Group representations and construction of minimal topological groups
For every continuous biadditive mapping \(\omega : E \times F \to A\) with Abelian topological groups \(E\), \(F\) and \(A\), the author defines a corresponding topological group \(M(\omega)\) and proves its minimality under natural restrictions. With the use of the evaluation mapping \(G \times G^* \to \mathbb{T}\) provided by the Pontryagin-van Kampen duality and the canonical duality \(E \times E^* \to \mathbb{R}\) for a normed space \(E\), it is shown that every locally compact Abelian group is a group retract of a minimal locally compact group. Furthermore, every Abelian topological group is a quotient of a perfectly minimal group. The latter gives a positive answer to a question of Arkhangel'skij for the abelian case. The author also solves a problem of Stoyanov by constructing a perfectly minimal group of countable weight which is not topologically isomorphic to any subgroup of the unitary group \(U(\mathbb{H})\) for a Hilbert space \(\mathbb{H}\).
scientific article; zbMATH DE number 735667
scientific article