In some calculations I am writing up, $\newcommand{\cR}{{\mathcal R}}$ I want to add - as a fairly throwaway remark - that any countable product (= $\ell^\infty$-direct sum) of matrix algebras can be embedded inside $\cR$, the hyperfinite ${\rm II}_1$. I think I have managed to hack out an explicit embedding, by realizing $\cR$ as an infinite tensor product of matrix algebras into which I embed successive blocks of my original product algebra, but the inductive construction seems both tedious and wasteful.

Assuming that I have not made a mistake, and that such product algebras do embed as von Neumann subalgebras of $\cR$, does anyone know of a reference for this, so that

(a) I don't have to tax the reader's patience by slogging through something tedious and routine

(b) I can refer the reader to a better construction than the one I currently have?

Moreover, my guess is that something slightly more general should hold: a **finite** Type ${\rm I}$ with separable predual should embed into $\cR$. Again, does anyone know of a reference, or a counter-example if I have made a mistake?