Talk:Ordered algebra

Property section is not correct

Hi, the result stated in the section "Properties" is incorrect or formulated in a highly misleading way: Consider the polynomial algebra in at least two variables with cone of positive elements defined as the set of all sums of squares of polynomials. This is an ordered algebra as defined in the introduction (the Archimedean order bit is somewhat tricky, but ultimately can be shown to hold by elementary analysis). However, this ordered algebra has more extremal normalized positive linear functionals than just the multiplicative ones (which are the evaluations at points of the plane), otherwise (skipping some steps here) every pointwise positive polynomial would be a sum of squares (which is not true, a counterexample is the Motzkin polynomial; this lead to Hilbert's 17th problem). I think there is an additional assumption missing, like the algebraic unit being an order unit? Maybe "ordered algebra with unit e" is to be interpreted this way, i.e. e is algebraic and order unit? But no one who doesn't already know the correct result is going to interpret it like this! I unfortunately do not have the book of Schaefer and Wolf at hand, so I can't check the reference. Please correct this!TSBM (talk) 22:15, 4 November 2021 (UTC)