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Hilbert Algebra (other)
In mathematics, Hilbert algebras and left Hilbert algebras occur in the theory of von Neumann algebra In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann algeb ...s in: * Commutation theorem for traces#Hilbert algebras * Tomita–Takesaki theory#Left Hilbert algebras {{Mathematical disambiguation Von Neumann algebras ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Neumann Algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann algebras were originally introduced by John von Neumann, motivated by his study of single operators, group representations, ergodic theory and quantum mechanics. His double commutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries. Two basic examples of von Neumann algebras are as follows: *The ring L^\infty(\mathbb R) of essentially bounded measurable functions on the real line is a commutative von Neumann algebra, whose elements act as multiplication operators by pointwise multiplication on the Hilbert space L^2(\mathbb R) of square-integrable functions. *The algebra \mathcal B(\mathcal H) of all bounded operators on a Hilbert space \mathcal H is a von Neumann algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutation Theorem For Traces
In mathematics, a commutation theorem for traces explicitly identifies the commutant of a specific von Neumann algebra acting on a Hilbert space in the presence of a Von Neumann algebra#Weights, states, and traces, trace. The first such result was proved by Francis Joseph Murray and John von Neumann in the 1930s and applies to the von Neumann algebra generated by a discrete group or by the dynamical system associated with a ergodic theory, measurable transformation preserving a probability measure. Another important application is in the theory of unitary representations of Haar measure, unimodular locally compact groups, where the theory has been applied to the regular representation and other closely related representations. In particular this framework led to an abstract version of the Plancherel theorem for unimodular locally compact groups due to Irving Segal and W. Forrest Stinespring, Forrest Stinespring and an abstract Plancherel theorem for spherical functions associate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tomita–Takesaki Theory
In the theory of von Neumann algebras, a part of the mathematical field of functional analysis, Tomita–Takesaki theory is a method for constructing modular automorphisms of von Neumann algebras from the polar decomposition of a certain involution. It is essential for the theory of type III factors, and has led to a good structure theory for these previously intractable objects. The theory was introduced by , but his work was hard to follow and mostly unpublished, and little notice was taken of it until wrote an account of Tomita's theory. Modular automorphisms of a state Suppose that ''M'' is a von Neumann algebra acting on a Hilbert space ''H'', and Ω is a cyclic and separating vector of ''H'' of norm 1. (Cyclic means that ''MΩ'' is dense in ''H'', and separating means that the map from ''M'' to ''MΩ'' is injective.) We write \phi for the vector state \phi(x) = (x\Omega, \Omega) of ''M'', so that ''H'' is constructed from \phi using the Gelfand–Naimark–Segal construct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |