The Development of Locally Compact Quantum Groups
by
Alfons Van Daele
University of Leuven

Let A be a C^*-algebra. Consider the minimal C^*-tensor product A\otimesA and its multiplier algebra M(A\otimesA). A comultiplication \Phi on A is a non-degenerate ^*-homomorphism from A to M(A\otimesA) satisfying coassociativity (\Phi\otimes\iota)\Phi = (\iota\otimes\Phi)\Phi. The typical example is the C^*-algebra C₀(G) of continuous complex functions tending to 0 at infinity on a locally compact group G. The dual of the product gives rise to a coproduct defined on C₀(G) by \Phi(f)(p, q)=f(pq). We identify C₀(G)\otimesC₀(G) with C₀(G ×G) and M(C₀(G)\otimesC₀(G)) with C_b(G ×G), the C^*-algebra of all bounded continuous complex functions on G×G.

If we think of a C^*-algebra A as a locally compact quantum space, then a pair (A, \Phi) of a C^*-algebra A and a comultiplication \Phi stands for a locally compact quantum semi-group. In this point of view, a locally compact quantum group is a C^*-algebra with a comultiplication with some extra group like properties.

In this talk, I will first discuss the notion of a compact quantum group (i.e. the case where A has an identity). Secondly, I will consider a class of locally compact quantum groups which contains the compact and the discrete quantum groups and which is self-dual (in the sense of Pontryagin duality). Finally, I will define and treat the (general) locally compact quantum groups. In each of these three cases, I will also give some historical comments. In particular, I will indicate how the present theory of locally compact quantum groups evolved out of the two other cases that we consider in this talk.

Date received: April 29, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacw-40.