Invariant Subsets and Homological Properties of Orlicz Modules over Group Algebras

Abstract

Let G be a locally compact group with left Haar measure. We study the closed convex left invariant subsets of L Φ (G) and characterize affine mappings from the space of nonnegative functions in L 1 (G) of norm 1 into L Φ (G) spaces. We apply the results to the study of the multipliers of L Φ (G). We also investigate the homological properties of L Φ (G) as a Banach left L 1 (G)-module such as projectivity, injectivity and flatness.

A criterion for nonzero multiplier for Orlicz spaces of an affine group $\mathbb{R}_{+}\times \mathbb{R}$

“…(1) By using [19,Theorem 5.4], one can see that T = 0 is the only compact multiplier for the pair (L Φ 1 (A), L Φ 2 (A)). (2) Let A be a weakly compact closed left invariant subspace of L Φ 1 (A) and T : A → L 1 (A) be a bounded linear operator commuting with left translations.…”

“…(2) Let A be a weakly compact closed left invariant subspace of L Φ 1 (A) and T : A → L 1 (A) be a bounded linear operator commuting with left translations. Then by [19,Theorem 5.5], T = 0.…”

“…{0}. Moreover in [19] these standard results were extended to L Φ (G) spaces by Üster and Öztop. In particular, they achieved some results for the multipliers in L Φ (G) spaces.…”

Transference and restriction of Fourier multipliers on Orlicz spaces

“…In this paper, we shall deal with Orlicz spaces defined on locally compact abelian groups. Recently, most of the properties of Orlicz spaces and weighted Orlicz spaces over a locally compact group G have been also investigated (see, e.g., [1, 30–32, 36, 37]).…”

Affine mappings and multipliers for weighted Orlicz spaces over an affine group $\R_{+}\times \R$

“…For L p spaces, in [16], Lau studied the affine mappings T between the subsets of Lebesgue spaces. In [27], Üster and Öztop studied continuous affine mappings on the subsets of Orlicz spaces. On the other hand the characterization of multipliers for weighted Lebesgue spaces has been given by Gaudry [10].…”