Consider a discrete group G and a bounded self-adjoint convolu-tion operator T on l 2 (G); let σ(T) be the spectrum of T. The spectral theorem gives a unitary isomorphism U between l 2 (G) and a direct sum n L 2 (∆n, ν), where ∆n ⊂ σ(T), and ν is a regular Borel measure supported on σ(T). Through this isomorphism T corresponds to multiplication by the identity function on each summand. We prove that a nonzero function f ∈ l 2 (G) and its transform Uf cannot be simultaneously concentrated on sets V ⊂ G, W ⊂ σ(T) such that ν(W) and the cardinality of V are both small. This can be regarded as an extension to this context of Heisenberg's classical uncertainty principle.

ABSTRACT. It is proved in this note, that a strongly continuous semigroup of (sub)positive contractions acting on an L p -space, for 1 Ú p Ú 1 p  ≥ 2, can be dilated by a strongly continuous group of (sub)positive isometries in a manner analogous to the dilation M. A. Akçoglu and L. Sucheston constructed for a discrete semigroup of (sub)positive contractions. From this an improvement of a von Neumann type estimation, due to R. R. Coifman and G. Weiss, on the transfer map belonging to the semigroup is deduced.

Let G be a compactly generated, locally compact group of polynomial growth. Removing a restrictive technical condition from a previous work, we show that the weighted group algebra L 1 ω (G) is a symmetric Banach * -algebra if and only if the weight function ω satisfies the GRScondition. This condition expresses in a precise technical sense that ω grows subexponentially.

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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