Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userPoisson boundary on full Fock space
69
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
In this paper we develop a framework to extend the theory of generalized stochastic processes in the Hida white noise space to more general probability spaces which include the grey noise space. To obtain a Wiener-It^o expansion we recast it as a moment problem and calculate the moments explicitly. We further show the importance of a family of topological algebras called strong algebras in this context. Furthermore we show the applicability of our approach to the study of stochastic processes.
We study two classes of operator algebras associated with a unital subsemigroup $P$ of a discrete group $G$: one related to universal structures, and one related to co-universal structures. First we provide connections between universal C*-algebras that arise variously from isometric representations of $P$ that reflect the space $mathcal{J}$ of constructible right ideals, from associated Fell bundles, and from induced partial actions. This includes connections of appropriate quotients with the strong covariance relations in the sense of Sehnem. We then pass to the reduced representation $mathrm{C}^*_lambda(P)$ and we consider the boundary quotient $partial mathrm{C}^*_lambda(P)$ related to the minimal boundary space. We show that $partial mathrm{C}^*_lambda(P)$ is co-universal in two different classes: (a) with respect to the equivariant constructible isometric representations of $P$; and (b) with respect to the equivariant C*-covers of the reduced nonselfadjoint semigroup algebra $mathcal{A}(P)$. If $P$ is an Ore semigroup, or if $G$ acts topologically freely on the minimal boundary space, then $partial mathrm{C}^*_lambda(P)$ coincides with the usual C*-envelope $mathrm{C}^*_{text{env}}(mathcal{A}(P))$ in the sense of Arveson. This covers total orders, finite type and right-angled Artin monoids, the Thompson monoid, multiplicative semigroups of nonzero algebraic integers, and the $ax+b$-semigroups over integral domains that are not a field. In particular, we show that $P$ is an Ore semigroup if and only if there exists a canonical $*$-isomorphism from $partial mathrm{C}^*_lambda(P)$, or from $mathrm{C}^*_{text{env}}(mathcal{A}(P))$, onto $mathrm{C}^*_lambda(G)$. If any of the above holds, then $mathcal{A}(P)$ is shown to be hyperrigid.
We apply Arvesons non-commutative boundary theory to dilate every Toeplitz-Cuntz-Krieger family of a directed graph $G$ to a full Cuntz-Krieger family for $G$. We do this by describing all representations of the Toeplitz algebra $mathcal{T}(G)$ that have unique extension when restricted to the tensor algebra $mathcal{T}_+(G)$. This yields an alternative proof to a result of Katsoulis and Kribs that the $C^*$-envelope of $mathcal T_+(G)$ is the Cuntz-Krieger algebra $mathcal O(G)$. We then generalize our dilation results further, to the context of colored directed graphs, by investigating free products of operator algebras. These generalizations rely on results of independent interest on complete injectivity and a characterization of representations with the unique extension property for free products of operator algebras.
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 $G$.
In this paper, we introduce the Fock space over $mathbb{C}^{infty}$ and obtain an isomorphism between the Fock space over $mathbb{C}^{infty}$ and Bose-Fock space. Based on this isomorphism, we obtain representations of some operators on the Bose-Fock space and answer a question in cite{coburn1985}. As a physical application, we study the Gibbs state.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
