Primary literature? What’s that, Gramps?
Prompted by something that I saw today, and by the same issue that was bugging me in this old post, I have a new idea for a series of blog posts. Though, like all the previous ideas, it will no doubt remain unwritten, thus demonstrating something about vaulting ambition which o’erleaps itself, or something.
Anyway: the starting point for this planned ramble/rant/exposition is the following point.
If you present the known result that for a discrete group G, amenability of G is equivalent to injectivity of VN(G) as a von Neumann algebra, and then say (paraphrasing)
This can fail for non-discrete groups. For instance, whenever G is a connected Lie group, Connes proved that VN(G) is injective. In particular VN(SL(2,R)) is injective…
then I have two suggestions or remarks.
(a) Although the result is indeed first stated and proved in Connes’s paper on injective factors, it is really done as an observation en passant. Moreover, the kernel of the result is really contained in an older paper of Dixmier on the von Neumann algebras of connected Lie groups (which itself apparently needs, in one place, to be patched by a subsequent paper of Pukanszky).
(b) If the example you want to highlight is SL(2,R), then invoking that part of Connes’s paper, or indeed the results of Dixmier, is massive overkill. Semisimple Lie groups are Type I — I am told this is originally due to Harish-Chandra. In the case of semisimple matrix groups, a short proof was later given by Stinespring, streamlining older arguments of Godement. From this it follows that VN(SL2,R)) is hyperfinite — in fact, this is really what Stinespring proves — and it is easy to see that hyperfinite von Neumann algebras are injective.
Which is to say: sometimes “proof by appeal to Fields-medal winning paper” is not just unsatisfying, but it might even be obfuscatory.
edit 09-02-11: typo corrected