Browsing by Author "Russo, Francesco G"

Now showing 1 - 2 of 2
  • Thumbnail Image
    Item
    A characterization of procyclic groups via complete exterior degree
    (Multidisciplinary Digital Publishing Institute (MDPI), 2024) Rodrigues, Bernardo; Russo, Francesco G
    We describe the nonabelian exterior square (Formula presented.) of a pro-p-group G (with p arbitrary prime) in terms of quotients of free pro-p-groups, providing a new method of construction of (Formula presented.) and new structural results for (Formula presented.). Then, we investigate a generalization of the probability that two randomly chosen elements of G commute: this notion is known as the “complete exterior degree” of a pro-p-group and we will use it to characterize procyclic groups. Among other things, we present a new formula, which simplifies the numerical aspects which are connected with the evaluation of the complete exterior degree
  • Thumbnail Image
    Item
    Ground state representations of topological groups
    (Springer Science and Business Media Deutschland GmbH, 2024) Neeb, Karl-Hermann; Russo, Francesco G
    Let α : R → Aut(G) define a continuous R-action on the topological group G. A unitary representation (π , H) of the extended group G := G α R is called a ground state representation if the unitary one-parameter group π (e, t) = eitH has a nonnegative generator H ≥ 0 and the subspace H0 := ker H of ground states generates H under G. In this paper, we introduce the class of strict ground state representations, where (π , H) and the representation of the subgroup G0 := Fix(α) on H0 have the same commutant. The advantage of this concept is that it permits us to classify strict ground state representations in terms of the corresponding representations of G0. This is particularly effective if the occurring representations of G0 can be characterized intrinsically in terms of concrete positivity conditions. To find such conditions, it is natural to restrict to infinite dimensional Lie groups such as (1) Heisenberg groups (which exhibit examples of non-strict ground state representations); (2) Finite dimensional groups, where highest weight representations provide natural examples; (3) Compact groups, for which our approach provides a new perspective on the classification of unitary representations; (4) Direct limits of compact groups, as a class of examples for which strict ground state representations can be used to classify large classes of unitary representations