Representation theory, Borel cross-sections, and minimal measures

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dc.contributor.authorMiller, Janice E.en
dc.contributor.committeechairOlin, Robert F.en
dc.contributor.committeememberBall, Joseph A.en
dc.contributor.committeememberMcCoy, Robert A.en
dc.contributor.committeememberAull, Charles E.en
dc.contributor.committeememberJohnson, Lee W.en
dc.contributor.departmentMathematicsen
dc.date.accessioned2014-03-14T21:15:08Zen
dc.date.adate2006-06-19en
dc.date.available2014-03-14T21:15:08Zen
dc.date.issued1993en
dc.date.rdate2006-06-19en
dc.date.sdate2006-06-19en
dc.description.abstractLet E be an analytic metric space, let X be a separable metric space with a regular Borel probability measure μ and let Π: E → X be a continuous map with μ(X \ Π(E)) =0. Schwartz’s lemma states that there exists a Borel cross-section for Π defined almost everywhere (μ). The equivalence classes of these Borel cross-sections are in one-to-one correspondence with the representations of the form Γ:C<sub>b</sub>(E) → L<sup>∞</sup>(μ) with Γ(f∘Π) = f for every f ∈ C<sub>b</sub>(X). The representations are also in one-to-one correspondence with equivalence classes of the minimal measures on E. Now let E, X, and μ be as above and let Π: E → X be an onto Borel map. There exists a Borel cross-section for Π defined almost everywhere (μ). The equivalence classes of the Borel cross-sections for Π are in one-to-one correspondence with the representations of the form Γ:B(E) → L<sup>∞</sup>(μ) with Γ(f∘Π) = f for every f in C<sub>b</sub>(X), where B(E) is the C*-algebra of the bounded Borel functions on E. The representations are also in one-to-one correspondence with equivalence classes of the minimal measures on E.en
dc.description.degreePh. D.en
dc.format.extentiv, 51 leavesen
dc.format.mediumBTDen
dc.format.mimetypeapplication/pdfen
dc.identifier.otheretd-06192006-125737en
dc.identifier.sourceurlhttp://scholar.lib.vt.edu/theses/available/etd-06192006-125737/en
dc.identifier.urihttp://hdl.handle.net/10919/38643en
dc.language.isoenen
dc.publisherVirginia Techen
dc.relation.haspartLD5655.V856_1993.M556.pdfen
dc.relation.isformatofOCLC# 29021824en
dc.rightsIn Copyrighten
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/en
dc.subject.lccLD5655.V856 1993.M556en
dc.subject.lcshRepresentations of algebrasen
dc.titleRepresentation theory, Borel cross-sections, and minimal measuresen
dc.typeDissertationen
dc.type.dcmitypeTexten
thesis.degree.disciplineMathematicsen
thesis.degree.grantorVirginia Polytechnic Institute and State Universityen
thesis.degree.leveldoctoralen
thesis.degree.namePh. D.en

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