An operator algebra is an algebra of bounded linear operators acting on a Hilbert space that is closed in a certain norm topology. When that algebra is closed with respect to the adjoint operation (an abstract conjugate transpose), we call it a C*-algebra. The prototypical examples of C*-algebras include the ring of n x n matrices over the complex numbers and the ring of complex-valued continuous functions on a compact Hausdorff space. The latter example gives an algebraic perspective for studying topological dynamics. In particular, one can build an operator algebra called a crossed product that encodes the dynamical information of a group of homeomorphisms acting on a topological space.
In the 1960s, W. Arveson determined that the action of a homeomorphism on a topological space is better encoded in a crossed product via the action of a semigroup on that space, rather than a group, which led to many important results in operator algebra theory.
I will discuss how and why operator algebraists have been returning in recent years to crossed products in the context of groups acting on non-adjoint closed operator algebras, and I will discuss a recent partial solution to when dynamics are encoded fully in this crossed product context.