Title: Diagonalizing matrices over operator algebras with many projections
Speaker: Manuel Reyes, Bowdoin College, Brunswick, Maine
Seminar Type: The OSU-OU Ring Theory Seminar
Abstract: The classical Spectral Theorem of linear algebra states implies that any commutative self-adjoint subalgebra of a square complex matrix algebra can be diagonalized by a unitary matrix. We will show that this form of the theorem generalizes to rings of matrices over certain operator algebras with many self-adjoint idempotents, called AW*-algebras. We will explain how this problem leads to the development of a dimension theory for the projections in a general AW*-algebra. If time permits, we will also discuss how this result applies to show that an AW*-algebra is determined up to isomorphism by its active lattice, a structure that can be viewed as a kind of "noncommutative complete Boolean algebra."
