Mathematical Sciences: Triangular AF Algebras and TriangularHyperfinite Algebras

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This Minority Research Initiation award is for the support of Professor Baker's work on nonselfadjoint operator algebras. He is particularly interested in subalgebras of approximately finite-dimensional C*-algebras (resp. von Neumann algebra factors) that play a role analogous to that of the upper triangular matrices in a full matrix algebra. The problems he will pursue have mainly to do with classification, isomorphism invariants, and isometric versus algebraic isomorphism. 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.