Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userQuasi Hyperrigidity and Weak Peak Points for Non-Commutative Operator Systems
67
0
0.0
(
0
)
Added by
Narayanan Namboodiri NEELAKANTAN
Publication date
2016
fields
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
In this article, we introduce the notions of weak boundary repre- sentation, quasi hyperrigidity and weak peak points in the non-commutative setting for operator systems in C* algebras. An analogue of Saskin theorem relating quasi hyperrigidity and weak Choquet boundary for particular classes of C* algebras is proved. We also show that, if an irreducible representation is a weak boundary representation and weak peak then it is a boundary repre- sentation. Several examples are provided to illustrate these notions. It is also observed that isometries on Hilbert spaces play an important role in the study of certain operator systems.
rate research
Read More
We adapt the theory of chordal Loewner chains to the operator-valued matricial upper-half plane over a $C^*$-algebra $mathcal{A}$. We define an $mathcal{A}$-valued chordal Loewner chain as a subordination chain of analytic self-maps of the $mathcal{A}$-valued upper half-plane, such that each $F_t$ is the reciprocal Cauchy transform of an $mathcal{A}$-valued law $mu_t$, such that the mean and variance of $mu_t$ are continuous functions of $t$. We relate $mathcal{A}$-valued Loewner chains to processes with $mathcal{A}$-valued free or monotone independent independent increments just as was done in the scalar case by Bauer (Lowners equation from a non-commutative probability perspective, J. Theoretical Prob., 2004) and Schei{ss}inger (The Chordal Loewner Equation and Monotone Probability Theory, Inf. Dim. Anal., Quantum Probability, and Related Topics, 2017). We show that the Loewner equation $partial_t F_t(z) = DF_t(z)[V_t(z)]$, when interpreted in a certain distributional sense, defines a bijection between Lipschitz mean-zero Loewner chains $F_t$ and vector fields $V_t(z)$ of the form $V_t(z) = -G_{ u_t}(z)$ where $ u_t$ is a generalized $mathcal{A}$-valued law. Based on the Loewner equation, we derive a combinatorial expression for the moments of $mu_t$ in terms of $ u_t$. We also construct non-commutative random variables on an operator-valued monotone Fock space which realize the laws $mu_t$. Finally, we prove a version of the monotone central limit theorem which describes the behavior of $F_t$ as $t to +infty$ when $ u_t$ has uniformly bounded support.
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.
Let $S = (S_1, ldots, S_d)$ denote the compression of the $d$-shift to the complement of a homogeneous ideal $I$ of $mathbb{C}[z_1, ldots, z_d]$. Arveson conjectured that $S$ is essentially normal. In this paper, we establish new results supporting this conjecture, and connect the notion of essential normality to the theory of the C*-envelope and the noncommutative Choquet boundary. The unital norm closed algebra $mathcal{B}_I$ generated by $S_1,ldots,S_d$ modulo the compact operators is shown to be completely isometrically isomorphic to the uniform algebra generated by polynomials on $overline{V} := overline{mathcal{Z}(I) cap mathbb{B}_d}$, where $mathcal{Z}(I)$ is the variety corresponding to $I$. Consequently, the essential norm of an element in $mathcal{B}_I$ is equal to the sup norm of its Gelfand transform, and the C*-envelope of $mathcal{B}_I$ is identified as the algebra of continuous functions on $overline{V} cap partial mathbb{B}_d$, which means it is a complete invariant of the topology of the variety determined by $I$ in the ball. Motivated by this determination of the C*-envelope of $mathcal{B}_I$, we suggest a new, more qualitative approach to the problem of essential normality. We prove the tuple $S$ is essentially normal if and only if it is hyperrigid as the generating set of a C*-algebra, which is a property closely connected to Arvesons notion of a boundary representation. We show that most of our results hold in a much more general setting. In particular, for most of our results, the ideal $I$ can be replaced by an arbitrary (not necessarily homogeneous) invariant subspace of the $d$-shift.
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.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
