Mathematics Sciences: Applications of Complete Boundedness to Operator Algebras

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9401540 Smith During the past few years the theory of operator algebras, both selfadjoint and non-selfadjoint has evolved at a rapid pace. Recent developments by many researchers are providing further ties between the selfadjoint and non-selfadjoint aspects of the theory. The proposed work focuses on the theory of tensor products of operator algebras, complete boundedness for operator algebras, cohomology theory, and its connection with algebraic topology. Schur multiplier theory, the structure of derivations and automorphisms of non-selfadjoint and selfadjoint algebras and compact ideal perturbations in semifinite algebras will also be studied. Operator theory is concerned with infinite generalizations of finite matrices. When operator multiplication and addition are defined on a collection of operators the resulting collection is called an operator algebra. Cohomology is a theory that examines when local pieces can be put together to form a global object. This project is concerned with cohomology on operator algebras. The goal is to contribute to an old conjecture which asserts that certain cohomologies of operator algebras are trivial. This would mean that local objects can always be pieced together into a global object. ***