Mathematical Sciences: C*-Algebras and the Geometry of Hilbert Space

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This is mathematical research in pursuit of two famous open problems having to do with pure states on operator algebras. The term state comes from quantum mechanics. In some formulations thereof, observables are represented by selfadjoint operators in an appropriate algebra of operators, and states are certain linear functionals on the operator algebra; thus, specifying a state assigns a numerical value to each of the observables. The pure states are those not given as convex combinations of other states. In principal, they determine everything about the algebra, but pinning down the connection more precisely has led to daunting difficulties. One of the problems the present project will address is that of extending the Stone-Weierstrass theorem, which may be thought of as a result about commutative operator algebras, to the general, noncommutative situation. The other problem is to show the uniqueness of pure state extensions from maximal diagonal subalgebras of the algebra of all bounded operators on a Hilbert space.