Topological Realizations of Calkin Algebras on Frechet Domains of Unbounded Operator Algebras

Abstract

Let $\mathcal{D}$ be a dense linear subspace of a separable Hilbert space and let ${\mathcal{L}}^{+}(\mathcal{D})$ be the maximal Op*-algebra on $\mathcal{D}$ endowed with the uniform topology ${\tau }_{\mathcal{D}}$. Suppose $\mathcal{D}$ is a Frechet space with respect to the graph topology of ${\mathcal{L}}^{+}(\mathcal{D})$. Let $\mathcal{C}(\mathcal{D})$ denote the two-sided *-ideal of all operators in ${\mathcal{L}}^{+}(\mathcal{D})$ which map bounded subsets of $\mathcal{D}$ into relatively compact subsets. We study the question of when the quotient algebra $\mathcal{A}(\mathcal{D}):={\mathcal{L}}^{+}(\mathcal{D})/\mathcal{C}(\mathcal{D})$, endowed with the quotient topology, has a topological realization as an Op*-algebra.