Noncommutative Boundaries Arising in Dynamics and Representations of the Cuntz Relations

Palle Jorgensen; James Tian

doi:10.1080/01630563.2019.1665544

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Noncommutative Boundaries Arising in Dynamics and Representations of the Cuntz Relations

Journal article

Peer reviewed

Noncommutative Boundaries Arising in Dynamics and Representations of the Cuntz Relations

Palle Jorgensen and James Tian

Numerical functional analysis and optimization, Vol.41(5), pp.571-620

04/03/2020

DOI: 10.1080/01630563.2019.1665544

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Abstract

In a number of recent papers, the idea of generalized boundaries has found use in fractal and in multiresolution analysis; many of the papers having a focus on specific examples. Parallel with this new insight, and motivated by quantum probability, there has also been much research which seeks to study fractal and multiresolution structures with the use of certain systems of noncommutative operators; noncommutative harmonic/stochastic analysis. This in turn entails combinatorial, graph operations, and branching laws. The most versatile, of these noncommutative algebras are the Cuntz algebras; denoted N for the number of isometry generators. N is at least 2. Our focus is on the representations of We aim to develop new noncommutative tools, involving both representation theory and stochastic processes. They serve to connect these parallel developments.In outline, boundaries, Poisson, or Martin, are certain measure spaces (often associated to random walk models), designed to encode the asymptotic behavior, e.g., how trajectories diverge when the number of steps goes to infinity. We stress that our present boundaries (commutative or noncommutative) are purely measure-theoretical objects. Although, as we show, in some cases our boundaries may be compared with more familiar topological boundaries.

attractors

bit-representations

boundaries

boundary representations

Cuntz algebras

dynamics

endomorphisms

finitely correlated states

fractals

fractional calculus

Hilbert space of equivalence classes

iterated function system

Kolmogorov consistency

martingales

noncommutative Markov chains

orthogonality

path space measures

Primary: 47L60

quantum probability

representations

reproducing kernel Hilbert space

Secondary: 60D05

self-reproducing kernel

Details

Title: Subtitle: Noncommutative Boundaries Arising in Dynamics and Representations of the Cuntz Relations
Creators: Palle Jorgensen - University of Iowa
James Tian - Mathematical Reviews, Ann Arbor, Michigan, USA
Resource Type: Journal article
Publication Details: Numerical functional analysis and optimization, Vol.41(5), pp.571-620
Publisher: Taylor & Francis
DOI: 10.1080/01630563.2019.1665544
ISSN: 0163-0563
eISSN: 1532-2467
Language: English
Date published: 04/03/2020
Academic Unit: Mathematics
Record Identifier: 9984240876402771

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