1996 Stelson Lecture - Gilles Pisier
Gilles Pisier got his Ph.D. under the direction of Laurent Schwartz in 1977. He currently holds the Owen chair of mathematics at Texas A&M and he is also a professor at Paris VI. Pisier's interests include functional analysis, probability, harmonic analysis, operator theory, and C*-algebras. Pisier was a primary speaker in the Functional Analysis Section of the 1983 ICM Congress in Warsaw, Poland. Pisier's achievements have been recognized by many prizes and distinctions, such as the Grands Prix de l'Academie des Sciences de Paris 1992, and Salem Prize 1979. He is a member of the French Academy of Sciences.
Similarity problems for representations of groups or operator algebras
We will survey some recent progress on certain similarity problems which can be posed in various contexts. In the group case the problem has a long history going back to Sz. Nagy and Dixmier and may be formulated as: on which groups G is every uniformly bounded representation p : G Æ B(H) "unitarizable" (i.e. similar to a unitary one)? For the disc algebra, the analogous problem was posed by Halmos in 1970 and was recently solved: there is a polynomially bounded operator which is not similar to a contraction. For C*-algebras, the corresponding problem (posed in 1955 by Kadison) is still open: is every bounded homomorphism on a C*-algebra similar to a contractive one? The more recent concept of a "completely bounded map" allows us to treat these problems in a common framework. In each case (either groups, uniform algebras, or C*-algebras), the appropriate notion of "amenability" plays an important role in the recent developments that will be presented.