Positive linear functional
From Free net encyclopedia
In mathematics, especially functional analysis, a linear functional f on a C*-algebra <math>\mathcal{A}</math> is positive if
- <math>f(A)\geq 0</math>
whenever A is a positive element of <math>\mathcal{A}</math>.
That is to say, a positive linear functional does not necessarily take positive values all the time, but only for positive elements, like the identity function <math>z\to z</math> for complex numbers. The reason a positive linear functional is defined on a C*-algebra is to be able to define what a positive element is. The significance of positive linear functionals lies in results such as Riesz representation theorem.
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Examples
- Consider the C*-algebra of complex square matrices. Then, the positive elements are the positive-definite matrices. The trace function defined on this C*-algebra is a positive functional, as the eigenvalues of any positive-definite matrix are positive, and so its trace is positive.
- Consider the C*-algebra Cc(X) of all continuous complex-valued functions of compact support on a locally compact Hausdorff space X. Consider a Borel regular measure μ on X, and a functional ψ defined by
- <math> \psi(f) = \int_X f(x) d \mu(x) \quad </math>
- for all f in Cc(X). Then, this functional is positive (the integral of any positive function is a positive number). Moreover, any positive functional on this space has this form, as follows from the Riesz representation theorem.
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