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According to the duality between geometry and algebra, various (opposite) categories of commutative algebras and similar objects are equivalent to certain corresponding categories of spaces.
For example, Gelfand duality establishes a contravariant equivalence of categories between the category of commutative unital C*-algebras and the category of compact Hausdorff topological spaces.
Thus, one can drop the commutativity condition and argue that studying arbitrary C*-algebras amounts to studying noncommutative general topology, a part of noncommutative geometry.
Analogously, in measure theory one works with commutative von Neumann algebras instead, which are equivalent to several other categories: compact strictly localizable enhanced measurable spaces, hyperstonean topological spaces and open maps, hyperstonean locales and open maps, measurable locales (and arbitrary morphisms of locales).
Thus, dropping the commutativity condition results in the category of von Neumann algebras (with ultraweakly continuous $*$-homomorphisms), and we can view this category as the category of noncommutative measurable spaces.
Thus, noncommutative measure theory is more-or-less the study of von Neumann algebras, more precisely those aspects of them that reproduce known phenomena in measure theory when specialized to the commutative case.
The notion of a spatial product? of (compact strictly localizable enhanced) measurable spaces can be expressed algebraically as the spatial tensor product of von Neumann algebras. This is quite different from the categorical product of measurable spaces, which corresponds to the Dauns tensor product of von Neumann algebras.
As another example, the theory of $L^p$-spaces generalizes to von Neumann algebras. This is a work of many authors, including von Neumann, Edward Nelson, Irving Segal, and Uffe Haagerup. It plays a crucial role in Tomita-Takesaki modular theory, which is essentially nothing else than the theory of $L^p$-spaces where $p$ is a purely imaginary number. In particular, it allows one to classify type III factors?.
The notion of a measure is replaced by that of a normal weight on a von Neumann algebra. A trace on a von Neumann algebra is a special type of a weight.
The notion of a measurable field of Hilbert spaces can be expressed using two equivalent categories: the category of W*-modules over von Neumann algebras, and the category of representations of von Neumann algebras on a Hilbert space.
Masamichi Takesaki, Theory of Operator Algebras I, II, III, Springer, I: 1979. vii+415 pp. ISBN: 0-387-90391-7; II: 2003. xxii+518 pp. ISBN: 3-540-42914-X; III: 2003. xxii+548 pp. ISBN: 3-540-42913-1.
Nik Weaver, Greg Kuperberg, A von Neumann Algebra Approach to Quantum Metrics/Quantum Relations, Memoirs of the AMS 215 1010 (2011) [ams:memo-215-1010]
Last revised on January 13, 2024 at 08:39:26. See the history of this page for a list of all contributions to it.