I'm not quite sure whether its what you want, but a paper of mine, Measurable Categories, on the LANL preprint server at http://www.arxiv.org/abs/math.CT/0309185 develops the theory of measurable categories, and a 2-category of measurable functors and measurable natural transformations. The categories are categories of measurable fields of Hilbert spaces on a measurable space, which for technical reasons must be the Borel space associated to a Lusin space whose points are measureable. The functors between them are induced by measureable fields of Hilbert spaces on the product of source and target and a uniformly $\sigma$-finite conditional distribution of measures with respect to the second projection. The natural transformations are more or less induced by a field of bounded operators between the fields of Hilbert spaces inducing the functors, though one has to handle some technical difficulties involving Radon-Nikodym derivatives to get the definition to work. So far as I know this paper, and a companion in which Crane and I use measurable categories to begin the representation theory of 2-groups (including such things as Baez's Poincare 2-group) is the only progress to date on categorifying things functional-analytic. D.N. Yetter On Tuesday, February 24, 2004, at 05:13 PM, Iain wrote:
I am looking for information on categorifications of C*-algebras. First, I know that there exists a theory for finite dimensional 2-Hilbert spaces due to J.Baez, but is there more recent work on infinite dimensional versions of 2Hilb? Second, has anyone categorified the GNS construction?
-I