We initiate the study of sets of p-multiplicity in locally compact groups and their operator versions. We show that a closed subset E of a second countable locally compact group G is a set of p-multiplicity if and only if is a set of operator p-multiplicity. We exhibit examples of sets of p-multiplicity, establish preservation properties for unions and direct products, and prove a p-version of the Stone-von Neumann Theorem.
Let G be a locally compact abelian group and ℳ be a semifinite von Neumann algebra with a faithful semifinite normal trace τ. We study Hilbert transforms associated with G-flows on ℳ and closed semigroups Σ of Ĝ satisfying the condition Σ ∪ (-Σ) = Ĝ. We prove that Hilbert transforms on such closed semigroups satisfy a weak-type estimate and can be extended as linear maps from L¹(ℳ,τ) into . As an application, we obtain a Matsaev-type result for p = 1: if x is a quasi-nilpotent compact operator...
In this note we define and explore, à la Godement, spectral subspaces of Banach space representations of the Fourier-Eymard algebra of a (nonabelian) locally compact group.
Let A be a semisimple commutative regular tauberian Banach algebra with spectrum . In this paper, we study the norm spectra of elements of and present some applications. In particular, we characterize the discreteness of in terms of norm spectra. The algebra A is said to have property (S) if, for all , φ has a nonempty norm spectrum. For a locally compact group G, let denote the C*-algebra generated by left translation operators on and denote the discrete group G. We prove that the Fourier...
We give a construction of an analytic series of uniformly bounded representations of a free group G, through the action of G on its Poisson boundary. These representations are irreducible and give as their coefficients all the spherical functions on G which tend to zero at infinity. The principal and the complementary series of unitary representations are included. We also prove that this construction and the other known constructions lead to equivalent representations.