Conditionally positive-definite functions in quantum probability theory

The author introduces the concepts of positive-definite and conditionally positive definite functions with values in the algebra of bounded maps of a $C^*$-algebra. An analog of Schoenberg's theorem is proved, a GNS-representation is obtained for conditionally positive-definite functions in terms of suitable cocycles, and this representation leads to a noncommutative generalization of the Lévy–Khinchin formula. Applications to the problem of continuous measurement in quantum mechanics are considered. A complete mathematical description is presented of continuous measurement processes, based on the analogy with the classical parts of probability theory — the theory of infinitely divisible distributions and functional limit theorems for processes with independent increments.