It is proved that the statement "there exists a counterexample to Naimark's problem which is generated by aleph (1) elements" is undecidable in standard set theory.Expand

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… Expand

Abstract Basic aspects of differential geometry can be extended to various non-classical settings: Lipschitz manifolds, rectifiable sets, sub-Riemannian manifolds, Banach manifolds, Wiener space,… Expand

Let M be an abelian von Neumann algebra andEan abelian operator M-bimodule. Then the domain of any ultraweakly closed derivation from M intoEis a Lipschitz algebra. Conversely, every Lipschitz… Expand

We propose a new definition of quantum metric spaces, or W*-metric spaces, in the setting of von Neumann algebras. Our definition effectively reduces to the classical notion in the atomic abelian… Expand

These notes are based on the six-hour Appalachian Set Theory workshop given by Ilijas Farah on February 9th, 2008 at Carnegie Mellon University. The first half of the workshop (Sections 1–4)… Expand

It is proved that ℬ(H) has a pure state whose restriction to any masa is not pure and this resolves negatively old conjectures of Kadison and Singer and of Anderson.Expand

The “noncommutative graphs” which arise in quantum error correction are a special case of the quantum relations introduced in Weaver (Quantum relations. Mem Am Math Soc 215(v–vi):81–140, 2012). We… Expand