Compatibility of a noncommutative probability space and a noncommutative probability space with amalgamation

A noncommutative de Finetti theorem: Invariance under quantum permutations is equivalent to freeness with amalgamation

We show that the classical de Finetti theorem has a canonical noncommutative counterpart if we strengthen `exchangeability' (i.e., invariance of the joint distribution of the random variables under the action of the permutation group) to invariance under the action of the quantum permutation group. More precisely, for an infinite sequence of noncommutative random variables, we prove that invariance of their joint distribution under quantum permutations is equivalent to the fact that the random variables are identically distributed and free with respect to the conditional expectation onto their…

R-transform Theory On a Tower of Amalgamated Noncommutative Probability Spaces

In [9], we observed Amalgamated R-transform Theory. Different from the original definition of Voiculescu and Speicher, we define R-transforms of operator-valued random variable(s) by operator-valued formal series. By doing that we can establish Amalgamated R-transform calculus. In this paper, we will concentrate on observing R-transforms on a tower of amalgamated noncommutative probability spaces. We will consider the R-transform calculus and relation between R-transforms on the tower and distributions with respect to the tower. As application, we will observe the compatibility of the given to…

非交换几何中联络的挠率与度量相容性的两种概念

On two notions of torsion and metric compatibility of connections in noncommutative geometry

本文比较了Beggs-Majid与Mesland-Rennie框架下联络的度量相容性与挠率概念，证明了在特定条件下两者的等价性，并据此确立了多种流形和微积分上Levi-Civita联络的存在性与唯一性。

Noncommutative reproducing kernel Hilbert spaces

The theory of positive kernels and associated reproducing kernel Hilbert spaces, especially in the setting of holomorphic functions, has been an important tool for the last several decades in a number of areas of complex analysis and operator theory. An interesting generalization of holomorphic functions, namely free noncommutative functions (e.g., functions of square-matrix arguments of arbitrary size satisfying additional natural compatibility conditions), is now an active area of research, with motivation and applications from a variety of areas (e.g., noncommutative functional calculus, fr…

Constructing equivalence bimodules between noncommutative solenoids: a two-pronged approach

We revisit and generalize the application of a method introduced by Latrémolière and Packer for constructing finitely generated projective modules over the noncommutative solenoid C*-algebras. By realizing them as direct limits of rotation algebras, the method constructs directed systems of equivalence bimodules between rotation algebras that satisfy the necessary compatibility conditions to build Morita equivalence bimodules between the direct limit C*-algebras. In the irrational case, we use a fixed projection in a matrix algebra over the rotation algebra satisfying a key condition to build …

A noncommutative view on topology and order

In this paper we put forward the definition of particular subsets on a unital C*-algebra, that we call isocones, and which reduce in the commutative case to the set of continuous non-decreasing functions with real values for a partial order relation defined on the spectrum of the algebra, which satisfies a compatibility condition with the topology (complete separateness). We prove that this space/algebra correspondence is a dual equivalence of categories, which is in fact only a mild generalization of the Gelfand-Naimark duality. Thus we can expect that general isocones could serve to define a…