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.