L p -spaces on locally compact groups

In this paper, some relations between L ^p -spaces on locally compact groups are found. Applying these results proves that for a locally compact group G, the convolution Banach algebras L ^p (G) ∩ L ¹(G) (1 < p ≤ ∞), and A _p (G) ∩ L ¹(G) (1 < p < ∞) are amenable if and only if G is discrete and amenable.     

Gaussian hemigroups on a locally compact group

A notion of Gaussian hemigroup is introduced and its relationship with the Gauss condition is studied. Moreover, a Lévy-type martingale characterization is proved for processes with independent (not necessarily stationary) increments satisfying the Gauss condition in a compact Lie group. The characterization is given in terms of a faithful finite dimensional representation of the group and its tensor square. For the proofs noncommutative Fourier theory is applied for the convolution hemigroups associated with the increment processes. This revised version was published online in June 2006 with corrections to the Cover Date.     

Derivations on group algebras of a locally compact abelian group

Monatshefte für Mathematik - Let G be a locally compact abelian group. In this paper, we study derivations on the Banach algebra $$L_0^\infty (G)^*$$ . We prove that any derivation on...     

Convolution roots and embeddings of probability measures on locally compact groups

We discuss some properties of nilpotent Lie groups and their application in proving the embedding theorem for infinitely divisible probability measures on locally compact groups.     

Dilation of Ritt operators on L p -spaces

For any Ritt operator T: L ^p (Ω) → L ^p (Ω), for any positive real number α, and for any x ∈ L ^p (Ω), we consider $${\left\| x \right\|_{T,\alpha }} = {\left\| {{{\left( {\sum\limits_{k = 1}^\infty {{k^{2\alpha - 1}}} {{\left| {{T^{k - 1}}{{(I - T)}^\alpha }x} \right|}^2}} \right)}^{\frac{1}{2}}}} \right\|_{{L^p}}}$$ . We show that if T is actually an R-Ritt operator, then the square functions \({\left\| {} \right\|_{T,\alpha }}\) are pairwise equivalent. Then we show that T and its adjoint T*: L ^p′ (Ω) → L ^p′ (Ω) both satisfy uniform estimates \({\left\| x \right\|_{T,1}} \leqslant {\left\| x \right\|_{{L^p}}}\) and \({\left\| y \right\|_{T*,1}} \leqslant {\left\| y \right\|_{{L^{p'}}}}\) for x ∈ L ^p (Ω) and y ∈ L ^p′ (Ω) if and only if T is R-Ritt and admits a dilation in the following sense: there exist a measure space \(\tilde \Omega \) , an isomorphism \(U:{L^p}\tilde \Omega \to {L^p}\tilde \Omega \) such that {U ⁿ : n ∈ ?} is bounded, as well as two bounded maps \({L^p}(\Omega )\buildrel J \over \longrightarrow {L^p}(\tilde \Omega )\buildrel Q \over \longrightarrow {L^p}(\Omega )\) such that T ⁿ = QU ⁿ J for any n ≥ 0. We also investigate functional calculus properties of Ritt operators and analogs of the above results on noncommutative L ^p -spaces.     

Structure of differential operators on a locally compact abelian group

Translated from Matematicheskie Zametki, Vol. 52, No. 4, pp. 3–14, October, 1992.     

Remarks on contractive projections in L p -spaces

The aim of this note is to study the structure of the range of a contractive projection in a non-separableL _p -space; 1≦p<+∞. Supported by National Science Foundation Grant GP-7475     

A test of balance of a locally compact group

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 40, No. 1, pp. 127–129, Janaury–February, 1988.     

Invariant subspaces of some function spaces on a locally compact group

Let G be a σ-compact and locally compact group. If f?L^∞(G) let U_f be the closed subspace of L^∞(G) generated by the left translations of f. Conditions are given which ensure that each function in U_f may be expanded in an essentially unique way as an absolutely convergent series of translations of f. In this case U_f contains subspaces which are isometrically isomorphic to l¹. If G is metrizable and nondiscrete there is a continuum Γ in L^∞(G) such that, for each f?Γ, U_f contains no non-zero continuous function, and for f, g?Γ with f ≠ g, U_f ∩ U_g = {0}. If G is non-compact, metrizable, and non-discrete there is a continuum Γ of bounded continuous functions on G such that, for each f?Γ, U_f contains no non-zero left uniformly continuous function, and for f, g?Γ with f ≠ g, U_f ∩ U_g = {0}. The subspaces U_f above are translation invariant but are not convolution invariant.