ITPFI factors with restricted growth

Question

You will have to excuse my lack of understanding of von Neumann algebras. I do not know if my question is trivial or nonsensical.

There are ITPFI factors of bounded type, and ITPFI factors of unbounded type. But is there anything in between? Specifically, is there a name for the class of factors which can be represented as $$ \otimes_{n=1}^\infty (M_n, \nu_n) $$ Where $M_n$ is of type $I_{a_n}$, and some growth is permitted in $a_n$, such as $a_n \in O(n)$ or $a_n \in o(2^n)$?

Answer 1

See

Thierry Giordano and David Handelman, Matrix-valued random walks and variations on property AT, Münster J. Math 1 (2008) 15-72;

also the original paper by Giordano and Skandalis (cited in the one above), which I presume you are aware of. Bounded ITPF1 correspond to random walks corresponding to a sequence of Poisson distributions. The size of the $a_n$ is largely irrelevant.