Abstract

Let A be a -algebra that is graded by a group G with fibres A_g for g in G.  Assume that we have found a C-algebra B_e whose “representations” are “equivalent” to the “integrable” “representations” of the unit fibre A_e.  Call a “representation” of A “integrable”, if its restriction to A_e is “integrable”.  Under some assumptions, the “integrable” “representations” of A are “equivalent ” to the “representations” of a certain C-algebra B constructed from B_e and the graded -algebra A.  The C-algebra B is the section C-algebra of a Fell bundle over G.  The words in quotation marks have to be interpreted carefully to make this true.  In particular, representations must be understood to take place on Hilbert modules, not just Hilbert spaces, and the equivalence is required natural with respect to induction of representations and isometric intertwiners.  Under some commutativity assumptions, the main result of my lecture has been proved by Savchuk and Schmüdgen, who also give several examples.  A sample of the result concerns Weyl algebras and twisted Weyl algebras in countably many generators.  These come with a canonical grading by the free Abelian group on countably many generators.