Daniel Spiegel
Harvard University
Title
Topological Adventures in Pure State Space
Abstract
A pure state of a C*-algebra is a generalization of a ray in a Hilbert space. Furthermore, one may act on a pure state with a unitary element to obtain another pure state. In the Hilbert space setting, the topology of the space of rays and the structure of the action by unitaries is a well-understood and beautiful topic. For example, the unitary group of a Hilbert space forms a fiber bundle over the projective Hilbert space.
In the first part of my talk, I will equip the pure state space of a C*-algebra with the norm topology and show how fiber bundles from the Hilbert space story generalize nicely to the C*-algebraic setting. In the second part, we will investigate the weak* topology on the space of pure states (which is in general not analogous to the topology on projective Hilbert space) and see that weak*-continuous families of pure states can be deformed in surprising ways by norm-continuous families of unitaries. In this vein, I will discuss some ideas behind our proof that the pure state space is weakly contractible with the weak* topology for a broad class of C*-algebras. The hero of both parts of the talk will be Ernest Michael (1925-2013) and his theory of continuous selections, which I will explain.