October 2nd, 1953 – July 17th, 2022

In the introduction of his 1976 “Classification of injective factors” paper (see bottom of page 73 in [C76]), Alain Connes makes the following remark: Another remarkable property of the factor is that if is any self-adjoint subset, the von Neumann subalgebra of generated by can be characterized by a bicommutation property, analogue to the bicommutation… Read More Q3: Bicommutant characterization of the hyperfinte II1 factor

Q2: CE and Vanishing 2-Cohomology for I resume here my comments on the eight questions pertaining to the hyperfinite II factor listed in an early September 2020 blog-entry. I have labelled the questions Q1-Q8 and have already discussed Q1. I will now discuss Q2, which is about an equivalent formulation of the Connes Embedding (CE)… Read More The Ubiquitous Hyperfinite II_1 factor

It has been almost four months since Vaughan Jones untimely death, yet I am still struggling to come to terms with the terrible reality that he is no longer with us, and to cope with the lasting pain of losing one of my closest friends. He would have been 68 this December 31st,  the very last… Read More Vaughan and the gift of friendship

By an “-ergodicity question” I mean a question about constructing embeddings of the hyperfinite II factor into another von Neumann factor that’s ergodic, in the sense that the action is ergodic, while at the same time verifies various other properties. I favor the term “ergodic embedding”, rather than previous terminology “irreducible inclusion”, or “inclusion with… Read More The Ubiquitous Hyperfinite II_1 Factor, Q1: R-ergodicity questions

Ever since it has been defined by Murray and von Neumann in their famous “Rings of operators” papers some eight decades ago ([MvN36]-[MvN43]), the hyperfinite II factor played a central role in operator algebras (both C and W). It is certainly a mathematical object of fundamental importance. Its construction as a limit of dyadic matrices,… Read More The Ubiquitous Hyperfinite II_1 Factor

It is well known that while in the first of their “rings-cycle”, [MvN36], Murray and von Neumann have explicitly formulated several problems (all of which having been clarified by now), in their subsequent papers they do not formally state any. There are however several problems that, while not spelled out as such, do come across… Read More Revisiting “On Rings of Operators IV” and the problems therein

1. The tightness conjectures I wanted to popularize here a conjecture that I have formulated in (5.1(b) of [P18]), asserting that if a II factor is stably single generated (SSG), i.e., if is single generated as a von Neumann algebra for any , then has an –tight decomposition, meaning that it contains hyperfinite subfactors such… Read More Bimodule decomposition of a II-1 factor and the SSG property