Date of Award
Doctor of Philosophy (PhD)
A determinant in algebraic K-theory is associated to any two Fredholm operators which commute modulo the trace class. This invariant is defined in terms of the Fredholm determinant, which itself extends the usual notion of matrix determinant. On the other hand, one may consider a homologically defined invariant known as joint torsion. This thesis answers in the affirmative a conjecture of R. Carey and J. Pincus, namely that these two invariants agree.
The strategy is to analyze how joint torsion transforms under an action by certain groups of pairs of invertible operators. This allows one to reduce the calculation to the finite dimensional setting, where joint torsion is shown to be trivial. The equality implies that joint torsion has continuity properties, satisfies the expected Steinberg relations, and depends only on the images of the operators modulo trace class. Moreover, we show that the determinant invariant of two commuting operators can be computed in terms of finite dimensional data.
The second main goal of this thesis is to investigate how joint torsion behaves under the functional calculus. We study the extent to which the functional calculus commutes, modulo operator ideals, with projections in a finitely summable Fredholm module. As an application, we recover in particular some results of R. Carey and J. Pincus on determinants of Toeplitz operators and Tate tame symbols. In addition, we obtain variational formulas and explicit integral formulas for joint torsion.
Migler, Joseph, "Determinants in K-theory and operator algebras" (2015). Mathematics Graduate Theses & Dissertations. 35.