Nica’s q-Convolution is Not Positivity Preserving

Rank one HCIZ at high temperature: interpolating between classical and free convolutions

“…with µ a (x) = 1 N N i=1 δ(x − a i ). By using the formula for the generating function of the Jack polynomials (10), we get the following differential operator form for the HCIZ rank one integral,…”

“…Despite Speicher's work, there have been several attempts to construct generalized convolutions, with or without an underlying notion of independence, that would, in particular, interpolate between the classical convolution and the free convolution. To cite a few important results, let us mention: the q-convolution of Nica [8] (see also [9] for a similar but different q-convolution) which interpolates between the classical convolution at q = 1 and the free convolution at q = 0 but which does not seem to preserve the positivity of the measures [10]; the t-convolution of Benaych-Georges and Lévy in [11], which interpolates between the classical convolution (t = 0) and the free convolution (t → ∞) but for which it is not possible to construct a transform that linearizes the convolution and from which one can define cumulants at any order. In this note, we construct another one-parameter convolution, called the c-convolution as a continuous interpolation between the classical convolution at c = 0 and the free convolution as c → ∞.…”

Rank one HCIZ at high temperature: interpolating between classical and free convolutions

“…By relaxing the assumptions, it has been however possible to construct other type of convolutions see for example [7]. Despite Speicher's work, there has been several attend to construct a generalized convolution, with or without an underlying notion of independence, that would in particular interpolate between the classical convolution and the free convolution, namely to cite few important results: the q-convolution of Nica [8] (see also [9] for a similar but different q-convolution) which interpolates between the classical convolution at q = 1 and the free convolution at q = 0 but which seems to not preserved the positivity of the measures [10]; the t-convolution of Benaych-Georges and Lévy in [11], which interpolated between the classical convolution (t = 0) and the free convolution (t → ∞) but for which it is not possible to construct a transform that linearizes the convolution and from which one can define cumulants at any order.…”

“…From the power sum relation (9), we can express the HCIZ integral in terms of the inverse Laplace transform L −1 p [. ], using (10), we have for t > 0:…”

Convolutions related to q-deformed commutativity

“…The definition of q-convolution was rather algebraic and it was not clear whether the resulting object is again a measure. It was 10 years later, in 2005, when F. Oravecz [19] gave a negative answer to this question. He proved that for 0 < q < 1, the q-convolution does not preserve positivity, since the Poisson type limit is not a measure.…”