ◽  
Locally Compact Abelian Group ◽  
Measure Algebra ◽  
Locally Compact ◽  
Integrable Functions ◽  
Commutative Banach Algebras

LetA1,A2be commutative semisimple Banach algebras andA1⊗∂A2be their projective tensor product. We prove that, ifA1⊗∂A2is a group algebra (measure algebra) of a locally compact abelian group, then so areA1andA2. As a consequence, we prove that, ifGis a locally compact abelian group andAis a comutative semi-simple Banach algebra, then the Banach algebraL1(G,A)ofA-valued Bochner integrable functions onGis a group algebra if and only ifAis a group algebra. Furthermore, ifAhas the Radon-Nikodym property, then the Banach algebraM(G,A)ofA-valued regular Borel measures of bounded variation onGis a measure algebra only ifAis a measure algebra.

scholarly journals A theorem on abstract Segal algebras over some commutative Banach algebras

1982 ◽  
Vol 25 (2) ◽  
pp. 293-301 ◽  
Author(s):  
U.B. Tewari ◽  
K. Parthasarathy
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Compact Abelian Group ◽  
Maximal Ideal ◽  
Banach Algebras ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Ideal Space ◽  
Locally Compact ◽  
Commutative Banach Algebras

Let B be a commutative, semi-simple, regular, Tauberian Banach algebra with noncompact maximal ideal space Δ(B). Suppose B has the property that there is a constant C such that for every compact subset K of Δ(B) there exists a f ∈ B with = 1 on K, ‖f‖B ≤ C and has compact support. We prove that if A is a proper abstract Segal algebra over B then for every positive integer n there exists f ∈ B such that fn ∉ A but fn+1 ∈ A. As a consequence of this result we prove that if G is a nondiscrete locally compact abelian group, μ a positive unbounded Radon measure on Γ (the dual group of G), 1 ≤ p < q < ∞ and , then .

The Wiener–Pitt phenomenon on semi-groups

1963 ◽  
Vol 59 (1) ◽  
pp. 11-24 ◽  
Author(s):  
T. A. Davis
Keyword(s):  
Abelian Group ◽  
Banach Algebra ◽  
Haar Measure ◽  
Group Algebra ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Bounded Complex ◽  
Additive Measures ◽  
Complex Valued

Let G be a locally compact Abelian group, written adoptively, with Haar measure m, L1(G) the group algebra of G, and M(G) the Banach algebra of all bounded, complex-valued, regular, countably additive measures on G. For a general account of L1(G) and M(G) see Rudin (7).

scholarly journals On weak spectral synthesis

1991 ◽  
Vol 43 (2) ◽  
pp. 279-282 ◽  
Author(s):  
K. Parthasarathy ◽  
Sujatha Varma
Keyword(s):  
Abelian Group ◽  
Group Algebra ◽  
Lie Group ◽  
Compact Abelian Group ◽  
Spectral Synthesis ◽  
Locally Compact Abelian Group ◽  
Fourier Algebra ◽  
Compact Lie Group ◽  
Locally Compact

Weak spectral synthesis fails in the group algebra and the generalised group algebra of any non compact locally compact abelian group and also in the Fourier algebra of any infinite compact Lie group.

scholarly journals Spectral mapping theorem for representations of measure algebras

1997 ◽  
Vol 40 (2) ◽  
pp. 261-266 ◽  
Author(s):  
H. Seferoǧlu
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Continuous Homomorphism ◽  
Locally Compact Abelian Group ◽  
Measure Algebra ◽  
Spectral Mapping Theorem ◽  
Spectral Mapping ◽  
Stieltjes Transform ◽  
Locally Compact ◽  
Unital Banach Algebra

Let G be a locally compact abelian group, M0(G) be a closed regular subalgebra of the convolution measure algebra M(G) which contains the group algebra L1(G) and ω: M0(G) → B be a continuous homomorphism of M0(G) into the unital Banach algebra B (possibly noncommutative) such that ω(L1(G)) is without order with respect to B in the sense that if for all b ∈ B, b.ω(L1(G)) = {0} implies b = 0. We prove that if sp(ω) is a synthesis set for L1(G) then the equality holds for each μ ∈ M0(G), where sp(ω) denotes the Arveson spectrum of ω, σB(.) the usual spectrum in B, the Fourier-Stieltjes transform of μ.

scholarly journals Note on the semi-simplicity of measure algebras

2019 ◽  
Vol 192 (4) ◽  
pp. 935-938
Author(s):  
László Székelyhidi
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Discrete Group ◽  
Locally Compact Abelian Group ◽  
Measure Algebra ◽  
Locally Compact ◽  
Group Case

AbstractIn this paper we prove that the measure algebra of a locally compact abelian group is semi-simple. This result extends the corresponding result of S. A. Amitsur in the discrete group case using a completely different approach.

On Multipliers of the Group Lp;w(G)-Algebras

2013 ◽  
Vol 59 (2) ◽  
pp. 253-268
Author(s):  
Ilker Eryilmaz ◽  
Cenap Duyar
Keyword(s):  
Abelian Group ◽  
Banach Algebra ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact

Abstract Let G be a locally compact abelian group (non-compact, non-discrete) with Haar measure and 1 ≤ p < ∞: The purpose of this paper is to study the space of multipliers on Lp;w (G) and characterize it as the algebra of all multipliers of the closely related Banach algebra of tempered elements in Lp;w (G).

Large families of singular measures having absolutely continuous convolution squares

1968 ◽  
Vol 64 (4) ◽  
pp. 1015-1022 ◽  
Author(s):  
Karl Stromberg
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Measure Algebra ◽  
Regular Measure ◽  
Locally Compact ◽  
Absolutely Continuous ◽  
Singular Measures ◽  
Large Families ◽  
Nikodym Derivative

In 1966, Hewitt and Zuckerman(3,4) proved that if G is a non-discrete locally compact Abelian group with Haar measure λ, then there exists a non-negative, continuous regular measure μon G that is singular to λ(μ ┴ λ) such that μ(G)= 1, μ * μ is absolutely continuous with respect to λ(μ * μ ≪ λ), and the Lebesgue-Radon-Nikodym derivative of μ * μ with respect to λ is in (G, λ) for all real p > 1. They showed also that such a μ can be chosen so that the support of μ * μ contains any preassigned σ-compact subset of G. It is the purpose of the present paper to extend this result to obtain large independent sets of such measures. Among other things the present results show that, for such groups, the radical of the measure algebra modulo the -algebra has large dimension. This answers a question (6.4) left open in (3).

scholarly journals On continuity of derivations and epimorphisms on some vector-valued group algebras

1997 ◽  
Vol 56 (2) ◽  
pp. 209-215
Author(s):  
Ramesh V. Garimella
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Group Algebras ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Zero Divisors ◽  
Integrable Functions ◽  
Vector Valued ◽  
Nontrivial Zero

For a locally compact Abelian group G and a commutative Banach algebra B, let L1(G, B) be the Banach algebra of all Bochner integrable functions. We show that if G is compact and B is a nonunital Banach algebra without nontrivial zero divisors, then (i) all derivations on L1(G, B) are continuous if and only if all derivations on B are continuous, and (ii) each epimorphism from a Banach algebra X onto L1(G, B) is continuous provided every epimorphism from X onto B is continuous. If G is noncompact then every derivation on L1(G, B) and every epimorphism from a commutative Banach algebra onto L1(G, B) are continuous. Our results extend the results of Neumann and Velasco for nonunital Banach algebras.

scholarly journals Wiener Tauberian theorems for vector-valued functions

1994 ◽  
Vol 17 (3) ◽  
pp. 475-478 ◽  
Author(s):  
K. Parthasarathy ◽  
Sujatha Varma
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Fourier Transforms ◽  
Weak Form ◽  
Locally Compact Abelian Group ◽  
Tauberian Theorems ◽  
Locally Compact ◽  
Integrable Functions ◽  
Vector Valued Functions ◽  
Vector Valued

Different versions of Wiener's Tauberian theorem are discussed for the generalized group algebraL1(G,A)(of integrable functions on a locally compact abelian groupGtaking values in a commutative semisimple regular Banach algebraA) usingA-valued Fourier transforms. A weak form of Wiener's Tauberian property is introduced and it is proved thatL1(G,A)is weakly Tauberian if and only ifAis. The vector analogue of Wiener'sL2-span of translates theorem is examined.

Sign in / Sign up

Export Citation Format

Share Document