Topological realizations of Calkin algebras on Fréchet domains of unbounded operator algebras
Let D be a dense linear subspace of a complex Hilbert space H, and let \(L^+(D)\) be the *-algebra of all linear transformations a on H such that \(D\subset D(a^*)\) and \(a^*D\subset D\). An Op*-algebra on D is a *-subalgebra of \(L^+(D)\). The graph topology from \(L^+(D)\) on D induces a uniform topology on \(L^+(D)\). Let C(D) denote the *-ideal of all operators in \(L^+(D)\) which map each bounded subset of D into a relatively compact subset of D. Then the Calkin algebra \(L^+(D)/C(D)\) with the quotient topology is considered. For each free ultrafilter of \({\mathbb{N}}^ a \)faithful *-representation of \(L^+(D)/C(D)\) with continuous inverse is constructed such that the image is an Op*-algebra. The continuity of this representation is studied, and in connection with former results of the author's [Rep. Math. Phys. 17, 359-371 (1980; Zbl 0475.47030)], where D is subject to some additional assumptions, necessary and sufficient continuity conditions are proved.
scientific article; zbMATH DE number 4028227