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.