Memoirs of the American Mathematical Society 2011; 140 pp; softcover Volume: 215 ISBN10: 0821853414 ISBN13: 9780821853412 List Price: US$78 Individual Members: US$46.80 Institutional Members: US$62.40 Order Code: MEMO/215/1010
 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 \(\mathcal{M}{\overline{\otimes}} \mathcal{B}(l^2)\). Table of Contents A von Neumann Algebra Approach to Quantum Metrics by Greg Kuperberg and Nik Weaver  Introduction
 Measurable and quantum relations
 Quantum metrics
 Examples
 Lipschitz operators
 Quantum uniformities
 Bibliography
Quantum Relations by Nik Weaver  Introduction
 Measurable relations
 Quantum relations
 Bibliography
 Notation index
 Subject index
