The chapter introduces the concept of positive functions on C*-algebras, building on Neumark's theorem. It begins by rephrasing the original formulation using C*-algebras and linear functions instead of Boolean algebras and measures. The main theorem states that a linear function \(\mu\) from a C*-algebra \(\mathcal{A}\) to operators on a Hilbert space \(\mathcal{H}\) can be expressed as \(\mu(A) = V^* \rho(A) V\) for some bounded linear transformation \(V\) and \(\star\)-representation \(\rho\) of \(\mathcal{A}\) if and only if \(\mu\) is completely positive. The proof of necessity involves showing that \(\mu^{(n)}\) is positive for all \(n\), while the proof of sufficiency demonstrates that \(\mu\) is completely positive if it satisfies certain positivity conditions in the tensor product space \(\mathcal{A} \otimes \mathcal{H}\). The chapter also discusses the implications of commutativity in the algebra and provides a counterexample to show that positivity does not always imply complete positivity. Finally, it presents conditions for complete positivity, including the fact that the center-valued trace of a W*-algebra of finite type is completely positive, and that a positive linear function on a C*-algebra is completely positive.The chapter introduces the concept of positive functions on C*-algebras, building on Neumark's theorem. It begins by rephrasing the original formulation using C*-algebras and linear functions instead of Boolean algebras and measures. The main theorem states that a linear function \(\mu\) from a C*-algebra \(\mathcal{A}\) to operators on a Hilbert space \(\mathcal{H}\) can be expressed as \(\mu(A) = V^* \rho(A) V\) for some bounded linear transformation \(V\) and \(\star\)-representation \(\rho\) of \(\mathcal{A}\) if and only if \(\mu\) is completely positive. The proof of necessity involves showing that \(\mu^{(n)}\) is positive for all \(n\), while the proof of sufficiency demonstrates that \(\mu\) is completely positive if it satisfies certain positivity conditions in the tensor product space \(\mathcal{A} \otimes \mathcal{H}\). The chapter also discusses the implications of commutativity in the algebra and provides a counterexample to show that positivity does not always imply complete positivity. Finally, it presents conditions for complete positivity, including the fact that the center-valued trace of a W*-algebra of finite type is completely positive, and that a positive linear function on a C*-algebra is completely positive.