Talk:Positive linear functional
In the line "Consider the C*-algebra Cc(X) of all continuous complex-valued functions of compact support on a locally compact Hausdorff space X." there is an error, since the space of all continuos functions with compact support is NOT complete, and hence not a C*-Algebra.
So what's the order on CC?
If you have a C*-algebra, you have a complex vector space, so what does one mean by a nonnegative complex number? Given the ordering of the C*-algebra described in C*-algebras#Properties of C*-algebras, is there reason to think all linear functionals on a C*-algebra send all self-adjoint elements of the algebra to real numbers? Whatever's going on, it's not explicit, but it's relied upon in the terminology of State (functional analysis), which uses the term positive linear functional without seeming to use that as a synonym for what it calls a self-adjoint linear functional. ᛭ LokiClock (talk) 11:53, 24 July 2014 (UTC)
- From the fact that in these course notes, pg 15, preservation of adjoints is considered a lemma and not an axiom, I gather that what is assumed for complex algebras is either that all elements in an ordered subspace go to real numbers or all positive elements of the ordered subspace go to nonnegative real numbers. ᛭ LokiClock (talk) 08:57, 25 July 2014 (UTC)