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Spectral flow, in its simplest incarnation, counts the net number of eigenvalues which change sign as one traverses a path of self-adjoint Fredholm operators in the set of of bounded operators B(H) on a Hilbert space. A generalization of this idea changes the setting to a semifinite von Neumann algebra N and uses the trace τ to measure the amount of spectrum which changes from negative to positive along a path; the operators are still self-adjoint, but the Fredholm requirement is replaced by its von Neumann algebras counterpart, Breuer-Fredholm. Our work is ensconced in this semifinite von Neumann algebra setting. We prove a uniqueness result in the case when N is a factor. In the case when the operators under consideration are bounded perturbations of a fixed unbounded operator with τ-compact resolvents, we give a different proof of a p-summable integral formula which calculates spectral flow, and fill in some of the gaps in the proof that spectral flow can be viewed as an intersection number if N = B(H). / Graduate / 0280

We investigate some properties of (geometric) fields in toposes of sheaves over Boolean spaces and establish the internal validity of a number of classical theorems from Algebraic Geometry and the theory of ordered fields. We then use our results to obtain, via sheaf representations, some know theorems about (von Neumann) regular rings as well as some new theorems for regular f-rings. By contrast with previous investigations in these last two subjects (Saracino and Weispfenning {39} and van den Dries {42}) a more natural approach, inspired by work of Macintyre {30}, Loullis {29}, Bunge-Reyes {7} and Bunge {4},{5} is employed here. In addition to sheaf theoretic methods we use a variety of logical methods from geometric logic, infinitary intuitionistic logic and model theory. We also prove some new theorems on the transfer of subobjects along certain morphisms and a "lifting theorem" taking truth from statements about global sections to their internal validity.

Thesis (Ph.D.)--Flinders University, School of Informatics and Engineering. / Typescript bound. Includes bibliographical references: (leaves 174-180) and index. Also available online.

Thesis--Freie Universität, Berlin. / Vita. Includes bibliographical references (p. 119-125).

Diss. : Philosophische Fakultät : Würzburg : 1982. - Bibliogr. p. 245-258. Index. -

Texte remanié de: Diss.--Philosophie--Köln, 1973. / Bibliogr. p. 448-463. Index.

Thesis (Ph. D.)--University of Washington, 2004. / Vita. Includes bibliographical references (p. 111-112).

Thesis--Freie Universität, Berlin. / Vita. Includes bibliographical references (p. 119-125).

Thesis (doctoral)--Universität Bonn, 1988. / Cover title: Integraldarstellungen und a priori-Abschätzungen für das [delta bar]-Neumann-Problem. Includes bibliographical references (p. 94-96).