Singularity structures for noncommutative spaces

Free noncommutative principal divisors and commutativity of the tracial fundamental group

We define the principal divisor of a free noncommuatative function. We use these divisors to compare the determinantal singularity sets of free noncommutative functions. We show that the divisor of a noncommutative rational function is the difference of two polynomial divisors. We formulate a nontrivial theory of cohomology, fundamental groups and covering spaces for tracial free functions. We show that the natural fundamental group arising from analytic continuation for tracial free functions is a direct sum of copies of $\mathbb{Q}$. Our results contrast the classical case, where the analogo…

Scattering in quantum dots via noncommutative rational functions

In the customary random matrix model for transport in quantum dots with $M$ internal degrees of freedom coupled to a chaotic environment via $N\ll M$ channels, the density $ρ$ of transmission eigenvalues is computed from a specific invariant ensemble for which explicit formula for the joint probability density of all eigenvalues is available. We revisit this problem in the large $N$ regime allowing for (i) arbitrary ratio $ϕ:= N/M \le 1$; and (ii) general distributions for the matrix elements of the Hamiltonian of the quantum dot. In the limit $ϕ\to 0$ we recover the formula for the density $ρ…

Modulation Spaces as a Smooth Structure in Noncommutative Geometry

We demonstrate that a class of modulation spaces are examples of a smooth structure on the noncommutative 2-torus in the sense of recent developments in KK-theory. In addition, we prove that this class of modulation spaces can be represented as corners in operator linking algebras.

The K-inductive Structure of the Noncommutative Fourier Transform

The noncommutative Fourier transform of the irrational rotation C*-algebra is shown to have a K-inductive structure (at least for a large concrete class of irrational parameters, containing dense $G_δ$'s). This is a structure for automorphisms that is analogous to Huaxin Lin's notion of tracially AF for C*-algebras, except that it requires more structure from the complementary projection.

非交换$C^*$-代数网络：在$C^*$-代数中学习具有强大乘积结构的神经网络

Noncommutative $C^*$-algebra Net: Learning Neural Networks with Powerful Product Structure in $C^*$-algebra

本文提出基于非交换$C^*$-代数的神经网络参数空间推广，利用其丰富乘积结构提升学习能力，可同时学习多个相关网络及群作用等变特征，并通过数值实验验证了有效性。

Tensor algebras and displacement structure. II. Noncommutative Szego polynomials

In this paper we continue to explore the connection between tensor algebras and displacement structure. We focus on recursive orthonormalization and we develop an analogue of the Szego type theory of orthogonal polynomials in the unit circle for several noncommuting variables. Thus, we obtain the recurrence equations and Christoffel-Darboux type formulas, as well as a Favard type result. Also we continue to study a Szego kernel for the N-dimnesional unit ball of an infinite dimensional Hilbert space.