Mathematical Sciences: Operator Algebras and Operator Theory

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The mathematical research project of Professors Larson and Smith concerns algebras of operators on Hilbert space and also on more general Banach spaces. Part of this work will focus on operator algebras that come from lattices of subspaces or projections, in particular on the question of when two such are conjugate by an invertible transformation. The property of relexivity for such algebras will also be studied. The latter consideration makes contact with single operator theory. Structural properties of operator algebras and of maps between them will be investigated. Of especial interest here are notions like complete boundedness that relate to tensoring with matrix algebras of arbitrarily high dimension. The operators that lie at the heart of this project may be thought of as a species of enriched numbers. They obey the same laws of arithmetic as numbers with two notable exceptions, namely that the result of multiplication depends on the order in which the factors are taken, and not every nonzero operator has an inverse. Mathematical analysis knows only two number systems, the real and complex, but there is an immense variety of operator algebras. Situations unsuitable for numerical representation can often be represented in some fashion by operator algebras, thanks to the greater degree of flexibility available with the latter. The price for this is that understanding the structure and classification of operator algebras requires considerable effort, as for instance the work to be undertaken in this project.