| dc.contributor.author | Wendt, Michael Albert. | en_US |
| dc.date.accessioned | 2014-10-21T12:37:31Z | |
| dc.date.issued | 1992 | en_US |
| dc.identifier.other | AAINN80155 | en_US |
| dc.identifier.uri | http://hdl.handle.net/10222/55327 | |
| dc.description | Our aim is to study X-families of Hilbert spaces for X a measure space; the ultimate goal being the understanding of the classical (von Neumann) direct integral in the context of indexed category theory. Indeed, the diagram, $\int\sp\oplus: {\bf Hilb}\sp{X}\ {\longleftarrow\atop\longrightarrow}\ {\bf Hilb} : \Delta,$ provides a useful summary of our goal. | en_US |
| dc.description | We first require a good category of measure spaces and introduce, Disint, the category of disintegrations. Disint does not have products (nor does any "useful" category of measure spaces) so we do not have the usual Pare-Schumacher style indexing. The diagram above cannot be interpreted as an adjunction. | en_US |
| dc.description | We must approximate the situation as best possible and we put forth three approximations. Specifically, we propose three notions of X-family of Hilbert spaces: (1) measurable fields of Hilbert spaces on X, (2) Hilbert sheaves on X, and (3) Hilbert families over X. We will describe each of these approaches in detail including substitution with respect to base category morphisms and $\int\sp\oplus$. Finally, we will discuss connections between the three ideas and list some possible future directions for this work. | en_US |
| dc.description | Thesis (Ph.D.)--Dalhousie University (Canada), 1992. | en_US |
| dc.language | eng | en_US |
| dc.publisher | Dalhousie University | en_US |
| dc.publisher | | en_US |
| dc.subject | Mathematics. | en_US |
| dc.title | On measurably indexed families of Hilbert spaces. | en_US |
| dc.type | text | en_US |
| dc.contributor.degree | Ph.D. | en_US |
