Maximal extensions of a linear functional

Extensions of a positive hermitian linear functional ω, defined on a dense *-subalgebra A0 of a topological *-algebra A[τ] are analyzed. It turns out that their maximal extensions as linear functionals or hermitian linear functionals are everywhere defined. The situation however changes deeply if one looks for positive extensions. The case of fully positive and widely positive extensions considered in [2] is revisited from this point of view. Examples mostly taken from the theory of integration are discussed.