Only the most basic knowledge of functional analysis will be assumed. In the first lecture I will go over the basics of operators and operator algebras on a (complex) Hilbert space and in particular the spectral theory and GNS representations. Here GNS (Gelfand-Naimark-Segal) stands for an operator-algebraist's way of saying that the class of concrete C* algebras is axiomatizable in an appropriate logic.
A special attention will be given to the Calkin algebra, C(H), on a separable infinite-dimensional Hilbert space H. The Calkin algebra is the quotient of the algebra of all bounded linear operators on H over its ideal of compact operators.
After studying the lattice of projections in C(H) and showing some of its amusing properties (e.g., that it is not a lattice) we shall move to automorphisms. I will construct an outer automorphism of the Calkin algebra using the Continuum Hypothesis (Phillips-Weaver) and outline the fact that assuming Todorcevic's (open coloring) Axiom all automorphisms are inner (Farah).
I will also construct (using less than CH) a pure state on the algebra of bounded operators on a Hilbert space not diagonalizable by any abelian subalgebra (Akemann-Weaver).
After other selected topics, the talks will end by discussing an enticing list of open problems.