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 Riesz space 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
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- 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.
Read more about this topic: Positive Linear Functional
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