The structure of order ideals and gaps in the Calkin algebra.

摘要

We study some set theoretic aspects of the algebra of bounded operators on a separable infinite dimensional Hilbert space H denoted by B (H). We also study the poset of projections in Calkin algebra which is the quotient C (H) = B (H)/ K (H), where K (H) is the ideal of compact operators.;In chapter 2 we study the gap structure of the poset P ( C (H)), we exhibit a difference from the gap structure of P (o)/Fin. We prove the existence of an analytic Hausdorff gap in P ( C (H)). As a consequence we obtain that under Todorcevic's Axiom and Martin's Axiom the gap spectrum of P ( C (H)) is strictly bigger than the gap spectrum of P (o)/Fin.;S. Solecki in 1999 characterizes Analytic P-ideals of subsets of o. He proves that an ideal J on o is an analytic P-ideal if and only if it is of the form Exh(&phis;) for some lower semicontinuous submeasure. The set of bounded positive operators of norm at most one BH+ ≤1 is a compact metrizable space with respect to the weak operator topology. It can also be naturally regarded as a partial order. We define P -ideals in BH+ ≤1 and in P ( B (H)). There are well known two sided ideals in B (H) such as the Trace class and Hilbert-Schmidt operators whose intersection with BH+ ≤1 are examples of the ideals in BH+ ≤1 that we define. Furthermore, the notion of ideal in P ( B (H)) generalizes the classical notion of ideal on o. In chapter 5 we prove a non commutative version of Solecki's Theorem and we see that Solecki's result can be derived from it.;In Set Theory two central objects of study are the power set of natural numbers P (o) and the Boolean algebra P (o)/Fin. The partial order of projections in B (H) denoted by P ( B (H)) and the projections in the Calkin algebra P ( C (H)) can be seen as quantum or non-commutative analogues of P (o) and P (o)/Fin respectively. Some statements about P (o)/Fin have a corresponding statement about the projections in the Calkin algebra P ( C (H)).