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This implies that for a C*-algebra, a positive linear functional sends any equal to for some to a real number, which is equal to its complex conjugate, and therefore all positive linear functionals preserve the self-adjointness of such .
A linear function on a preordered vector space is called positive if it satisfies either of the following equivalent conditions: . implies (); if then () (). [1]; The set of all positive linear forms on a vector space with positive cone , called the dual cone and denoted by , is a cone equal to the polar of .
The statement of the theorem for positive linear functionals on C c (X), the space of compactly supported complex-valued continuous functions, is as follows: Theorem Let X be a locally compact Hausdorff space and ψ {\displaystyle \psi } a positive linear functional on C c ( X ) .
Another approach to measure theory is to restrict to locally compact Hausdorff spaces, and only consider the measures that correspond to positive linear functionals on the space of continuous functions with compact support (some authors use this as the definition of a Radon measure). This produces a good theory with no pathological problems ...
In functional analysis, a state of an operator system is a positive linear functional of norm 1. States in functional analysis generalize the notion of density matrices in quantum mechanics, which represent quantum states, both mixed states and pure states. Density matrices in turn generalize state vectors, which only represent pure states.
Positive linear functionals on () correspond to (positive) regular Borel measures on , by a different form of the Riesz representation theorem. ( Rudin 1966 , Chapter 2) If X {\displaystyle X} is infinite, then C ( X ) {\displaystyle {\mathcal {C}}(X)} is not reflexive , nor is it weakly complete .
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.
Let be a real vector space, be a vector subspace, and be a convex cone.. A linear functional: is called -positive, if it takes only non-negative values on the cone : ().A linear functional : is called a -positive extension of , if it is identical to in the domain of , and also returns a value of at least 0 for all points in the cone :