Representations of Multiplier Algebras in Spaces of Completely Bounded Maps

If G is a locally compact group, then the measure algebra M(G) and the completely bounded multipliers of the Fourier algebra McbA(G) can be seen to be dual objects to one another in a sense which generalises Pontryagin duality for abelian groups. We explore this duality in terms of representations of these algebras in spaces of completely bounded maps. This article is intended to give a tour of the growing body of work on representing multiplier algebras in spaces of maps on the C*-algebra B(H), where H is a Hilbert space. The ideas here begin in the 1980s with the work of Størmer [37], on representations of measure algebras of abelian groups; work of Ghahramani [12], on a representation of group algebras for general groups; and unpublished work of Haagerup [13], on a representation of completely bounded multipliers of the Fourier algebra of a general locally compact group. Størmer’s and Ghahramani’s results have been expanded upon and improved in the work of Neufang [22], Neufang, Ruan and Spronk [23] and Smith and Spronk [33]. The results of Haagerup were rediscovered by Spronk [34, 35]. These ideas are currently being reinterpreted by Neufang, Ruan and Spronk, and may yield results on multipliers of “quantum groups”. Date: March 2, 2005. 2000 Mathematics Subject Classification. Primary 46L07, 22D20, 43A20; Secondary 22D10,22D25.