Tensor Algebras and Displacement Structure. III. Asymptotic properties

Tensor algebras, displacement structure, and the Schur algorithm

In this paper we explore the connection between tensor algebras and displacement structure. We describe a scattering experiment in this framework, we obtain a realization of the elements of the tensor algebra as transfer maps of a certain class of nonstationary linear systems, and we describe a Schur algorithm for the Schur elements of the tensor algebra.

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.

Tensor algebras and displacement structure. IV. Invariant kernels

In this paper we investigate the class of invariant positive definite kernels on the free semigroup on N generators. We provide a combinatorial description of the positivity of the kernel in terms of Dyck paths and then we find a displacement equation that encodes the invariance property of the kernel.

具有非负曲率的Alexandrov空间与熵张量的位移凸性

Alexandrov spaces with non negative curvature and displacement convexity of the entropy tensor

将熵张量推广到曲率有下界的Alexandrov空间，通过构造满足上循环条件和第二变分公式的平行平凡化，证明非负曲率等价于熵张量的矩阵位移凸性。

Random Sets and Invariants for (Type II) Continuous Tensor Product Systems of Hilbert Spaces

In a series of papers Tsirelson constructed from measure types of random sets and generalised random processes a new range of examples for continuous tensor product systems of Hilbert spaces introduced by Arveson for classifying $E_0$-semigroups. This paper establishes the converse: Each continuous tensor product systems of Hilbert spaces comes with measure types of distributions of random (closed) sets in [0,1] or $R_+$. These measure types are stationary and factorise over disjoint intervals. In a special case of this construction, the corresponding measure type is an invariant of the produc…

Realization of rigid C$^*$-tensor categories via Tomita bimodules

Starting from a (small) rigid C$^*$-tensor category $\mathscr{C}$ with simple unit, we construct von Neumann algebras associated to each of its objects. These algebras are factors and can be either semifinite (of type II$_1$ or II$_\infty$, depending on whether the spectrum of the category is finite or infinite) or they can be of type III$_λ$, $λ\in (0,1]$. The choice of type is tuned by the choice of Tomita structure (defined in the paper) on certain bimodules we use in the construction. Moreover, if the spectrum is infinite we realize the whole tensor category directly as endomorphisms of th…