A coalgebraic description of the hyperfinite II_1 revisited

Question

Back here I was asking for a coalgebraic characterisation of the hyperfinite $II_1$ factor. Recall the latter's construction by forming the inductive limit of a chain of matrix algebras $R \to M_2(R) \to M_{2^2}(R) \to ...$, where a matrix is sent to two copies of itself placed in the diagonal blocks, zero elsewhere. Then completion in the weak topology gives the hyperfinite factor.

The trace is halved each step along the chain. Traces for projections in the inductive limit are dyadic rationals in [0, 1], while in the hyperfinite factor they are the whole real interval [0, 1]. Now, the dyadic rationals are the initial algebra for the endofunctor on Bipointed Set, $X \mapsto X \vee X$, identifying the second point of the first copy with the first point of the second copy. The [0, 1] interval is the terminal coalgebra for the same functor and the Dedekind and Cauchy completion of the initial algebra. Perhaps results such as Adamek's Final Coalgebras are Ideal Completions of Initial Algebras may be extended here.

This put me on the quest of characterising the hyperfinite factor coalgebraically. So now the question: what relevant facts are known about the endofunctor on algebras over the reals: $X \to M_2(X)$? I believe the hyperfinite factor is a fixed point. Presumably there is a need to be clear over isomorphism versus Morita equivalence. Might it be that the hyperfinite factor is the greatest fixed point up to isomorphism? Maybe I should be looking in the category of a certain kind of algebra.

Answer 1

This is an interesting question, but the motivation is a bit misaligned. $C^*$ algebras are a non-commutative or quantum generalization of compact Hausdorff spaces and von Neumann algebras are a non-commutative or quantum generalization of (not too unreasonable) measurable spaces. However, both of these generalizations are contravariant. Your motivation is a covariant comparison between von Neumann algebras and topological spaces, which is problematic.

The von Neumann algebra or $C^*$-algebra $M_2(\mathbb{C})$ is now famously known as a "qubit"; it is a great non-commutative analogue of $\mathbb{C} \oplus \mathbb{C}$, which is of course the complex functional algebra of a classical bit. The endofunctor that you ask about is a geometric product of $X$ and a qubit, and the morphism in your question is a geometric projection back to $X$ with a qubit fiber. So the completion that you ask about is thus a geometric product with a quantum Cantor set. I forget what the fiber is called in the $C^*$-algebra setting, but I remember that, unlike a classical Cantor set, its isomorphism type depends on the sizes of the matrices. In the von Neumann case, this quantum Cantor set is interpreted as a measurable space, and then it is always the hyperfinite $II_1$ factor and does not depend on the matrix sizes.

I think that the hyperfinite factor is not the only fixed point of tensoring with a qubit. Let $S$ be any set, let $F$ be the set of functions from $S$ to a bit (or any finite set), and then let $M$ be the von Neumann closure of the local operators on $\ell^2(F)$. Here a local operator is one that affects only finitely many values of $f \in F$. If $S$ is an infinite set of any cardinality, then $M$ goes to itself when you tensor it with a qubit.