Speaker: George Elliott

Title: Not every functor is manifestly covariant

Abstract: Usually, a functor used for C^*-algebra classification is manifestly covariant---one does not have to prove that it is a functor (for C^*-algebra maps). One interesting exception is the Bratteli diagram, used for classifying AF algebras. In fact, this is a functor, but one has to prove it. (This is the main content of Glimm's classification of UHF algebras, as in this case the Bratteli diagram fairly obviously contains sufficient information to determine the algebra; in the case of more general AF algebras, not only did Bratteli (essentially) have to show that the diagram was a functor, but he had to show, by an intertwining argument that was much more subtle than in the UHF case, that it still contained sufficient information.)

More precisely, by "the" Bratteli diagram is meant a choice of Bratteli diagram---corresponding of course to a chosen inductive limit decomposition of the AF algebra. The interesting fact (not quite explicitly stated by Bratteli) is that, with respect to an arbitrary choice of Bratteli diagram for each AF algebra, and with a suitable notion of morphism for Bratteli diagrams (with respect to which isomorphic diagrams are just ones that are equivalent in the sense of Bratteli), the Bratteli diagram is a functor on the class of AF algebras (with C^*-algebra maps). The proof of this uses the fact proved by Glimm that finite-dimensional C^*-algebras have stable relations (more precisely, are weakly semiprojective), and in particular the fact that close projections are equivalent.

There is a manifestly covariant functor from Bratteli diagrams to ordered abelian groups (namely, just take the inductive limit!). The combined functor, as it happens, is also manifestly covariant (and is the well-known Murray-von Neumann, or K-theory, functor).

Interestingly, while certain inductive limit countable groups, involving products of alternating groups, also admit a Bratteli diagram as a functor, the corresponding ordered group is no longer manifestly a functor. (Although it is of course still a functor, and classifies these particular groups, it does not seem possible to imitate the K-theoretical construction of the C^*-algebra setting.)

Another (so far) non-manifest functor useful for classification is a modification (due to Zhiqiang Li) of the Dadarlat-Loring order structure on the total K-theory group, for dimension drop interval algebras for which the dimension drops at the two ends are not equal. Much as for the inductive limit construction of the ordered group invariant for an AF algebra, one gets an ordered total K-theory functor for inductive limits of (general) dimension drop interval algebras, and in the real rank zero case this (non-manifest) invariant turns out to be complete. (For simple inductive limits, as shown by Jiang and Su, the total K-theory is not needed, just the same invariant as is used for simple limits of circle algebras. In the non-simple case, not surprisingly, the total K-theory is needed, and, somewhat surprisingly, perhaps, the Dadarlat-Loring order structure, in the distinct dimension drop case, is not sufficient.)