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On convolution operators with small support which are far from being convolution by a bounded measure

Edmond Granirer — 1994

Colloquium Mathematicae

Let C V p ( F ) be the left convolution operators on L p ( G ) with support included in F and M p ( F ) denote those which are norm limits of convolution by bounded measures in M(F). Conditions on F are given which insure that C V p ( F ) , C V p ( F ) / M p ( F ) and C V p ( F ) / W are as big as they can be, namely have l as a quotient, where the ergodic space W contains, and at times is very big relative to M p ( F ) . Other subspaces of C V p ( F ) are considered. These improve results of Cowling and Fournier, Price and Edwards, Lust-Piquard, and others.

On group representations whose C * algebra is an ideal in its von Neumann algebra

Edmond E. Granirer — 1979

Annales de l'institut Fourier

Let τ be a continuous unitary representation of the locally compact group G on the Hilbert space H τ . Let C τ * [ V N τ ] be the C * [ W * ] algebra generated by ( L 1 ( G ) ) and M τ ( C τ * ) = φ V N τ ; φ C τ * + C τ * φ C τ * . The main result obtained in this paper is Theorem 1: If G is σ -compact and M τ ( C τ * ) = V N τ then supp τ is discrete and each π in supp τ in CCR. We apply this theorem to the quasiregular representation τ = π H and obtain among other results that M π H ( C π H * ) = V N π H implies in many cases that G / H is a compact coset space.

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