Lethbridge Analysis Seminar: Berndt Brenken

A C-algebra is a norm closed algebra of bounded operators on a Hilbert space which is closed under taking adjoints. Types of operators such as unitaries, projections, and partial isometries are described. A uni- tary operator implements a -automorphism, and representing groups via unitary operators on Hilbert space lead to considering operator algebras incorporating dynamics from group actions. Similarly one can consider the map implemented by a partial isometry. Although not a homomorphism it is a linear map with order preserving properties. We consider semigroups, a class containing groups and inverse semigroups, and introduce order structures involving matrices. The algebra generated by a partial isometry can be described as the algebra arising from a complete positive map acting on a -semigroup.