Mathematical Sciences: Operator Algebras

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The central theme of this project is the tower of normed matrix spaces that sits over certain kinds of normed linear spaces, for instance subspaces of the algebra of bounded operators on Hilbert space. Completely bounded maps and tensor products will be studied in this setting. Additional aspects of the project include work on Hilbert modules over function algebras, and investigation of invariant subspaces for operators composed of an ergodic transformation and a multiplication. The mathematical research to be pursued here is aimed at fundamental questions in the theory of operator algebras. Operators may be thought of as enriched numbers, obeying the same laws of arithmetic as ordinary numbers with two notable exceptions: multiplication of operators depends upon the order in which the factors are taken, and not every non-zero operator has an inverse. In the finite-dimensional situation, operators are just square matrices of numbers, the basic objects of linear algebra, and even at this stage the richer structure becomes apparent. Infinite-dimensionally, notions of size and distance, as measured by what is called the norm of an operator, become crucial. How these norms behave when subjected to various constructions is one of the foci of this project. Another focus is the study of certain algebras of operators that come from higher-dimensional complex geometry.