We extend the results by Froelich and Spronk and Turowska on the connection between operator synthesis and spectral synthesis for A(G) to second countable locally compact groups G. This gives us another proof that one-point subset of G is a set of spectral synthesis and that any closed subgroup is a set of local spectral synthesis. Furthermore we show that ``non-triangular'' sets are strong operator Ditkin sets and we establish a connection between operator Ditkin sets and Ditkin sets. These results are applied to prove that any closed subgroup of $G$ is a local Ditkin set.
We introduce and study the notion of reduced spectral synthesis, which unifies the concepts of spectral synthesis and uniqueness in locally compact groups. We exhibit a number of examples and prove that every non-discrete locally compact group with an open abelian subgroup has a subset that fails reduced spectral synthesis. We introduce compact operator synthesis as an operator algebraic counterpart of this notion and link it with other exceptional sets in operator algebra theory, studied previously. We show that a closed subset $E$ of a second countable locally compact group $G$ satisfies red…
The second part of our work on operator synthesis deals with individual operator synthesis of elements in some tensor products, in particular in Varopoulos algebras, and its connection with linear operator equations. Using a developed technique of ``approximate inverse intertwining'' we obtain some generalizations of the Fuglede and the Fuglede-Weiss theorems. Additionally, we give some applications to spectral synthesis in Varopoulos algebras and to partial differential equations.
The interplay between the invariant subspace theory and spectral synthesis for locally compact abelian group discovered by Arveson is extended to include other topics as harmonic analysis for Varopoulos algebras and approximation by projection-valued measures. We propose a ''coordinate'' approach which nevertheless does not use the technique of pseudo-integral operators, as well as a coordinate free one which allows to extend to non-separable spaces some important results and constructions of [W.Arveson, Operator Alegebras and Invariant subspaces, Ann. of Math. (2) 100 (1974)] and solve some p…
Using techniques from TRO equivalence of masa bimodules we prove various transference results: We show that when $α$ is a group homomorphism which pushes forward the Haar measure of $G$ to a measure absolutely continuous with respect to the Haar measure on $H$, then $(α\timesα)^{-1}$ preserves sets of compact operator synthesis, and conversely when $α$ is onto. We also prove similar preservation results for operator Ditkin sets and operator M-sets, obtaining preservation results for M-sets as corollaries. Some of these results extend or complement existing results of Ludwig, Shulman, Todorov a…
We initiate the study of the completely bounded multipliers of the Haagerup tensor product $A(G)\otimes_{\rm h} A(G)$ of two copies of the Fourier algebra $A(G)$ of a locally compact group $G$. If $E$ is a closed subset of $G$ we let $E^{\sharp} = \{(s,t) : st\in E\}$ and show that if $E^{\sharp}$ is a set of spectral synthesis for $A(G)\otimes_{\rm h} A(G)$ then $E$ is a set of local spectral synthesis for $A(G)$. Conversely, we prove that if $E$ is a set of spectral synthesis for $A(G)$ and $G$ is a Moore group then $E^{\sharp}$ is a set of spectral synthesis for $A(G)\otimes_{\rm h} A(G)$. …
The purpose of this paper is to make a comprehensive connection between the basic results and properties derived from the two kinds of topologies (namely the $(ε,λ)-$topology introduced by the author and the stronger locally $L^{0}-$convex topology recently introduced by Filipovi$\acute{c}$ et. al) for a random locally convex module. First, we give an extremely simple proof of the known Hahn-Banach extension theorem of $L^{0}-$linear functions as well as its continuous variants. Then we give the essential relations between the hyperplane separation theorems in [Filipovi$\acute{c}$ et. al, J. F…
In this paper we introduce a new way of deforming convolution algebras and Fourier algebras on locally compact groups. We demonstrate that this new deformation allows us to reveal some informations of the underlying groups by examinining Banach algebra properties of deformed algebras. More precisely, we focus on representability as an operator algebra of deformed convolution algebras on compact connected Lie groups with connection to the real dimension of the underlying group. Similary, we investigate complete representability as an operator algebra of deformed Fourier algebras on some finitel…
We show that for a connected Lie group $G$, its Fourier algebra $A(G)$ is weakly amenable only if $G$ is abelian. Our main new idea is to show that weak amenability of $A(G)$ implies that the anti-diagonal, $\checkΔ_G=\{(g,g^{-1}):g\in G\}$, is a set of local synthesis for $A(G\times G)$. We then show that this cannot happen if $G$ is non-abelian. We conclude for a locally compact group $G$, that $A(G)$ can be weakly amenable only if it contains no closed connected non-abelian Lie subgroups. In particular, for a Lie group $G$, $A(G)$ is weakly amenable if and only if its connected component of…
In the second part of the paper we consider a convolution of probability measures on spaces of locally finite configurations (subsets of a phase space) as well as their connection with the convolution of the corresponding correlation measures and functionals. In particular, the convolution of Gibbs measures is studied. We describe also a connection between invariant measures with respect to some operator and properties of the corresponding image of this operator on correlation functions.
Let $G$ be a unimodular type I second countable locally compact group and $\hat G$ its unitary dual. We introduce and study a global pseudo-differential calculus for operator-valued symbols defined on $G\times\hat G$, and its relations to suitably defined Wigner transforms and Weyl systems. We also unveil its connections with crossed products $C^*$-algebras associated to certain $C^*$-dynamical systems, and apply it to the spectral analysis of covariant families of operators. Applications are given to nilpotent Lie groups, in which case we relate quantizations with operator-valued and scalar-v…
We shall prove that a moduli space of flat irreducible Lie algebroid connections over a compact manifold has locally a natural structure of a smooth differentiable space. This is a generalization of some well known results for the moduli space of holomorphic structures on a complex vector bundle over a compact complex manifold.
By using a coset of closed subgroup, we define a Fourier like transform for locally compact abelian (LCA) topological groups. We studied two wavelet multipliers and Landau-Pollak-Slepian operators on locally compact abelian topological groups associated to the transform and show that the transforms are $L^p$bounded linear operators, and are in Schatten p-class for $1\leq p\leq \infty$. Finally, we determine their trace class and also obtain a connection with the generalized Landau-Pollak-Slepian operators.
Let $\G$ be a locally compact group satisfying some technical requirements and $\wG$ its unitary dual. Using the theory of twisted crossed product $C^*$-algebras, we develop a twisted global quantization for symbols defined on $\G\times\wG$ and taking operator values. The emphasis is on the representation-theoretic aspect. For nilpotent Lie groups, the connection is made with a scalar quantization of the cotangent bundle $T^*(\G)$ and with a Quantum Mechanical theory of observables in the presence of variable magnetic fields.
In a recent paper by M. Mantoiu and M. Ruzhansky, a global pseudo-differential calculus has been developed for unimodular groups of type I. In the present article we generalize the main results to arbitrary locally compact groups of type I. Our methods involve the use of Plancherel's theorem for non-unimodular groups. We also make connections with a $C^*$-algebraic formalism, involving dynamical systems, and give explicit constructions for the group of affine transformations of the real line.
