scholarly journals Order Structure of the Figà-Talamanca-Herz Algebra

ISRN Geometry ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-13
Author(s):  
Marzieh Shams Yousefi ◽  
Massoud Amini ◽  
Fereshteh Sady
Keyword(s):  
Dual Space ◽  
Space Structure ◽  
Locally Compact Group ◽  
Order Structure ◽  
Locally Compact ◽  
Completely Positive ◽  
Algebra Isomorphism ◽  
Positive Linear Map ◽  
Operator Space Structure ◽  
And Algebra

We study the interplay between the order structure and the -operator space structure of Figà-Talamanca-Herz algebra of a locally compact group . We show that for amenable groups, an order and algebra isomorphism of Figà-Talamanca-Herz-algebras yields an isomorphism or anti-isomorphism of the underlying groups. We also give a bound for the norm of a -completely positive linear map from Figà-Talamanca-Herz algebra to its dual space.

scholarly journals Interpolation in wavelet spaces and the HRT-conjecture

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Eirik Berge
Keyword(s):  
Compact Group ◽  
Wavelet Transforms ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Space Structure ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Positive Type ◽  
Time Frequency ◽  
Hrt Conjecture

AbstractWe investigate the wavelet spaces $$\mathcal {W}_{g}(\mathcal {H}_{\pi })\subset L^{2}(G)$$ W g ( H π ) ⊂ L 2 ( G ) arising from square integrable representations $$\pi :G \rightarrow \mathcal {U}(\mathcal {H}_{\pi })$$ π : G → U ( H π ) of a locally compact group G. We show that the wavelet spaces are rigid in the sense that non-trivial intersection between them imposes strong restrictions. Moreover, we use this to derive consequences for wavelet transforms related to convexity and functions of positive type. Motivated by the reproducing kernel Hilbert space structure of wavelet spaces we examine an interpolation problem. In the setting of time–frequency analysis, this problem turns out to be equivalent to the HRT-conjecture. Finally, we consider the problem of whether all the wavelet spaces $$\mathcal {W}_{g}(\mathcal {H}_{\pi })$$ W g ( H π ) of a locally compact group G collectively exhaust the ambient space $$L^{2}(G)$$ L 2 ( G ) . We show that the answer is affirmative for compact groups, while negative for the reduced Heisenberg group.

scholarly journals ON HOMOLOGICAL PROPERTIES FOR SOME MODULES OF UNIFORMLY CONTINUOUS FUNCTIONS OVER CONVOLUTION ALGEBRAS

2011 ◽  
Vol 84 (2) ◽  
pp. 177-185
Author(s):  
RASOUL NASR-ISFAHANI ◽  
SIMA SOLTANI RENANI
Keyword(s):  
Compact Group ◽  
Group Algebra ◽  
Dual Space ◽  
Continuous Functions ◽  
Locally Compact Group ◽  
Measure Algebra ◽  
Locally Compact ◽  
Convolution Algebras ◽  
Uniformly Continuous

AbstractFor a locally compact group G, let LUC(G) denote the space of all left uniformly continuous functions on G. Here, we investigate projectivity, injectivity and flatness of LUC(G) and its dual space LUC(G)* as Banach left modules over the group algebra as well as the measure algebra of G.

Weak Containment and Induced Representations of Groups

1962 ◽  
Vol 14 ◽  
pp. 237-268 ◽  
Author(s):  
J. M. G. Fell
Keyword(s):  
Dual Space ◽  
Locally Compact Group ◽  
Equivalence Classes ◽  
Unitary Representations ◽  
Unitary Equivalence ◽  
Locally Compact ◽  
Induced Representations ◽  
Weak Containment ◽  
Irreducible Unitary Representations ◽  
Special Case

Let G be a locally compact group and G† its dual space, that is, the set of all unitary equivalence classes of irreducible unitary representations of G. An important tool for investigating the group algebra of G is the so-called hull-kernel topology of G†, which is discussed in (3) as a special case of the relation of weak containment. The question arises: Given a group G, how do we determine G† and its topology? For many groups G, Mackey's theory of induced representations permits us to catalogue all the elements of G†. One suspects that by suitably supplementing this theory it should be possible to obtain the topology of G† at the same time. It is the purpose of this paper to explore this possibility. Unfortunately, we are not able to complete the programme at present.

Isomorphisms between the second duals of group algebras of locally compact groups

1996 ◽  
Vol 119 (4) ◽  
pp. 657-663 ◽  
Author(s):  
Hamid-Reza Farhadi
Keyword(s):  
Compact Group ◽  
Group Algebra ◽  
Locally Compact Group ◽  
Group Algebras ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Algebra Isomorphism ◽  
Continuous Automorphism

AbstractLet G be a locally compact group and L1(G) be the group algebra of G. We show that G is abelian or compact if every continuous automorphism of L1(G)** maps L1(G) onto L1(G) This characterizes all groups with this property and answers a question raised by F. Ghahramani and A. T. Lau in [7]. We also show that if G is a compact group and θ is any (algebra) isomorphism from L1(G)** onto L1(H)**, then H is compact and θ maps L1(G) onto L1(H).

Bipositive Isomorphisms Between Beurling Algebras and Between their Second Dual Algebras

2017 ◽  
Vol 69 (1) ◽  
pp. 3-20 ◽  
Author(s):  
F. Ghahramani ◽  
S. Zadeh
Keyword(s):  
Compact Group ◽  
Banach Algebras ◽  
Locally Compact Group ◽  
Order Structure ◽  
Algebra Structure ◽  
Locally Compact ◽  
Second Dual ◽  
Beurling Algebras ◽  
Continuous Weight

AbstractLet G be a locally compact group and let ω be a continuous weight on G. We show that for each of the Banach algebras L1(G,ω ), M(G,ω ), LUC(G,ω -1)*, and L1(G, ω)**, the order structure combined with the algebra structure determines the weighted group.

scholarly journals Order structure on certain classes of ideals in group algebras and amenability

2002 ◽  
Vol 66 (1) ◽  
pp. 69-73
Author(s):  
Yuji Takahashi
Keyword(s):  
Probability Measure ◽  
Maximal Element ◽  
Locally Compact Group ◽  
Group Algebras ◽  
Order Structure ◽  
Maximal Elements ◽  
Locally Compact ◽  
Discrete Probability ◽  
Cardinal Numbers ◽  
Set Inclusion

Let G be a separable, locally compact group and let d (G) be the set of all closed left ideals in L1(G) which have the form Jμ = {f − f ∗ μ: f ∈ L1(G)}− for some discrete probability measure μ. It is shown that if d (G) has a unique maximal element with respect to the order structure by set inclusion, then G is amenable. This answers a problem of G.A. Willis. We also examine cardinal numbers of the sets of maximal elements in d (G) for nonamenable groups.

scholarly journals ON ITERATED POWERS OF POSITIVE DEFINITE FUNCTIONS

2015 ◽  
Vol 92 (3) ◽  
pp. 440-443
Author(s):  
MEHRDAD KALANTAR
Keyword(s):  
Compact Group ◽  
Strong Operator Topology ◽  
Positive Definite Function ◽  
Positive Definite ◽  
Locally Compact Group ◽  
Definite Function ◽  
Completely Positive Maps ◽  
Locally Compact ◽  
Completely Positive ◽  
Positive Maps

We prove that if ${\it\rho}$ is an irreducible positive definite function in the Fourier–Stieltjes algebra $B(G)$ of a locally compact group $G$ with $\Vert {\it\rho}\Vert _{B(G)}=1$, then the iterated powers $({\it\rho}^{n})$ as a sequence of unital completely positive maps on the group $C^{\ast }$-algebra converge to zero in the strong operator topology.

scholarly journals The predual of the space of convolutors on a locally compact group

1998 ◽  
Vol 57 (3) ◽  
pp. 409-414 ◽  
Author(s):  
Michael Cowling
Keyword(s):  
Compact Group ◽  
Maximal Ideal ◽  
Lebesgue Space ◽  
Dual Space ◽  
Locally Compact Group ◽  
Ideal Space ◽  
Convolution Operators ◽  
Locally Compact ◽  
Algebra Of Functions ◽  
Definition Of

Let Cvp(G) be the space of convolution operators on the Lebesgue space LP(G), for an arbitrary locally compact group G. We describe Cvp(G) as a dual space, whose predual, is a Banach algebra of functions on G, under pointwise operations, with maximal ideal space G. This involves a variation of Herz's definition of AP(G); the benefit of this new definition is that all of Cvp(G) is obtained as the dual in the nonamenable setting. We also discuss further developments of this idea.

scholarly journals Covariant functions of characters of compact subgroups

2021 ◽  
Vol 12 (3) ◽  
Author(s):  
Arash Ghaani Farashahi
Keyword(s):  
Compact Group ◽  
Harmonic Analysis ◽  
Banach Spaces ◽  
Systematic Study ◽  
Dual Space ◽  
Quotient Space ◽  
Compact Subgroup ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Conjugate Exponent

AbstractThis paper presents a systematic study for abstract harmonic analysis on classical Banach spaces of covariant functions of characters of compact subgroups. Let G be a locally compact group and H be a compact subgroup of G. Suppose that $$\xi :H\rightarrow \mathbb {T}$$ ξ : H → T is a character, $$1\le p<\infty$$ 1 ≤ p < ∞ and $$L_\xi ^p(G,H)$$ L ξ p ( G , H ) is the set of all covariant functions of $$\xi$$ ξ in $$L^p(G)$$ L p ( G ) . It is shown that $$L^p_\xi (G,H)$$ L ξ p ( G , H ) is isometrically isomorphic to a quotient space of $$L^p(G)$$ L p ( G ) . It is also proven that $$L^q_\xi (G,H)$$ L ξ q ( G , H ) is isometrically isomorphic to the dual space $$L^p_\xi (G,H)^*$$ L ξ p ( G , H ) ∗ , where q is the conjugate exponent of p. The paper is concluded by some results for the case that G is compact.

scholarly journals Some groups with T1 primitive ideal spaces

1985 ◽  
Vol 38 (1) ◽  
pp. 55-64 ◽  
Author(s):  
A. L. Carey ◽  
W. Moran
Keyword(s):  
Normal Subgroup ◽  
Lie Groups ◽  
Lie Group ◽  
Locally Compact Group ◽  
Primitive Ideal ◽  
Ideal Space ◽  
Solvable Lie Groups ◽  
Locally Compact ◽  
Simply Connected ◽  
Connected Lie Group

AbstractLet G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

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