For any topological group the dual object is defined as the set of equivalence classes of irreducible unitary representations of equipped with the Fell topology. If is compact, is discrete. In an earlier paper we proved that is discrete for every metrizable precompact group, i.e. a dense subgroup of a compact metrizable group. We generalize this result to the case when is an almost metrizable precompact group.
Let ⟨G,X,α⟩ be a G-space, where G is a non-Archimedean (having a local base at the identity consisting of open subgroups) and second countable topological group, and X is a zero-dimensional compact metrizable space. Let be the natural (evaluation) action of the full group of autohomeomorphisms of the Cantor cube. Then (1) there exists a topological group embedding ; (2) there exists an embedding , equivariant with respect to φ, such that ψ(X) is an equivariant retract of with respect to φ...
An embedding X ⊂ G of a topological space X into a topological group G is called functorial if every homeomorphism of X extends to a continuous group homomorphism of G. It is shown that the interval [0, 1] admits no functorial embedding into a finite-dimensional or metrizable topological group.
Let G be an abelian topological group. The Lévy continuity theorem says that if G is an LCA group, then it has the following property (PL) a sequence of Radon probability measures on G is weakly convergent to a Radon probability measure μ if and only if the corresponding sequence of Fourier transforms is pointwise convergent to the Fourier transform of μ. Boulicaut [Bo] proved that every nuclear locally convex space G has the property (PL). In this paper we prove that the property (PL) is inherited...
A symplectic Lie group is a Lie group with a left-invariant symplectic form. Its Lie algebra structure is that of a quasi-Frobenius Lie algebra. In this note, we identify the groupoid analogue of a symplectic Lie group. We call the aforementioned structure a -symplectic Lie groupoid; the “" is motivated by the fact that each target fiber of a -symplectic Lie groupoid is a symplectic manifold. For a Lie groupoid , we show that there is a one-to-one correspondence between quasi-Frobenius Lie algebroid...