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Register a new userBimodules over ${rm VN}(G)$, harmonic operators and the non-commutative Poisson boundary
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Starting with a left ideal $J$ of $L^1(G)$ we consider its annihilator $J^{perp}$ in $L^{infty}(G)$ and the generated ${rm VN}(G)$-bimodule in $mathcal{B}(L^2(G))$, ${rm Bim}(J^{perp})$. We prove that ${rm Bim}(J^{perp})=({rm Ran} J)^{perp}$ when $G$ is weakly amenable discrete, compact or abelian, where ${rm Ran} J$ is a suitable saturation of $J$ in the trace class. We define jointly harmonic functions and jointly harmonic operators and show that, for these classes of groups, the space of jointly harmonic operators is the ${rm VN}(G)$-bimodule generated by the space of jointly harmonic functions. Using this, we give a proof of the following result of Izumi and Jaworski - Neufang: the non-commutative Poisson boundary is isomorphic to the crossed product of the space of harmonic functions by $G$.
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This paper is concerned with weak* closed masa-bimodules generated by A(G)-invariant subspaces of VN(G). An annihilator formula is established, which is used to characterise the weak* closed subspaces of B(L^2(G)) which are invariant under both Schur multipliers and a canonical action of M(G) on B(L^2(G)) via completely bounded maps. We study the special cases of extremal ideals with a given null set and, for a large class of groups, we establish a link between relative spectral synthesis and relative operator synthesis.
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.
This article is devoted to studying the non-commutative Poisson boundary associated with $Big(Bbig(mathcal{F}(mathcal{H})big), P_{omega}Big)$ where $mathcal{H}$ is a separable Hilbert space (finite or infinite-dimensional), $dim mathcal{H} > 1$, with an orthonormal basis $mathcal{E}$, $Bbig(mathcal{F}(mathcal{H})big)$ is the algebra of bounded linear operators on the full Fock space $mathcal{F}(mathcal{H})$ defined over $mathcal{H}$, $omega = {omega_e : e in mathcal{E} }$ is a sequence of positive real numbers such that $sum_e omega_e = 1$ and $P_{omega}$ is the Markov operator on $Bbig(mathcal{F}(mathcal{H})big)$ defined by begin{align*} P_{omega}(x) = sum_{e in mathcal{E}} omega_e l_e^* x l_e, x in Bbig(mathcal{F}(mathcal{H})big), end{align*} where, for $e in mathcal{E}$, $l_e$ denotes the left creation operator associated with $e$. The non-commutative Poisson boundary associated with $Big(Bbig(mathcal{F}(mathcal{H})big), P_{omega}Big)$ turns out to be an injective factor of type $III$ for any choice of $omega$. Moreover, if $mathcal{H}$ is finite-dimensional, we completely classify the Poisson boundary in terms of its Connes $S$-invarinat and curiously they are type $III _{lambda }$ factors with $lambda$ belonging to a certain small class of algebraic numbers.
We examine the common null spaces of families of Herz-Schur multipliers and apply our results to study jointly harmonic operators and their relation with jointly harmonic functionals. We show how an annihilation formula obtained in J. Funct. Anal. 266 (2014), 6473-6500 can be used to give a short proof as well as a generalisation of a result of Neufang and Runde concerning harmonic operators with respect to a normalised positive definite function. We compare the two notions of support of an operator that have been studied in the literature and show how one can be expressed in terms of the other.
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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