This paper discusses positive functions on C*-algebras, focusing on the concept of complete positivity. The author, W. Forrest Stinespring, begins by rephrasing Neumark's theorem in terms of C*-algebras and linear functions. He defines a positive function as one that maps positive elements of the algebra to positive operators. A function is completely positive if it is positive when applied to matrices of elements of the algebra. The main theorem states that a linear function from a C*-algebra to operators on a Hilbert space is completely positive if and only if it can be expressed as a composition of a bounded linear transformation and a *-representation. The proof of necessity shows that if a function has this form, it is completely positive. The proof of sufficiency demonstrates that if a function is completely positive, it can be expressed in this form. The paper also provides a counterexample showing that positivity does not necessarily imply complete positivity. It then discusses conditions for complete positivity, including theorems about W*-algebras and C*-algebras. Theorems show that certain functions, such as center-valued traces on W*-algebras and positive functions on C*-algebras, are completely positive. The paper concludes with a theorem stating that positive operator-valued linear functions on commutative C*-algebras are completely positive. The paper references several mathematical works by other authors.This paper discusses positive functions on C*-algebras, focusing on the concept of complete positivity. The author, W. Forrest Stinespring, begins by rephrasing Neumark's theorem in terms of C*-algebras and linear functions. He defines a positive function as one that maps positive elements of the algebra to positive operators. A function is completely positive if it is positive when applied to matrices of elements of the algebra. The main theorem states that a linear function from a C*-algebra to operators on a Hilbert space is completely positive if and only if it can be expressed as a composition of a bounded linear transformation and a *-representation. The proof of necessity shows that if a function has this form, it is completely positive. The proof of sufficiency demonstrates that if a function is completely positive, it can be expressed in this form. The paper also provides a counterexample showing that positivity does not necessarily imply complete positivity. It then discusses conditions for complete positivity, including theorems about W*-algebras and C*-algebras. Theorems show that certain functions, such as center-valued traces on W*-algebras and positive functions on C*-algebras, are completely positive. The paper concludes with a theorem stating that positive operator-valued linear functions on commutative C*-algebras are completely positive. The paper references several mathematical works by other authors.