A comparison between the max and min norms on C∗(Fn)⊗C∗(Fn).

Abstract. Let Fn, n > 2, be the free group on n generators, denoted by U1,U2, . . . ,Un. Let C¤(Fn) be the full C¤-algebra of Fn. Let X be the vector subspace of the algebraic tensor product C(Fn) ­ C¤(Fn), spannedby 1 ­ 1,U1 ­ 1, . . . ,Un ­ 1, 1 ­ U1, . . . , 1 ­ Un. Let k · kmin and k · kmax be the minimal and maximal C¤ tensor norms on C¤(Fn)­C¤(Fn), and use the same notation for the corresponding (matrix) norms induced on Mk(C)­X, k 2 N. Identifying X with the subspace of C¤(F2n) obtained by mapping U1­ 1, . . . , 1­Un into the 2n generators and the identity into the identity, we get a matrix norm k · kC¤(F2n) which dominates the k · kmax norm on Mk(C)­X. In this paper we prove that, with N = 2n + 1 = dimX, we have kXkmax 6 kXkC¤(F2n) 6 (N2 − N)1/2kXkmin, X 2 Mk(C) ­

Singularity of the radial subalgebra of ℒ(FN) and the Pukánszky invariant

Let ℒ(F N ) be the von Neumann algebra of the free group with N generators x 1 ,⋯,x N , N≥2 and let A be the abelian von Neumann subalgebra generated by x 1 +x 1 -1 +⋯+x N +x N -1 acting as a left convolutor on ℓ 2 (F N ). The radial algebra A appeared in the harmonic analysis of the free group as a maximal abelian subalgebra of ℒ(F N ), the von Neumann algebra of the free group. The aim of this paper is to prove that A is singular (which means that there are no unitaries u in ℒ(F N ) except those coming from A such that u * Au⊆A). This is done by showing that the Pukánszky invariant of A is infinite, where the Pukánszky invariant of A is the type of the commutant of the algebra A in B(ℓ 2 (F N )) generated by A and x 1 +x 1 -1 +⋯+x N +x N -1 regarded also as a right convolutor on ℓ 2 (F N

A non-commutative, analytic version of Hilbert's 17th problem in type II1 von Neumann algebras

We prove a non-commutative version of the Hilbert's 17th problem, giving a characterization of the class of non-commutative polynomials in n-undeterminates that have positive trace when evaluated in n-selfadjoint elements in arbitrary II1 von Neumann algebra. As a corollary we prove that Connes's embedding conjecture is equivalent to a statement that can be formulated entirely in the context of finite matrices

Non-commutative Markov processes in free groups factors, related to Berezin's quantization and automorphic forms.

Non-commutative Markov processes in free groups factors, related to Berezin's quantization and automorphic form

Fast But Not Furious. When Sped Up Bit Rate of Information Drives Rule Induction

The language abilities of young and adult learners range from memorizing specific items to finding statistical regularities between them (item-bound generalization) and generalizing rules to novel instances (category-based generalization). Both external factors, such as input variability, and internal factors, such as cognitive limitations, have been shown to drive these abilities. However, the exact dynamics between these factors and circumstances under which rule induction emerges remain largely underspecified. Here, we extend our information-theoretic model (Radulescu et al., 2019), based on Shannon’s noisy-channel coding theory, which adds into the “formula” for rule induction the crucial dimension of time: the rate of encoding information by a time-sensitive mechanism. The goal of this study is to test the channel capacity-based hypothesis of our model: if the input entropy per second is higher than the maximum rate of information transmission (bits/second), which is determined by the channel capacity, the encoding method moves gradually from item-bound generalization to a more efficient category-based generalization, so as to avoid exceeding the channel capacity. We ran two artificial grammar experiments with adults, in which we sped up the bit rate of information transmission, crucially not by an arbitrary amount but by a factor calculated using the channel capacity formula on previous data. We found that increased bit rate of information transmission in a repetition-based XXY grammar drove the tendency of learners toward category-based generalization, as predicted by our model. Conversely, we found that increased bit rate of information transmission in complex non-adjacent dependency aXb grammar impeded the item-bound generalization of the specific a_b frames, and led to poorer learning, at least judging by our accuracy assessment method. This finding could show that, since increasing the bit rate of information precipitates a change from item-bound to category-based generalization, it impedes the item-bound generalization of the specific a_b frames, and that it facilitates category-based generalization both for the intervening Xs and possibly for a/b categories. Thus, sped up bit rate does not mean that an unrestrainedly increasing bit rate drives rule induction in any context, or grammar. Rather, it is the specific dynamics between the input entropy and the maximum rate of information transmission