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.

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.

Actions of a Locally Compact Group with Zero

1971 ◽  
Vol 23 (3) ◽  
pp. 413-420 ◽  
Author(s):  
T. H. McH. Hanson
Keyword(s):  
Topological Group ◽  
Compact Group ◽  
Topological Semigroup ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Definition Of ◽  
Isolated Point ◽  
Locally Compact Topological Group

In [2] we find the definition of a locally compact group with zero as a locally compact Hausdorff topological semigroup, S, which contains a non-isolated point, 0, such that G = S – {0} is a group. Hofmann shows in [2] that 0 is indeed a zero for S, G is a locally compact topological group, and the unit, 1, of G is the unit of S. We are to study actions of S and G on spaces, and the reader is referred to [4] for the terminology of actions.If X is a space (all are assumed Hausdorff) and A ⊂ X, A* denotes the closure of A. If {xρ} is a net in X, we say limρxρ = ∞ in X if {xρ} has no subnet which converges in X.

Ideals with bounded approximate units in L1-algebras of locally compact groups

2000 ◽  
Vol 128 (1) ◽  
pp. 65-77
Author(s):  
EBERHARD KANIUTH
Keyword(s):  
Compact Group ◽  
Locally Compact Group ◽  
Primitive Ideal ◽  
Locally Compact Groups ◽  
Ideal Space ◽  
Compact Groups ◽  
Locally Compact ◽  
Approximate Unit

We show that for an arbitrary locally compact group G and for E in a certain class of closed subsets of the primitive ideal space of L1(G), the kernel k(E) has a bounded approximate unit. This generalizes some well-known previous results.

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.

scholarly journals THE GLIMM IDEAL SPACE OF A TWO-STEP NILPOTENT LOCALLY COMPACT GROUP

2001 ◽  
Vol 44 (3) ◽  
pp. 505-526 ◽  
Author(s):  
Eberhard Kaniuth ◽  
William Moran
Keyword(s):  
Compact Group ◽  
Sufficient Conditions ◽  
Locally Compact Group ◽  
Primitive Ideal ◽  
Ideal Space ◽  
Locally Compact ◽  
C Algebra ◽  
Primary 22D25 ◽  
Secondary 22D10 ◽  
Necessary And Sufficient

AbstractFor a two-step nilpotent locally compact group $G$, we determine the Glimm ideal space of the group $C^*$-algebra $C^*(G)$ and its topology. This leads to necessary and sufficient conditions for $C^*(G)$ to be quasi-standard. Moreover, some results about the Glimm classes of points in the primitive ideal space $\mathrm{Prim}(C^*(G))$ are obtained.AMS 2000 Mathematics subject classification: Primary 22D25. Secondary 22D10

Feichtinger's Segal algebra on homogeneous spaces

2015 ◽  
Vol 26 (08) ◽  
pp. 1550054 ◽  
Author(s):  
K. Parthasarathy ◽  
N. Shravan Kumar
Keyword(s):  
Compact Group ◽  
Homogeneous Spaces ◽  
Compact Subgroup ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Segal Algebra ◽  
Definition Of

Let K be a compact subgroup of a locally compact group G. We extend to the context of homogeneous spaces, G/K, the definition of Feichtinger's Segal algebra. The functorial properties of the Segal algebra are proved.

Bipositive and Isometric Isomorphisms of Some Convolution Algebras

1965 ◽  
Vol 17 ◽  
pp. 839-846 ◽  
Author(s):  
R. E. Edwards
Keyword(s):  
Compact Group ◽  
Lebesgue Space ◽  
Inductive Limit ◽  
Continuous Functions ◽  
Locally Compact Group ◽  
Natural Topology ◽  
Locally Compact ◽  
Convolution Algebras ◽  
Image Position ◽  
Complex Valued

Throughout this paper the term "space" will mean "Hausdorff locally compact space" and the term '"group" will mean "Hausdorff locally compact group." If G is a group and 1 ≤ p < ∞, Lp(G) denotes the usual Lebesgue space formed relative to left Haar measure on G. It is well known that L1(G) is an algebra under convolution, and that the same is true of Lp(G) whenever G is compact. We introduce also the space Cc(G) of complex-valued continuous functions f on G for each of which the support (supp f), is compact. The "natural" topology of CC(G) is obtained by regarding CC(G) as the inductive limit of its subspaces

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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