A von Neumann Algebra Approach to Quantum Metrics/Quantum Relations - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook A von Neumann Algebra Approach to Quantum Metrics/Quantum Relations write by Greg Kuperberg. This book was released on 2012. A von Neumann Algebra Approach to Quantum Metrics/Quantum Relations available in PDF, EPUB and Kindle. In A von Neumann Algebra Approach to Quantum Metrics, Kuperberg and Weaver propose a new definition of quantum metric spaces, or W*-metric spaces, in the setting of von Neumann algebras. Their definition effectively reduces to the classical notion in the atomic abelian case, has both concrete and intrinsic characterizations, and admits a wide variety of tractable examples. A natural application and motivation of their theory is a mutual generalization of the standard models of classical and quantum error correction. In Quantum Relations Weaver defines a ``quantum relation'' on a von Neumann algebra $\mathcal{M}\subseteq\mathcal{B}(H)$ to be a weak* closed operator bimodule over its commutant $\mathcal{M}'$. Although this definition is framed in terms of a particular representation of $\mathcal{M}$, it is effectively representation independent. Quantum relations on $l^\infty(X)$ exactly correspond to subsets of $X^2$, i.e., relations on $X$. There is also a good definition of a ``measurable relation'' on a measure space, to which quantum relations partially reduce in the general abelian case. By analogy with the classical setting, Weaver can identify structures such as quantum equivalence relations, quantum partial orders, and quantum graphs, and he can generalize Arveson's fundamental work on weak* closed operator algebras containing a masa to these cases. He is also able to intrinsically characterize the quantum relations on $\mathcal{M}$ in terms of families of projections in $\mathcal{M}{\overline{\otimes}} \mathcal{B}(l^2)$.
A Von Neumann Algebra Approach to Quantum Metrics
A Von Neumann Algebra Approach to Quantum Metrics - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook A Von Neumann Algebra Approach to Quantum Metrics write by Greg Kuperberg. This book was released on 2012. A Von Neumann Algebra Approach to Quantum Metrics available in PDF, EPUB and Kindle. We define a "quantum relation" on a von Neumann algebra M⊆B(H) to be a weak* closed operator bimodule over its commutant M′. Although this definition is framed in terms of a particular representation of M, it is effectively representation independent. Quantum relations on l∞(X) exactly correspond to subsets of X2, i.e., relations on X. There is also a good definition of a "measurable relation" on a measure space, to which quantum relations partially reduce in the general abelian case. By analogy with the classical setting, we can identify structures such as quantum equivalence relations, quantum partial orders, and quantum graphs, and we can generalize Arveson's fundamental work on weak* closed operator algebras containing a masa to these cases. We are also able to intrinsically characterize the quantum relations on M in terms of families of projections in M⊗ ̄B(l2).
A von Neumann Algebra Approach to Quantum Metrics/Quantum Relations
A von Neumann Algebra Approach to Quantum Metrics/Quantum Relations - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook A von Neumann Algebra Approach to Quantum Metrics/Quantum Relations write by Greg Kuperberg. This book was released on 2011. A von Neumann Algebra Approach to Quantum Metrics/Quantum Relations available in PDF, EPUB and Kindle.
Lipschitz Algebras (Second Edition)
Lipschitz Algebras (Second Edition) - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Lipschitz Algebras (Second Edition) write by Nik Weaver. This book was released on 2018-05-14. Lipschitz Algebras (Second Edition) available in PDF, EPUB and Kindle. 'The book is very well-written by one of the leading figures in the subject. It is self-contained, includes relevant recent advances and is enriched by a large number of examples and illustrations. In addition to the general bibliography, each chapter includes a section of notes, which details the authorship of the main results, and provides useful hints for further readings. Undoubtedly, this edition will be received by researchers with the same success as the first one.'European Mathematical SocietyThis is the standard reference on algebras of Lipschitz functions, written by the leading figure in the field. The second edition includes new chapters on nonlinear Banach space geometry, differentiability in metric measure spaces, and quantum metrics. This latest material reflects the importance of spaces of Lipschitz functions in a diverse range of current research directions. Every functional analyst should have some knowledge of this subject.
Dimer Models and Calabi-Yau Algebras
Dimer Models and Calabi-Yau Algebras - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Dimer Models and Calabi-Yau Algebras write by Nathan Broomhead. This book was released on 2012-01-23. Dimer Models and Calabi-Yau Algebras available in PDF, EPUB and Kindle. In this article the author uses techniques from algebraic geometry and homological algebra, together with ideas from string theory to construct a class of 3-dimensional Calabi-Yau algebras. The Calabi-Yau property appears throughout geometry and string theory and is increasingly being studied in algebra. He further shows that the algebras constructed are examples of non-commutative crepant resolutions (NCCRs), in the sense of Van den Bergh, of Gorenstein affine toric threefolds. Dimer models, first studied in theoretical physics, give a way of writing down a class of non-commutative algebras, as the path algebra of a quiver with relations obtained from a `superpotential'. Some examples are Calabi-Yau and some are not. The author considers two types of `consistency' conditions on dimer models, and shows that a `geometrically consistent' dimer model is `algebraically consistent'. He proves that the algebras obtained from algebraically consistent dimer models are 3-dimensional Calabi-Yau algebras. This is the key step which allows him to prove that these algebras are NCCRs of the Gorenstein affine toric threefolds associated to the dimer models.
