Positive linear functional

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In mathematics, especially functional analysis, a linear functional f on a C*-algebra is positive if

whenever A is a positive element of .

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 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.

[edit] 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 C_c(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

for all f in C_c(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.

[edit] See also

• positive element