A property of locally compact groups

Helge Glöckner & Jörg Winkelmann

Abstract.

A locally convex topological vector space V is nuclear if the following property is fulfilled: For every open neighbourhood U of the origin there is an open neighbourhood V such that for every finite sequence g1,..,gn of elements in V there are signs si (i.e. each si equals either +1 or -1) such that:

$\displaystyle g_1^{s_1}\cdot\ldots\cdot g_n^{s_n}\in U $

Banaszczyk proposed to generalize this notion of being nuclear from topological vector spaces to (not necessarily commutative) topological groups. In this context he conjectured that the above stated property holds for every locally compact topological group. We prove that every locally compact topological groups fulfills a slightly weaker property, namely that for every such a finite sequence there exists signs si and a permutation of 1,..,n such that the desired relation holds after permuting the factors of the product.
Appeared in:

Advances in Lie Groups. Rsearch and Exposition in Mathematics. 25 205--210 (2002).
Heldermann Verlag, Berlin.

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