Page "State (functional analysis)" Paragraph 8

A proof can be sketched as follows: Let Ω be the weak *- compact set of positive linear functionals on A with norm ≤ 1, and C ( Ω ) be the continuous functions on Ω.

A can be viewed as a closed linear subspace of C ( Ω ) ( this is Kadison's function representation ).

By Hahn – Banach, f extends to a g in C ( Ω )* with || g ||