Transformations of measurable sets by automorphism groups.
Let G be an infinite locally compact abelian group and X be a Banach space. We show that if every bounded Fourier multiplier T on L²(G) has the property that is bounded on L²(G,X) then X is isomorphic to a Hilbert space. Moreover, we prove that if 1 < p < ∞, p ≠ 2, then there exists a bounded Fourier multiplier on which is not completely bounded. Finally, we examine unconditionality from the point of view of Schur multipliers. More precisely, we give several necessary and sufficient conditions...
We show that if ϕ is a continuous homomorphism between weighted convolution algebras on ℝ⁺, then its extension to the corresponding measure algebras is always weak* continuous. A key step in the proof is showing that our earlier result that normalized powers of functions in a convolution algebra on ℝ⁺ go to zero weak* is also true for most measures in the corresponding measure algebra. For some algebras, we can determine precisely which measures have normalized powers converging to zero weak*. We...