HYPERGROUP ALGEBRAS AS TOPOLOGICAL ALGEBRAS

2014 ◽  
Vol 90 (3) ◽  
pp. 486-493
Author(s):  
S. MAGHSOUDI ◽  
J. B. SEOANE-SEPÚLVEDA
Keyword(s):  
Haar Measure ◽  
Lebesgue Space ◽  
Topological Algebra ◽  
Topological Algebras ◽  
Locally Compact ◽  
Continuous Functionals ◽  
Locally Convex Topology ◽  
Compact Hypergroup ◽  
Locally Convex ◽  
Hypergroup Algebras

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}K$ be a locally compact hypergroup endowed with a left Haar measure and let $L^1(K)$ be the usual Lebesgue space of $K$ with respect to the left Haar measure. We investigate some properties of $L^1(K)$ under a locally convex topology $\beta ^1$. Among other things, the semireflexivity of $(L^1(K), \beta ^1)$ and of sequentially$\beta ^1$-continuous functionals is studied. We also show that $(L^1(K), \beta ^1)$ with the convolution multiplication is always a complete semitopological algebra, whereas it is a topological algebra if and only if $K$ is compact.

Local analytic structure in certain commutative topological algebras

1972 ◽  
Vol 6 (2) ◽  
pp. 161-167 ◽  
Author(s):  
R.J. Loy
Keyword(s):  
Power Series ◽  
Topological Algebra ◽  
Fréchet Space ◽  
Analytic Structure ◽  
Frechet Space ◽  
Topological Algebras ◽  
Space Topology ◽  
Formal Power ◽  
Locally Convex Topology ◽  
Locally Convex

Let B be a topological algebra with Fréchet space topology, A an algebra with locally convex topology and an algebra of formal power series over A in n commuting indeterminates which carries a Fréchet space topology. In a previous paper the author showed, for the case n = 1, that a homomorphism of B into whose range contains polynomials is necessarily continuous provided the coordinate projections of into A satisfy a certain equicontinuity condition. This result is here extended to the case of general n, and also to weaker topological assumptions.

scholarly journals On the Generalized Weighted Lebesgue Spaces of Locally Compact Groups

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
I. Akbarbaglu ◽  
S. Maghsoudi
Keyword(s):  
Convex Space ◽  
Topological Algebra ◽  
Locally Convex Space ◽  
Locally Compact Group ◽  
Compact Groups ◽  
Topological Properties ◽  
Locally Compact ◽  
The Family ◽  
Locally Convex Topology ◽  
Locally Convex

Let be a locally compact group with a fixed left Haar measure and be a system of weights on . In this paper, we deal with locally convex space equipped with the locally convex topology generated by the family of norms . We study various algebraic and topological properties of the locally convex space . In particular, we characterize its dual space and show that it is a semireflexive space. Finally, we give some conditions under which with the convolution multiplication is a topological algebra and then characterize its closed ideals and its spectrum.

A CONVOLUTION-INDUCED TOPOLOGY ON THE ORLICZ SPACE OF A LOCALLY COMPACT GROUP

2015 ◽  
Vol 99 (1) ◽  
pp. 1-11
Author(s):  
IBRAHIM AKBARBAGLU ◽  
SAEID MAGHSOUDI
Keyword(s):  
Compact Group ◽  
Orlicz Space ◽  
Haar Measure ◽  
Locally Compact Group ◽  
Young Function ◽  
Strict Topology ◽  
Locally Compact ◽  
Induced Topology ◽  
Locally Convex Topology ◽  
Locally Convex

Let $G$ be a locally compact group with a fixed left Haar measure. In this paper, given a strictly positive Young function ${\rm\Phi}$, we consider $L^{{\rm\Phi}}(G)$ as a Banach left $L^{1}(G)$-module. Then we equip $L^{{\rm\Phi}}(G)$ with the strict topology induced by $L^{1}(G)$ in the sense of Sentilles and Taylor. Some properties of this locally convex topology and a comparison with weak$^{\ast }$, bounded weak$^{\ast }$ and norm topologies are presented.

scholarly journals STRICT TOPOLOGY AS A MIXED TOPOLOGY ON LEBESGUE SPACES

2011 ◽  
Vol 84 (3) ◽  
pp. 504-515 ◽  
Author(s):  
SAEID MAGHSOUDI ◽  
RASOUL NASR-ISFAHANI
Keyword(s):  
Topological Group ◽  
Compact Space ◽  
Group Algebra ◽  
Lebesgue Space ◽  
Strict Topology ◽  
Lebesgue Spaces ◽  
Locally Compact ◽  
Locally Convex Topology ◽  
Locally Convex ◽  
Strong Dual

AbstractLetXbe a locally compact space, and 𝔏∞0(X,ι) be the space of all essentially boundedι-measurable functionsfonXvanishing at infinity. We introduce and study a locally convex topologyβ1(X,ι) on the Lebesgue space 𝔏1(X,ι) such that the strong dual of (𝔏1(X,ι),β1(X,ι)) can be identified with$({\frak L}_0^\infty (X,\iota ),\|\cdot \|_\infty )$. Next, by showing thatβ1(X,ι) can be considered as a natural mixed topology, we deduce some of its basic properties. Finally, as an application, we prove thatL1(G) , the group algebra of a locally compact Hausdorff topological groupG, equipped with the convolution multiplication is a complete semitopological algebra under theβ1(G) topology.

Translation-Invariant Operators On Lp(G), 0 < p < 1 (II)

1977 ◽  
Vol 29 (3) ◽  
pp. 626-630 ◽  
Author(s):  
Daniel M. Oberlin
Keyword(s):  
Compact Group ◽  
Haar Measure ◽  
Lebesgue Space ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Translation Invariant ◽  
Translation Operators ◽  
Left And Right

For a locally compact group G, let LP(G) be the usual Lebesgue space with respect to left Haar measure m on G. For x ϵ G define the left and right translation operators Lx and Rx by Lx f(y) = f(xy), Rx f(y) = f(yx)(f ϵ Lp(G),y ϵ G). The purpose of this paper is to prove the following theorem.

Infrasequential Topological Algebras

1979 ◽  
Vol 22 (4) ◽  
pp. 413-418 ◽  
Author(s):  
T. Husain
Keyword(s):  
Linear Functional ◽  
Topological Algebra ◽  
Topological Algebras ◽  
Locally Convex ◽  
Locally Convex Algebra

The notion of sequential topological algebra was introduced by this author and Ng [3], Among a number of results concerning these algebras, we showed that each multiplicative linear functional on a sequentially complete, sequential, locally convex algebra is bounded ([3], Theorem 1). From this it follows that every multiplicative linear functional on a sequential F-algebra (complete metrizable) is continuous ([3], Corollary 2).

On multipliers of Fourier transforms

1972 ◽  
Vol 71 (1) ◽  
pp. 63-66 ◽  
Author(s):  
Louis Pigno
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Lebesgue Space ◽  
Fourier Transforms ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Continuous Complex ◽  
Complex Valued

In this paper G is a locally compact Abelian group, φ a complex-valued function defined on the dual Γ, Lp(G) (1 ≤ p ≤ ∞) the usual Lebesgue space of index p formed with respect to Haar measure, C(G) the set of all bounded continuous complex-valued functions on G, and C0(G) the set of all f ∈ C(G) which vanish at infinity.

scholarly journals Multiplicative functionals and a class of topological algebras

1977 ◽  
Vol 17 (3) ◽  
pp. 391-399 ◽  
Author(s):  
Gerard A. Joseph
Keyword(s):  
Linear Functional ◽  
Topological Algebra ◽  
Null Sequence ◽  
Topological Algebras ◽  
Multiplicative Functionals ◽  
Locally Convex ◽  
Locally Convex Algebra

Every multiplicative linear functional on a pseudocomplete locally convex algebra satisfying the “sequential” property of Husain and Ng is bounded (a topological algebra is called “sequential” if every null sequence contains an element whose powers converge to zero). Characterizations of such algebras are given, with some examples.

Isomorphisms of Multiplier Algebras

1968 ◽  
Vol 20 ◽  
pp. 1165-1172 ◽  
Author(s):  
G. I. Gaudry
Keyword(s):  
Haar Measure ◽  
Lebesgue Space ◽  
Dirac Measure ◽  
Locally Compact ◽  
Radon Measures ◽  
Continuous Complex ◽  
Complex Valued ◽  
Multiplier Algebras ◽  
Identity Elements

Suppose that G1 and G2 are two locally compact Hausdorff groups with identity elements e and e’ and with respective left Haar measures dx and dy. Let 1 ≦ p ≦ ∞, and Lp(Gi) be the usual Lebesgue space over Gi formed relative to left Haar measure on Gi. We denote by M(Gi) the space of Radon measures, and by Mbd(Gi) the space of bounded Radon measures on Gi. If a ϵ Gi we write ϵa for the Dirac measure at the point a. Cc(Gi) will denote the space of continuous, complex-valued functions on Gi with compact supports, whilst Cc+ (Gi) will denote that subset of Cc(Gi) consisting of those functions which are real-valued and non-negative.

Spans of translates in Lp(G)

1965 ◽  
Vol 5 (2) ◽  
pp. 216-233 ◽  
Author(s):  
R. E. Edwards
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Lebesgue Space ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Linear Combinations ◽  
Character Group ◽  
Weak Closure

Throughout this paper, G denotes a Hausdorff locally compact Abelian group, X its character group, and Lp(G) (1 ≦ p ≦ ∞) the usual Lebesgue space formed relative to the Haar measure on G. If f ∈ Lp(G), we denote by Tp[f] the closure (or weak closure, if p = ∞) in Lp(G) of the set linear combinations of translates of f.

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