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| Anno Accademico |
2022/2023 |
| Corso di Studi |
Dott. - MI (1385) Modelli e Metodi Matematici per l'Ingegneria / Mathematical Models and Methods in Engineering |
| Anno di Corso |
1 |
| Codice Identificativo |
058411 |
| Denominazione Insegnamento |
COMPLETELY POSITIVE MAPS ON OPERATOR ALGEBRAS |
| Tipo Insegnamento |
MONODISCIPLINARE |
| Crediti Formativi Universitari (CFU) |
5.0 |
| Programma sintetico |
Interdisciplinary course: "Completely Positive maps on Operator Algebras"
The course is intended as an introduction to the themes of C* and von Neumann operator algebras and completely maps among them with application to
i) construction of approaches to KMS equilibria in Quantum Statistical Mechanics,
ii) construction of Markovian semigroups in Quantum Probability,
iii) spectral approach to approximation theory of maps (nuclearity, amenability),
iv) construction of channels in Quantum Information.
Sections:
1. Bounded and closed operators in Hilbert spaces
2. Examples: projections, isometries, unitaries, Laplace-Beltrami and Schroedinger operators)
3. Spectral Theorem for compact and general self-adjoint operators
4. C* and von Neumann algebras of operators in Hilbert spaces and their states
5. Examples: quantum channels among matrix algebras, type I von Neumann algebras, Ultra Hyper
Finite algebras of Quantum Spin Systems, CCR/CAR algebras and quasi-free states, type II algebras of free groups, algebra of the Kroneker foliation (noncommutative torus).
6. Completely positive maps between operator algebras. Stinespring Dilation Theorem.
Continuous groups and semigroups and their generators (derivations and dissipations).
7. Constructions of ergodic semigroups converging to equilibria of Quantum Spin Systems
8. Construction of generators of Markovian semigroups of Quantum Levy Processes
9. Spectral growth characterization of nuclearity, weak amenability (Banach-Tarski paradox)
10. von Neumann Gamma property and spectral gap/Poincaré inequality of elementary Dirichlet forms. |
| Settori Scientifico Disciplinari (SSD) |
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Scaglione
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Nome
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Programma dettagliato
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Da (compreso)
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A (escluso)
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Cipriani Fabio Eugenio Giovanni
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