Bounded elements of C*-inductive locally convex spaces

Abstract

The notion of bounded element of C*-inductive locally convex spaces (or C*- inductive partial *-algebras) is introduced and discussed in two ways: The first one takes into account the inductive structure provided by certain families of C*-algebras; the second one is linked to the natural order of these spaces. A particular attention is devoted to the relevant instance provided by the space of continuous linear maps acting in a rigged Hilbert space

Topological aspects of quasi *-algebras with sufficiently many *-representations

“…Proposition 5. 4 Let M be sufficient and suppose that τ F = τ F . Then, A[τ F ] is a locally convex space with the following properties:…”

“…The study of this structure and the analysis performed also in [1,4,5,7,9] made it clear that the most regular situation occurs when the locally convex quasi *-algebra (A[τ ], A 0 ) under consideration possesses a sufficiently rich family I A 0 (A) of invariant positive sesquilinear forms on A × A (see below for definitions); they allow a GNS construction similar to that defined by a positive linear functional on a *-algebra A 0 . The basic idea where this paper moves from is to consider a quasi *-algebra (A, A 0 ) where one can introduce a locally convex topology by means of the set of sesquilinear forms I A 0 (A) itself.…”

Singular Perturbations and Operators in Rigged Hilbert Spaces

“…We recall that if X ∈ L B (D, D × ) such that X ≥ 0, then X A ≥ 0 for any A ∈ W (X) [7]. S. di Bella and C. Trapani…”

“…Hence, X ∈ B(H) (see also [7,Theorem 3.5]) and then it is regular. (ii): Let {ξ n } be a sequence of vectors of D such that ξ n → 0 and θ X (ξ n − ξ m , ξ n − ξ m ) → 0 as m, n → +∞.…”

Groups, Jacobi functions, and rigged Hilbert spaces

“…The action of F ∈ Ψ × on each vector ϕ ∈ Ψ gives a complex number often denoted as ϕ|F , following the Dirac notation. It is not our intention here to review properties and applications of RHS, the reader may visit for this purpose the extensive literature on the subject [5][6][7][8][9][10][11][12][13], but we would like to mention that recently they have received renewed attention, see [14][15][16][17][18][19], including their relations with partial inner products [20] and frames [21]. We limit us to remember that our primary purpose is to provide of mathematical sense to the Dirac formulation of quantum mechanics and its continuous basis of eigenvectors of observables.…”