We introduce the notion of a conditionally free product and conditionally free convolution. We describe this convolution both from a combinatorial point of view, by showing its connection with the lattice of non-crossing partitions, and from an analytic point of view, by presenting the basic formula for its R-transform. We calculate explicitly the distributions of the conditionally free Gaussian and

We prove non-commutative versions of Wiener’s Lemma on absolutely convergent Fourier series (a) for the case of twisted convolution and (b) for rotation algebras. As an application we solve some open problems about Gabor frames, among them the problem of Feichtinger and Janssen that is known in the literature as the “irrational case”.

Abstract. We investigate the symbolic calculus for a large class of matrix algebras that are defined by the off-diagonal decay of infinite matrices. Applications are given to the symmetry of some highly non-commutative Banach algebras, to the analysis of twisted convolution, and to the theory of localized frames.

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.

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

No abstract

Introduction. Let G be an arbitrary locally compact group [^4(G)], B(G) the [Fourier] Fourier-Stieltjes algebra of G and M(G) the Banach algebra of bounded Radon measures on G (see definitions in what follows). We prove in §1 of this paper that we w*-topology z w * and the multiplier topology z M coincide on the unit sphere S = {u e B(G)\ \\u\\ = 1} of B(G\ where u a -> u in z M if and only if ||(w a -w)v|| -» 0 for each v e A(G). This result proves a conjecture of McKennon [10, p. 49]. It improves a result of Derighetti [1] and McKennon [10] (that z w * = z uc on S, where z uc is the topology of uniform convergence on compacta) which in turn improves a theorem of Raikov [13] and Yoshizawa [17] (that z w * = z uc on the positive definite face of S). Applying this result we show in theorem B x that for any compact K a G the Banach space A K (G) = {u e A(G); supp u a K} has the Radon-Nikodym property and consequently a strong Krein-Milman theorem, for closed bounded convex subsets of A K (G) 9 follows. Theorem B 2 of this section consists of a long list of topologies which coincide on S. §3 consists of a measure theoretical selfcontained proof of a result of McKennon [10] which states that the w* and the ZAmultiplier topology on S = {ju e M(G); ||//|| = 1} coincide (//« -> ft in the latter if and only if || (/ua -fi)*f\\p -• 0 for each fe LP). The reader familiar with [10, pp.21-25 and 32-33] will find, we think, that our proof is simpler, more natural and self-contained. Finally we investigate in §2, subsets of Bf(G) (the space of multipliers of A p (G)) on which the topologies z M and a(Bf, L l ) coincide. As a consequence a necessary and sufficient condition for a subset of Ap(G) to be norm compact is given (in case G is amenable). In view of [8] the results seem to be of interest even for the nonamenable case. Definitions and notations.Let G be a locally compact group with unit e. C(G) (C 00 (G)) [CQ(G)] will denote the space of complex bounded continuous functions (with compact support) [which vanish at infinity]. À or dx will denote a left Haar measure on G. \\f\\ p = ($\f\ p dx) v * will denote the