MATHEMATICAL ANALYSIS WITH APPLICATIONS Seminar

 

PROGRAM

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Plan rada Seminar iz matematičke analize sa primenama za MART 2023.

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Četvrtak, 23.03.2023. u 16:00, Online
Camillo Trapani, Dipartimento di Matematica e Informatica, Universita di Palermo, Italy
SOME TOPOLOGICAL ASPECTS OF QUASI *-ALGEBRAS OF UNBOUNDED OPERATORS
The studies of locally convex quasi *-algebras, as it has developed in the last forty years, makes of them a full-fledged theory. In this lecture we quickly sketch the basic aspects of them focusing, in particular, on their representation theory, whose natural arena is constituted by families of unbounded operators in Hilbert space. Namely, $\mathrm O^*$-algebras and partial $\mathrm O^*$-algebras, the natural generalization of bounded operator algebras to the unbounded case, are directly involved. For this reason a particular attention deserve quasi *-algebras of unbounded operators. They appear often as the completion of a *-algebra $\mathfrak M_0$ of bounded operators in a Hilbert space $\mathcal H$ under a certain topology $\tau$ (e.g. the so called uniform topology $\tau^*_u$) of the space
\[
\mathcal L^{\dagger}(\mathcal{D,H})=\{X\text{ closable},\ D(X)=\mathcal D;\ D(X^*)\supset\mathcal D\}
\]
where $\mathcal D$ is a dense subspace of $\mathcal H$.
Some results concerning locally convex quasi *-algebras that allow a faithful representation as a uniformly complete quasi *algebra of operators will also be presented in G. Bellomonte, C. Trapani, Topological aspects of quasi *-algebras with sufficiently many *-representations (in preparation).

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