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2014 •
In this paper, we answer the question of equivalence, or singularity, of two given quasi-stationary Markov measures on one-sided infinite words, and the corresponding question of equivalence of associated Cuntz algebra Ø_N representations. We do this by associating certain monic representations of Ø_N to quasi-stationary Markov measures, and then proving that equivalence for pairs of measures is decided by unitary equivalence of the corresponding pair of representations.
Ergodic Theory and Dynamical Systems
Representations of Cuntz algebras associated to quasi-stationary Markov measures2014 •
In this paper, we answer the question of equivalence, or singularity, of two given quasi-stationary Markov measures on one-sided infinite words, as well as the corresponding question of equivalence of associated Cuntz algebra${\mathcal{O}}_{N}$-representations. We do this by associating certain monic representations of${\mathcal{O}}_{N}$to quasi-stationary Markov measures and then proving that equivalence for a pair of measures is decided by unitary equivalence of the corresponding pair of representations.
2018 •
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 non-commutative operators; non-commutative harmonic/stochastic analysis. This in turn entails combinatorial, graph operations, and branching laws. The most versatile, of these non-commutative algebras are the Cuntz algebras; denoted $\mathcal{O}_{N}$, $N$ for the number of isometry generators. $N$ is at least 2. Our focus is on the representations of $\mathcal{O}_{N}$. We aim to develop new non-commutative 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...
2021 •
In this note we define one more way of quantization of classical systems. The quantization we consider is an analogue of classical Jordan-Schwinger (J.-S.) map which has been known and used for a long time by physicists. The difference, comparing to J.-S. map, is that we use generators of Cuntz algebra $\mathcal{O}_{\infty}$ (i.e. countable family of mutually orthogonal partial isometries of separable Hilbert space) as a "building blocks" instead of creation-annihilation operators. The resulting scheme satisfies properties similar to Van Hove prequantization i.e. exact conservation of Lie bracket and linearity.
Bulletin of The American Mathematical Society
Harmonic analysis of fractal measures induced by representations of a certain C$^*$-algebra1993 •
Journal of Functional Analysis
Harmonic Analysis and Fractal Limit-Measures Induced by Representations of a Certain C*Algebra1994 •
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1996 •
Journal of Mathematical Physics
Models of q‐algebra representations: q‐integral transforms and ‘‘addition theorems’’1994 •
Pacific Journal of Mathematics
q-canonical commutation relations and stability of the Cuntz algebra1994 •
arXiv (Cornell University)
Quantum deformed algebras : Coherent states and special functions2013 •
Letters in Mathematical Physics
Unbounded Representations of q-Deformation of Cuntz Algebra2008 •
2013 •
Journal of Functional Analysis
Noncommutative Pressure and the Variational Principle in Cuntz–Krieger-type C*-Algebras2002 •
Annales de la faculté des sciences de Toulouse Mathématiques
Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function2014 •
Annales Henri Poincaré
The Asymptotic Behaviour of the Fourier Transforms of Orthogonal Polynomials II: L.I.F.S. Measures and Quantum Mechanics2007 •
Journal of Mathematical Physics
Algebras of distributions suitable for phase‐space quantum mechanics. I1988 •