Master's Essay Lecture

Sam Harris, Pure Mathematics, University of Waterloo

"Kadison Similarity Problem and the Similarity Degree"

Kadison's Similarity Problem is a well-known open problem in the theory of representations of C*-algebras, dating back to the 1950s.  The problem asks whether every bounded homomorphism from a C*-algebra into B(H) is similar to a *-homomorphism.  While positive solutions have been found in some special cases, the problem remains open in general.  The main point of discussion will be the theory of similarity degree of unital operator algebras, formulated by Gilles Pisier.  We will see that the similarity degree is intimately related to Kadison's Similarity Problem; moreover, we can use similarity degree to obtain a promising avenue of approach to the problem.