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An algebraic framework for group duality. (English) Zbl 0933.16043
A multiplier Hopf algebra $$A$$ is a generalization of the notion of a Hopf algebra obtained if we drop the assumption that $$A$$ has an identity and if we allow the comultiplication $$\Delta$$ to have values in the multiplier algebra $$M(A\otimes A)$$. The motivating example is the algebra of complex functions with finite support on a group with comultiplication defined as the dual to the product in the group. There is a natural notion of left and right invariance for linear functionals for multiplier Hopf algebras (the analog of integrals in Hopf algebra theory). It is shown that if such invariant functionals exist, they are unique up to a scalar and faithful. If the multiplier Hopf algebra $$(A,\Delta)$$ is regular (i.e., it has invertible antipode) and has invariant functionals, a natural dual $$(\widehat A,\widehat\Delta)$$ is constructed, and this is also proved to be a regular multiplier Hopf algebra with invariant functionals. There is a natural isomorphism between the dual of $$(\widehat A,\widehat\Delta)$$ and $$(A,\Delta)$$. Several aspects of abstract harmonic analysis like the Fourier transform and Plancherel’s formula can be generalized in this framework. This duality generalizes the usual duality for finite dimensional Hopf algebras, and also the duality between discrete quantum groups and compact quantum groups. Thus the duality between compact Abelian groups and discrete Abelian groups is a particular case. An extension of this duality is obtained in the non-Abelian case, but within one category. Several nice examples of Hopf algebras are presented.

##### MSC:
 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 17B37 Quantum groups (quantized enveloping algebras) and related deformations 22D35 Duality theorems for locally compact groups 46L89 Other “noncommutative” mathematics based on $$C^*$$-algebra theory 43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
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