The analysis of Lie groups depends to a large extent on their maximal tori. For a compact connected topological group G, the subgroups analogous to the maximal tori are the maximal connected Abelian subgroups. As in Hofmann and Morris [7] we call them maximal protori. We sharpen some results of [7] by showing that each maximal protorus is in a natural way the projective limit of maximal tori Tα in the corresponding Gα, where G= projGα. This sharpened characterization together with some methods of Moskowitz [4], [10] will be used to show that a number of well‐known theorems concerning Lie groups extend in a natural way to all compact connected groups.
For a locally compact group G, the von Neumann kernel, n(G), is the intersection of the kernels of the finite dimensional (continuous) unitary representations of G. In this paper we calculate n(G) explicitly for a general connected locally compact group and for certain classes of non-connected groups.
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