Measurable Categories |
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Authors: | D N Yetter |
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Institution: | (1) Department of Mathematics, Kansas State University, Manhattan, KS 66506, USA |
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Abstract: | We develop the theory of categories of measurable fields of Hilbert spaces and bounded fields of operators. We examine classes
of functors and natural transformations with good measure theoretic properties, providing in the end a rigorous construction
for the bicategory used in 3] and 4] as the basis for a representation theory of (Lie) 2-groups. Two important technical results are established along the way:
first it is shown that all invertible additive bounded functors (and thus a fortiori all invertible *-functors) between categories of measurable fields of Hilbert spaces are induced by invertible measurable
transformations between the underlying Borel spaces and second we establish the distributivity of Hilbert space tensor product
over direct integrals over Lusin spaces with respect to σ-finite measures. The paper concludes with a general definition of measurable bicategories. |
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Keywords: | Primary 18D05 Secondary 18D10 28A20 |
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