Operators on a non locally compact group algebra

The Fourier algebra for locally compact groupoids

We introduce and investigate using Hilbert modules the properties of the Fourier algebra A(G) for a locally compact groupoid G. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compact groups proved by P. Eymard.

Induced representations of locally C*-algebras

In this paper, by analogy with the case of C*-algebras, we define the notion of induced representation of a locally C*-algebra, and then we prove a imprimitivity theorem for induced representations of locally C*-algebras.

A note on amenability of locally compact quantum groups

In this short note we introduce a notion called "quantum injectivity" of locally compact quantum groups, and prove that it is equivalent to amenability of the dual. Particularly, this provides a new characterization of amenability of locally compact groups.

Young's inequality for locally compact quantum groups

In this paper, we generalize Young's inequality for locally compact quantum groups and obtain some results for extremal pairs of Young's inequality and extremal functions of Hausdorff-Young inequality.

Lacunary sections for locally compact groupoids

It is proved that every second countable locally Hausdorff and locally compact continuous groupoid has a Borel set of units that meets every orbit and is what is called "lacunary," a property that implies that the intersection with every orbit is countable.

Ideal and weak Amenability of Fréchet locally $C^*$-algebra

The notion of Fréchet locally $C^*$-algebra generalizes the notion of $C^*$-algebra. In this paper, we first present some definitions and basic facts about locally $C^*$-algebra, and then we introduce and study the notion of ideal and weak amenability for these algebras. Also, we show that every Fréchet locally $C^*$-algebra is ideally amenable.

The Fourier algebra for locally compact groupoids

We introduce and investigate using Hilbert modules the properties of the {\em Fourier algebra} $A(G)$ for a locally compact groupoid $G$. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compact groups proved by P. Eymard.

A note on the uniqueness of involution in locally C*-algebras

In the present note we show that the involution in locally C*-algebras is uniquely determined.

Amenability and covariant injectivity of locally compact quantum groups

As is well known, the equivalence between amenability of a locally compact group $G$ and injectivity of its von Neumann algebra $\mathcal{L}(G)$ does not hold in general beyond inner amenable groups. In this paper, we show that the equivalence persists for all locally compact groups if $\mathcal{L}(G)$ is considered as a $\mathcal{T}(L_2(G))$-module with respect to a natural action. In fact, we prove an appropriate version of this result for every locally compact quantum group.

Structure theory of homologically trivial and annihilator locally C*-algebras

We study the structure of certain classes of homologically trivial locally C*-algebras. These include algebras with projective irreducible Hermitian A-modules, biprojective algebras, and superbiprojective algebras. We prove that, if A is a locally C*-algebra, then all irreducible Hermitian A-modules are projective if and only if A is a direct topological sum of elementary C*-algebras. This is also equivalent to A being an annihilator (dual, complemented, left quasi-complemented, or topologically modular annihilator) topological algebra. We characterize all annihilator $σ$-C*-algebras and descr…

Morphisms of locally compact groupoids endowed with Haar systems

The purpose of this paper is extend the notion of morphism of groupoids introduced by Zakrzewski to locally compact $σ$-compact groupoids endowed with Haar systems and to use the extension to construct a covariant functor from this category to a category of $C^{\ast}$-algebras in the sense of Woronowicz.

On Morita equivalence of group actions on locally C*-algebras

We extend to the context of locally C*-algebras a result of F. Combes [Proc. London Math. Soc. 49(1984), 289-306].

Quantum Locally Compact Metric Spaces

We introduce the notion of a quantum locally compact metric space, which is the noncommutative analogue of a locally compact metric space, and generalize to the nonunital setting the notion of quantum metric spaces introduced by Rieffel. We then provide several examples of such structures, including the Moyal plane, as well as compact quantum metric spaces and locally compact metric spaces. This paper provides an answer to the question raised in the literature about the proper notion of a quantum metric space in the nonunital setup and offers important insights into noncommutative geometry for…

Abstract Structure of Measure Algebras on Coset Spaces of Compact Subgroups in Locally Compact Groups

This paper presents a systematic operator theory approach for abstract structure of Banach measure algebras over coset spaces of compact subgroups. Let $H$ be a compact subgroup of a locally compact group $G$ and $G/H$ be the left coset space associated to the subgroup $H$ in $G$. Also, let $M(G/H)$ be the Banach measure space consists of all complex measures over $G/H$. We then introduce an operator theoretic characterization for the abstract notion of involution over the Banach measure space $M(G/H)$.

Scattered locally C*-algebras

In this paper, we introduce the notion of a scattered locally $C^{\ast }$ -algebra and we give conditions for a locally $C^{\ast }$-algebra to be scattered. Given an action $α$ of a locally compact group $G$ on a scattered locally $C^{\ast }$-algebra $A[τ_{Γ}]$, it is natural to ask under what conditions the crossed product $A[τ_{Γ}]\times_{α}G$ is also scattered. We obtain some results concerning this question.

Covariant completely positive linear maps between locally C*-algebras

We prove a covariant version of the KSGNS (Kasparov, Stinespring, Gel'fand,Naimark,Segal) construction for completely positive linear maps between locally $C^{*}$-algebras. As an application of this construction, we show that a covariant completely positive linear map $ρ$ from a locally $C^{*}$-algebra $A$ to another locally $C^{*}$ -algebra $B$ with respect to a locally $C^{*}$-dynamical system $(G,A,α)$ extends to a completely positive linear map on the crossed product $A\times_{α^{}}G$.

Cyclicity of the left regular representation of a locally compact group

We combine harmonic analysis and operator algebraic techniques to give a concise argument that the left regular representation of a locally compact group is cyclic if and only if the group is first countable, a result first proved by Greenleaf and Moskowitz.

Idempotent states on locally compact groups and quantum groups

This is a short survey on idempotent states on locally compact groups and locally compact quantum groups. The central topic is the relationship between idempotent states, subgroups and invariant C*-subalgebras. We concentrate on recent results on locally compact quantum groups, but begin with the classical notion of idempotent probability measure. We also consider the `intermediate' case of idempotent states in the Fourier--Stieltjes algebra: this is the dual case of idempotent probability measures and so an instance of idempotent states on a locally compact quantum group.

A note on the $luc-$compactification of locally compact groups

Let $G$ be a locally compact non-compact topological group. We show that $G^{luc}$ ($G^*$) is an F-space if and only if $G$ is a discrete group.

A Positive Quantization on Type I Locally Compact Groups

Let $\G$ be a unimodular type I second countable locally compact group and $\wG$ its unitary dual. Motivated by a recent pseudo-differential calculus, we develop a positive Berezin-type quantization with operator-valued symbols defined on $\wG\times\G$.

Meta-centralizers of non locally compact group algebras

Meta-centralizers of non-locally compact group algebras are studied. Theorems about their representations with the help of families of generalized measures are proved. Isomorphisms of group algebras are investigated in relation with meta-centralizers.

Compact quantum subgroups and left invariant C*-subalgebras of locally compact quantum groups

We show that there is a one-to-one correspondence between compact quantum subgroups of a co-amenable locally compact quantum group $\mathbb{G}$ and certain left invariant C*-subalgebras of $C_0(\mathbb{G})$. We also prove that every compact quantum subgroup of a co-amenable quantum group is co-amenable. Moreover, there is a one-to-one correspondence between open subgroups of an amenable locally compact group $G$ and non-zero, invariant C*-subalgebras of the group C*-algebra $C^*(G)$.

Stepanov and Weyl almost periodic functions on locally compact Abelian groups

We study Stepanov and Weyl almost periodic functions on locally compact Abelian groups, which naturally appear in the theory of aperiodic order.

Weighted $L^p-$spaces on nilpotent, locally compact groups

Our paper begins with a revision of spectral theory for commutative Banach algebras, which enables us to prove the $L^p_ω-$conjecture for locally compact abelian groups. We follow an alternative approach to the one known in the literature. In particular, we do not resort to any structural theorems for locally compact groups. Subsequently, we discuss nilpotent, locally compact groups. The climax of the paper is the proof of the $L^p_ω-$conjecture for these groups.

Grand Lebesgue Spaces are really Banach algebras relative to the convolution on unimodular locally compact groups

We prove that the Grand Lebesgue Space, builded on a unimodular locally compact topological group, forms a Banach algebra relative to the convolution.

Left invariant measures on locally compact fan loops

In this article left invariant measures and functionals on locally compact nonassociative fan loops are investigated. For this purpose necessary properties of topological fan loops, estimates and approximations of functions on them are studied. An existence of nontrivial left invariant measures on locally compact fan loops is proved. Abundant families of fan loops are provided with the help of different types of their products.

Fundamental Theorems of Fourier-Stieltjes Transform Defined by Induced Representation on Locally Compact Group

This work addresses an extension of Fourier-Stieltjes transform of a vector measure defined on compact groups to locally compact groups, by using a group representation induced by a representation of one of its compact subgroups.

$L^T_p-$ functions on locally compact groups

Let $G$ be a locally compact group and $1\leq p<\infty$. Based on some important earlier works, in this paper the concept of $L_p^T-$function is introduced. Then the structure of the space $L^{T}_p(G)$, which is consisting of all $L_p^T-$functions, is investigated. As an important result, $L^{T}_p(G)^+$ is completely characterized, for the class of amenable locally compact groups. Furthermore, it is verified some applications of Plancherel Theorem on $L^{T}_2(G)$, in the class of abelian locally compact groups.

The Krengel's theorem for compact operators between locally solid vector lattices

Suppose $X$ is a locally solid vector lattice. It is known that there are several non-equivalent notions for compact operators on $X$. Furthermore, notion of the $AM$-property in $X$ as an extension for the $AM$-spaces in Banach lattices has been considered, recently. In this paper, we establish a variant of the known Krengel's theorem for different types of compact operators between locally solid vector lattices.

紧致局部多面体流形的微分同胚群

Diffeomorphism groups of compact locally polyhedral manifolds

研究紧致光滑局部多面体流形的光滑微分同胚群，将其构造为正规李群。

On an isomorphism theorem for the Feichtinger's Segal algebra on locally compact groups

In this article we observe that a locally compact group $G$ is completely determined by the algebraic properties of its Feichtinger's Segal algebra $S_0(G).$ Let $G$ and $H$ be locally compact groups. Then any linear (not necessarily continuous) bijection of $S_0(G)$ onto $S_0(H)$ which preserves the convolution and pointwise products is essentially a composition with a homeomorphic isomorphism of $H$ onto $G.$

Interpolation sets and the size of quotients of function spaces on a locally compact group

We devise a fairly general method for estimating the size of quotients between algebras of functions on a locally compact group. This method is based on the concept of interpolation sets and unifies the approaches followed by many authors to obtain particular cases. Among the applications we find, we obtain that the quotients WAP(G)/B(G) (G being a locally compact group in the class [IN] or a nilpotent locally compact group) and CB(G)/LUC(G) (G being any non-compact non-discrete locally compact group) contain a linearly isometric copy of \ell_\infty(κ(G)) where κ(G) is the compact covering num…

Orlicz Modules over Coset Spaces of Compact Subgroups in Locally compact Groups

Let $H$ be a compact subgroup of a locally compact group $G$ and let $m$ be the normalized $G$-invariant measure on homogeneous space $G/H$ associated with Weil's formula. Let $φ$ be a Young function satisfying $Δ_2$-condition. We introduce the notion of left module action of $L^1(G/H, m)$ on the Orlicz spaces $L^φ(G/H, m).$ We also introduce a Banach left $L^1(G/H, m)$-submodule of $L^φ(G/H, m).$

Fourier-Stieltjes transform defined by induced representation on locally compact groups

In this work we extend the Fourier-Stieltjes transform of a vector measure and a continuous function defined on compact groups to locally compact groups. To do so, we consider a representation L of a normal compact subgroup K of a locally compact group G, and we use a representation of G induced by that of L. Then, we define the Fourier-Stieltjes transform of a vector measure and that of a continuous function with compact support defined on G from the representation of G. Then, we extend the Shur orthogonality relation established for compact groups to locally compact groups by using the repre…

The weak type $(1,p)$ for convolution operators on locally compact groups

In this paper we provide necessary and sufficient conditions for the $\textnormal{weak}(1,p)$ boundedness, $1< p<\infty,$ of convolution operators on locally compact (Hausdorff) topological groups. So, we generalize a classical result due to Sobolev-Hardy-Littlewood and Stepanov. Applications to Fourier multipliers on Lie groups also are given.

Modulation preserving operators on locally compact abelian groups

Let $G$ be a locally compact abelian group and $Λ$ be a closed subgroup of the dual group $\widehat{G}$. In this paper, we investigate modulation preserving operators with respect to $Λ$, and give a characterization of them in terms of range operators.

The weak compactification of locally compact groups

We further investigate the weak topology generated by the irreducible unitary representations of a group $G$. A deep result due to Ernest \cite{Ernest1971} and Hughes \cite{Hughes1973} asserts that every weakly compact subset of a locally compact (LC) group $G$ is compact in the LC-topology, generalizing thereby a previous result of Glicksberg \cite{glicks1962} for abelian locally compact (LCA) groups. Here, we first survey some recent findings on the weak topology and establish some new results about the preservation of several compact-like properties when going from the weak topology to the …