The fifth question Q5 on the list of problems concerning the hyperfinite II factor that I have listed in my blog entry “The ubiquitous hyperfinite II factor”, back in September 2020, was whether is quasidiagonal or not. This problem was first raised in [B06], then re-iterated as Problem 10.4.9 in the Brown-Ozawa book [Bo08], with… Read More Q5: Is R quasidiagonal?
Recall from prior Blog-entries that the set Q.7 of problems with the title “Classifying smooth and discrete dynamics on ” consists of the following two questions: Q7.1 Classify ergodic automorphisms of the hyperfinite II factor (or equivalently ergodic outer actions of the integers ) up to conjugacy. Q7.2 Classify one parameter actions (flows) of the… Read More Q7.1: ON THE CLASSIFICATION OF CONNES-STORMER BERNOULLI SHIFTS
I have titled the set of questions Q7 on the hyperfinite II factor “Classifying smooth and discrete dynamics on ”. I meant to describe in it the following two problems: Q7.1 Classify single aperiodic automorphisms of the hyperfinite II factor (or equivalently outer actions of the integers ) up to conjugacy. Q7.2 Classify up to… Read More Q7.2: Classification of one parameter groups of automorphisms on the hyperfinite II1 factor
After a long pause in posting on this blog, I resume here my discussion of the eight sets of problems about the hyperfinite II factor that I have initiated in a blog entry on 9/5/2020. I will discuss the fourth set of problems, Q4, whose main question asks whether the free group can be embedded… Read More Q4: Can one embed F_2 discretely into U(R)?
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
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