We extend the results by Froelich and Spronk and Turowska on the connection between operator synthesis and spectral synthesis for A(G) to second countable locally compact groups G. This gives us another proof that one-point subset of G is a set of spectral synthesis and that any closed subgroup is a set of local spectral synthesis. Furthermore, we show that "nontriangular" sets are strong operator Ditkin sets and we establish a connection between operator Ditkin sets and Ditkin sets. These results are applied to prove that any closed subgroup of G is a local Ditkin set.
Consider a right-invariant sub-Laplacian L on an exponential solvable Lie group G, endowed with a left-invariant Haar measure. Depending on the structure of G and possibly also that of L, L may admit differentiable L p -functional calculi, or may be of holomorphic L p -type for a given p{2, as recent studies of specific classes of groups G and sub-Laplacians L have revealed. By``holomorphic L p -type'' we mean that every L p -spectral multiplier for L is necessarily holomorphic in a complex neighborhood of some point in the L 2 -spectrum of L. This can only arise if the group algebra L 1 (G) is non-symmetric. In this article we prove that, for large classes of exponential groups, including all rank one AN-groups, a certain Lie algebraic condition, which characterizes the non-symmetry of L 1 (G) [37], also suffices for L to be of holomorphic L 1 -type. Moreover, if this condition, which was first introduced by J. Boidol [6] in a different context, holds for generic points in the dual g* of the Lie algebra of G, then L is of holomorphic L p -type for every p{2. Besides the non-symmetry of L 1 (G), also the closedness of coadjoint orbits plays a crucial role. We also discuss an example of a higher rank AN-group. This example and our results in the rank one case suggest that sub-Laplacians on exponential Lie groups may be of holomorphic L 1 -type if and only if there exists a closed coadjoint orbit 0/g* such that the points of 0 satisfy Boidol's condition. In the course of the proof of our main results, whose principal strategy is similar as in [8], we develop various tools which may be of independent interest and largely apply to more general Lie groups. Some of them are certainly known as`f olklore'' results. For instance, we study subelliptic estimates on representation spaces, the relation between spectral multipliers and unitary representations, and develop some``holomorphic'' and``continuous'' perturbation theory for images of subLaplacians under``smoothly varying'' families of irreducible unitary representations.
Let G be a compactly generated group of polynomial growth and ω a weight function on G. For a large class of weights we characterize symmetry of the weighted group algebra L 1 (G, ω). In particular, if the weight ω is sub-exponential, then the algebra L 1 (G, ω) is symmetric. For these weights we develop a functional calculus on a total part of L 1 (G, ω) and use it to prove the Wiener property. (2000): 43A20, 22D15, 22D12.
Mathematics Subject Classification
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