Search results
Results from the WOW.Com Content Network
The significance of positive linear functionals lies in results such as Riesz–Markov–Kakutani representation theorem. When V {\displaystyle V} is a complex vector space, it is assumed that for all v ≥ 0 , {\displaystyle v\geq 0,} f ( v ) {\displaystyle f(v)} is real.
Any positive linear functionals on dominated by is of the form = (), for some positive operator in () ′ with in the operator order. This is a version of the Radon–Nikodym theorem . For such g {\displaystyle g} , one can write f {\displaystyle f} as a sum of positive linear functionals: f = g + g ′ {\displaystyle f=g+g'} .
The Gelfand representation or Gelfand isomorphism for a commutative C*-algebra with unit is an isometric *-isomorphism from to the algebra of continuous complex-valued functions on the space of multiplicative linear functionals, which in the commutative case are precisely the pure states, of A with the weak* topology.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
A proof can be sketched as follows: Let be the weak*-compact set of positive linear functionals on with norm ≤ 1, and () be the continuous functions on . A {\displaystyle A} can be viewed as a closed linear subspace of C ( Ω ) {\displaystyle C(\Omega )} (this is Kadison 's function representation ).
The states of the quantum mechanical system are defined to be the states of the C*-algebra (in other words the normalized positive linear functionals ). The value ω ( A ) {\displaystyle \omega (A)} of a state ω {\displaystyle \omega } on an element A {\displaystyle A} is the expectation value of the observable A {\displaystyle A} if the ...
A weight ω on a von Neumann algebra is a linear map from the set of positive elements (those of the form a*a) to [0,∞]. A positive linear functional is a weight with ω(1) finite (or rather the extension of ω to the whole algebra by linearity). A state is a weight with ω(1) = 1. A trace is a weight with ω(aa*) = ω(a*a) for all a.
There are many closely related variations of the theorem, as the linear functionals can be complex, real, or positive, the space they are defined on may be the unit interval or a compact space or a locally compact space, the continuous functions may be vanishing at infinity or have compact support, and the measures can be Baire measures or ...