A Duality of Locally Compact Groups That Does Not Involve the Haar Measure - Kuznetsova, Yulia (MATHEMATICA SCANDINAVICA, 2015)
We present a simple and intuitive framework for duality of locally compacts groups, which is not based on the Haar measure. This is a map, functorial on a non-degenerate subcategory, on the category of coinvolutive Hopf C^*-algebras, and a similar map on the category of coinvolutive Hopf-von Neumann algebras. In the C^*-version, this functor sends C_0(G) to C^*(G) and vice versa, for every locally compact group G. As opposed to preceding approaches, there is an explicit description of commutative and co-commutative algebras in the range of this map (without assumption of being isomorphic to their bidual): these algebras have the form C_0(G) or C^*(G) respectively, where G is a locally compact group. The von Neumann version of the functor puts into duality, in the group case, the enveloping von Neumann algebras of the algebras above: C_0(G)^** and C^*(G)^**.