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Register a new userUniversality of free random variables: atoms for non-commutative rational functions
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We address the following question: what can one say, for a tuple $(Y_1,dots,Y_d)$ of normal operators in a tracial operator algebra setting with prescribed sizes of the eigenspaces for each $Y_i$, about the sizes of the eigenspaces for any non-commutative polynomial $P(Y_1,dots,Y_d)$ in those operators? We show that for each polynomial $P$ there are unavoidable eigenspaces, which occur in $P(Y_1,dots,Y_d)$ for any $(Y_1,dots,Y_d)$ with the prescribed eigenspaces for the marginals. We will describe this minimal situation both in algebraic terms - where it is given by realizations via matrices over the free skew field and via rank calculations - and in analytic terms - where it is given by freely independent random variables with prescribed atoms in their distributions. The fact that the latter situation corresponds to this minimal situation allows to draw many new conclusions about atoms in polynomials of free variables. In particular, we give a complete description of atoms in the free commutator and the free anti-commutator. Furthermore, our results do not only apply to polynomials, but much more general also to non-commutative rational functions.
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We formulate a free probabilistic analog of the Wasserstein manifold on $mathbb{R}^d$ (the formal Riemannian manifold of smooth probability densities on $mathbb{R}^d$), and we use it to study smooth non-commutative transport of measure. The points of the free Wasserstein manifold $mathscr{W}(mathbb{R}^{*d})$ are smooth tracial non-commutative functions $V$ with quadratic growth at $infty$, which correspond to minus the log-density in the classical setting. The space of smooth tracial non-commutative functions used here is a new one whose definition and basic properties we develop in the paper; they are scalar-valued functions of self-adjoint $d$-tuples from arbitrary tracial von Neumann algebras that can be approximated by trace polynomials. The space of non-commutative diffeomorphisms $mathscr{D}(mathbb{R}^{*d})$ acts on $mathscr{W}(mathbb{R}^{*d})$ by transport, and the basic relationship between tangent vectors for $mathscr{D}(mathbb{R}^{*d})$ and tangent vectors for $mathscr{W}(mathbb{R}^{*d})$ is described using the Laplacian $L_V$ associated to $V$ and its pseudo-inverse $Psi_V$ (when defined). Following similar arguments to arXiv:1204.2182, arXiv:1701.00132, and arXiv:1906.10051 in the new setting, we give a rigorous proof for the existence of smooth transport along any path $t mapsto V_t$ when $V$ is sufficiently close $(1/2) sum_j operatorname{tr}(x_j^2)$, as well as smooth triangular transport.
The purpose of this short note was to outline the current status, then in 2011, of some research programs aiming at a categorification of parts of A.Connes non-commutative geometry and to provide an outlook on some possible subsequent developments in categorical non-commutative geometry.
In this paper we show how questions about operator algebras constructed from stochastic matrices motivate new results in the study of harmonic functions on Markov chains. More precisely, we characterize coincidence of conditional probabilities in terms of (generalized) Doob transforms, which then leads to a stronger classification result for the associated operator algebras in terms of spectral radius and strong Liouville property. Furthermore, we characterize the non-commutative peak points of the associated operator algebra in a way that allows one to determine them from inspecting the matrix. This leads to a concrete analogue of the maximum modulus principle for computing the norm of operators in the ampliated operator algebras.
A Herglotz function is a holomorphic map from the open complex unit disk into the closed complex right halfplane. A classical Herglotz function has an integral representation against a positive measure on the unit circle. We prove a free analytic analogue of the Herglotz representation and describe how our representations specialize to the free probabilistic case. We also show that the set of representable Herglotz functions arising from noncommutative conditional expectations must be closed in a natural topology.
After an introduction to some basic issues in non-commutative geometry (Gelfand duality, spectral triples), we present a panoramic view of the status of our current research program on the use of categorical methods in the setting of A.Connes non-commutative geometry: morphisms/categories of spectral triples, categorification of Gelfand duality. We conclude with a summary of the expected applications of categorical non-commutative geometry to structural questions in relativistic quantum physics: (hyper)covariance, quantum space-time, (algebraic) quantum gravity.
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