Let G be a locally compact group, let $\Omega:G\times G\to \mathbb{C}^*$ be a
2-cocycle, and let $\Phi$ be a Young function. In this paper, we consider the
Orlicz space $L^\Phi(G)$ and investigate its algebraic property under the
twisted convolution $\circledast$ coming from $\Omega$. We find sufficient
conditions under which $(L^\Phi(G),\circledast)$ becomes a Banach algebra or a
Banach $*$-algebra; we call it a {\it twisted Orlicz algebra}. Furthermore, we
study its harmonic analysis properties, such as symmetry, existence of
functional calculus, regularity, and having Wiener property, mostly for the
case when $G$ is a compactly generated group of polynomial growth. We apply our
methods to several important classes of polynomial as well as subexponential
weights and demonstrate that our results could be applied to variety of cases
For a locally compact group $G$ with left Haar measure and a Young function ${\rm\Phi}$, we define and study the weighted Orlicz algebra $L_{w}^{{\rm\Phi}}(G)$ with respect to convolution. We show that $L_{w}^{{\rm\Phi}}(G)$ admits no bounded approximate identity under certain conditions. We prove that a closed linear subspace $I$ of the algebra $L_{w}^{{\rm\Phi}}(G)$ is an ideal in $L_{w}^{{\rm\Phi}}(G)$ if and only if $I$ is left translation invariant. For an abelian $G$, we describe the spectrum (maximal ideal space) of the weighted Orlicz algebra and show that weighted Orlicz algebras are semisimple.
We deduce continuity, compactness and invariance properties for
quasi-Banach Orlicz modulation spaces on
ℝ
d
{\mathbb{R}^{d}}
. We characterize such spaces
in terms of Gabor expansions and by their images under the Bargmann transform.
Let G be a locally compact group, let Ω : G × G → C * be a 2-cocycle, and let (Φ,Ψ) be a complementary pair of strictly increasing continuous Young functions. We continue our investigation in [14] of the algebraic properties of the Orlicz space L Φ (G) with respect to the twisted convolution ⊛ coming from Ω. We show that the twisted Orlicz algebra (L Φ (G), ⊛) posses a bounded approximate identity if and only if it is unital if and only if G is discrete. On the other hand, under suitable condition on Ω, (L Φ (G), ⊛) becomes an Arens regular, dual Banach algebra. We also look into certain cohomological properties of (L Φ (G), ⊛), namely amenability and Connes-amenability, and show that they rarely happen. We apply our methods to compactly generated group of polynomial growth and demonstrate that our results could be applied to variety of cases.
We deduce continuity, compactness and invariance properties for quasi-Banach Orlicz modulation spaces. We characterize such spaces in terms of Gabor expansions and by their images under the Bargmann transform.
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