scholarly journals Pseudo-contractibility Of Weighted Lp -Algebras

2013 ◽  
Vol 21 (3) ◽  
pp. 5-16
Author(s):  
Fatemeh Abtahi
Keyword(s):  
Weight Function ◽  
Compact Group ◽  
Sufficient Conditions ◽  
Locally Compact Group ◽  
Necessary And Sufficient Conditions ◽  
Locally Compact ◽  
Necessary And Sufficient ◽  
Group 1

Abstract 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 L1pω(G). Here, we establish necessary and sufficient conditions for L1pω(G) to be φ-contractible, pseudo-contractible or contractible. Moreover we give some similar results about LP(G, ω).

Convolution on L p-spaces of a locally compact group

2013 ◽  
Vol 63 (2) ◽  
Author(s):  
Fatemeh Abtahi ◽  
Rasoul Nasr-Isfahani ◽  
Ali Rejali
Keyword(s):  
Compact Group ◽  
Sufficient Conditions ◽  
Function Spaces ◽  
Locally Compact Group ◽  
Necessary And Sufficient Conditions ◽  
Locally Compact ◽  
Necessary And Sufficient

AbstractWe 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 ∈ L p(G). Here, we study the existence of f * g for all f, g ∈ L p(G) in the case where 0 < p ≤ 2. Also, for 0 < p < ∞, we offer some necessary and sufficient conditions for L p(G) * L p(G) to be contained in certain function spaces on 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

On invariant convex subsets in algebras defined on a locally compact group G

2012 ◽  
Vol 49 (3) ◽  
pp. 301-314
Author(s):  
Ali Ghaffari
Keyword(s):  
Compact Group ◽  
Sufficient Conditions ◽  
Locally Compact Group ◽  
Left Translation ◽  
Measure Algebra ◽  
Locally Compact ◽  
Translation Invariant ◽  
Second Dual ◽  
Necessary And Sufficient ◽  
Convex Subsets

Suppose that A is either the Banach algebra L1(G) of a locally compact group G, or measure algebra M(G), or other algebras (usually larger than L1(G) and M(G)) such as the second dual, L1(G)**, of L1(G) with an Arens product, or LUC(G)* with an Arenstype product. The left translation invariant closed convex subsets of A are studied. Finally, we obtain necessary and sufficient conditions for LUC(G)* to have 1-dimensional left ideals.

scholarly journals GROUPOID -ALGEBRAS WITH HAUSDORFF SPECTRUM

2013 ◽  
Vol 88 (2) ◽  
pp. 232-242 ◽  
Author(s):  
GEOFF GOEHLE
Keyword(s):  
Haar System ◽  
Sufficient Conditions ◽  
Orbit Space ◽  
Necessary And Sufficient Conditions ◽  
Locally Compact ◽  
Fell Topology ◽  
Necessary And Sufficient ◽  
Convergent Sequences

AbstractSuppose that $G$ is a second countable, locally compact Hausdorff groupoid with abelian stabiliser subgroups and a Haar system. We provide necessary and sufficient conditions for the groupoid ${C}^{\ast } $-algebra to have Hausdorff spectrum. In particular, we show that the spectrum of ${C}^{\ast } (G)$ is Hausdorff if and only if the stabilisers vary continuously with respect to the Fell topology, the orbit space ${G}^{(0)} / G$ is Hausdorff, and, given convergent sequences ${\chi }_{i} \rightarrow \chi $ and ${\gamma }_{i} \cdot {\chi }_{i} \rightarrow \omega $ in the dual stabiliser groupoid $\widehat{S}$ where the ${\gamma }_{i} \in G$ act via conjugation, if $\chi $ and $\omega $ are elements of the same fibre then $\chi = \omega $.

scholarly journals Multinorms and Approximate Amenability of Weighted Group Algebras

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Saman Ghaderkhani
Keyword(s):  
Weight Function ◽  
Compact Group ◽  
Banach Algebra ◽  
Locally Compact Group ◽  
Group Algebras ◽  
Locally Compact ◽  
Approximate Amenability

Let G be a locally compact group, and take p,q with 1≤p,q<∞. We prove that, for any left (p,q)-multiinvariant functional on L∞(G) and for any weight function ω≥1 on G, the approximate amenability of the Banach algebra L1(G,ω) implies the left (p,q)-amenability of G, but in general the opposite is not true. Our proof uses the notion of multinorms. We also investigate the approximate amenability of M(G,ω).

scholarly journals On the stable rank and real rank of group $C^*$-algebras of nilpotent locally compact groups

2005 ◽  
Vol 97 (1) ◽  
pp. 89
Author(s):  
Robert J. Archbold ◽  
Eberhard Kaniuth
Keyword(s):  
Abelian Group ◽  
Compact Group ◽  
Locally Compact Group ◽  
Stable Rank ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Real Rank ◽  
Locally Compact ◽  
C Algebras ◽  
Necessary And Sufficient

It is shown that if $G$ is an almost connected nilpotent group then the stable rank of $C^*(G)$ is equal to the rank of the abelian group $G/[G,G]$. For a general nilpotent locally compact group $G$, it is shown that finiteness of the rank of $G/[G,G]$ is necessary and sufficient for the finiteness of the stable rank of $C^*(G)$ and also for the finiteness of the real rank of $C^*(G)$.

On the equivalence of boundedness for multiple Hardy-Littlewood averages and related operators

2018 ◽  
Vol 123 (2) ◽  
pp. 273-296
Author(s):  
Dah-Chin Luor
Keyword(s):  
Weight Function ◽  
Geometric Mean ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Maximal Operators ◽  
Averaging Operators ◽  
One Dimensional ◽  
Almost Everywhere ◽  
Carry Over ◽  
Necessary And Sufficient

Necessary and sufficient conditions for the weight function $u$ are obtained, which provide the boundedness for a class of averaging operators from $L_p^+$ to $L_{q,u}^+$. These operators include the multiple Hardy-Littlewood averages and related maximal operators, geometric mean operators, and geometric maximal operators. We show that, under suitable conditions, the boundedness of these operators are equivalent. Our theorems extend several one-dimensional results to multi-dimensional cases and to operators with multiple kernels. We also show that in the case $p<q$, some one-dimensional results do not carry over to the multi-dimensional cases, and the boundedness of $T$ from $L_p^+$ to $L_{q,u}^+$ holds only if $u=0$ almost everywhere.

scholarly journals INVOLUTIONS ON THE SECOND DUALS OF GROUP ALGEBRAS AND A MULTIPLIER PROBLEM

2007 ◽  
Vol 50 (1) ◽  
pp. 153-161 ◽  
Author(s):  
H. Farhadi ◽  
F. Ghahramani
Keyword(s):  
Sufficient Conditions ◽  
Continuous Functions ◽  
Locally Compact Group ◽  
Group Algebras ◽  
Locally Compact ◽  
Dual Algebra ◽  
Amenable Subgroup ◽  
Second Dual ◽  
Necessary And Sufficient ◽  
Bounded Continuous Functions

AbstractWe show that if a locally compact group $G$ is non-discrete or has an infinite amenable subgroup, then the second dual algebra $L^1(G)^{**}$ does not admit an involution extending the natural involution of $L^1(G)$. Thus, for the above classes of groups we answer in the negative a question raised by Duncan and Hosseiniun in 1979. We also find necessary and sufficient conditions for the dual of certain left-introverted subspaces of the space $C_b(G)$ (of bounded continuous functions on $G$) to admit involutions. We show that the involution problem is related to a multiplier problem. Finally, we show that certain non-trivial quotients of $L^1(G)^{**}$ admit involutions.

Solution of Some Weight Problems for the Riemann–Liouville and Weyl Operators

1998 ◽  
Vol 5 (6) ◽  
pp. 565-574
Author(s):  
A. Meskhi
Keyword(s):  
Weight Function ◽  
Liouville Operator ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weyl Operator ◽  
Weight Problems ◽  
Necessary And Sufficient

Abstract The necessary and sufficient conditions are found for the weight function 𝑣, which provide the boundedness and compactness of the Riemann–Liouville operator 𝑅 α from 𝐿𝑝 to . The criteria are also established for the weight function 𝑤, which guarantee the boundedness and compactness of the Weyl operator 𝑊 α from to 𝐿𝑞.

scholarly journals Finite dimensional H-invariant spaces

1997 ◽  
Vol 56 (3) ◽  
pp. 353-361
Author(s):  
K.E. Hare ◽  
J.A. Ward
Keyword(s):  
Compact Group ◽  
Closed Subspace ◽  
Sufficient Conditions ◽  
Trigonometric Polynomials ◽  
Left Translation ◽  
Almost Periodic ◽  
Locally Compact ◽  
Finite Dimensional ◽  
Necessary And Sufficient ◽  
Invariant Spaces

A subset V of M(G) is left H-invariant if it is invariant under left translation by the elements of H, a subset of a locally compact group G. We establish necessary and sufficient conditions on H which ensure that finite dimensional subspaces of M(G) when G is compact, or of L∞(G) when G is locally compact Abelian, which are invariant in this weaker sense, contain only trigonometric polynomials. This generalises known results for finite dimensional G-invariant subspaces. We show that if H is a subgroup of finite index in a compact group G, and the span of the H-translates of μ is a weak*-closed subspace of L∞(G) or M(G) (or is closed in Lp(G)for 1 ≤ p < ∞), then μ is a trigonometric polynomial.We also obtain some results concerning functions that possess the analogous weaker almost periodic condition relative to H.

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