Linear functionals can be decomposed as linear combinations of positive ones?

Question

I am reading Arveson's Notes on Extensions of $C^*$-algebras. In proving theorem 1, he needs to establish some results concerning bounded linear functionals. However, he said it suffices to prove for positive linear functionals, which I do not quite understand.

So are bounded linear functionals over a $C^*$-algebra always the linear combinations of 4 positive ones?

Thanks!

Answer 1

I seem to remember you saying that you have access to Murphy's book?

If so, then the result you want is Theorem 3.3.10 (see also Theorem 3.3.6). The key trick is that there is a positive, real-linear isometry from the real vector space $A_{sa}$ into $C_{\mathbb R}(\Omega)$ for a suitable $\Omega$.

If you do not have access then I could copy out the details.

Answer 2

Certainly not, if it's a complex C^* algebra: the difference of positive functionals is real on the hermitian elements.