Stopping times are a powerful tool in the theory of stochastic processes, so it is natural to ask whether they have a counterpart in the theory of non-commutative processes. This paper is a part answer to that question. We show that the ‘formalism’ of stopping times carries over to a non-commutative context and prove an Optional Stopping Theorem.

Let Kr denote the set of r-tuples n = (n1, n2, …, nr), r ≥ 1, where the components ni are positive integers. Let {X, Xn, n ∈ Kr} be a family of independent and identically distributed random variables with positive mean EX = μ < + ∞ and finite variance VX = σ2 < + ∞. In a recent work, M. Maejima and T. Mori [2] have shown that, if X is integer valued, aperiodic and E∣X∣3 < + ∞, then, for r = 2 or 3,

where

the summation being extended over all members j = (j1,j2, …, jr) of Kr that satisfy jt ≤ nt for all 1 ≤ t ≤ r.

In this paper we are concerned with the following question: Given a function class , the space of L2-one periodic complex valued functions, when does sampling a function give optimal information for approximation? To make this question precise, we introduce the quantity

, where A is any mapping from ℝN and I is any continuous linear mapping from into ℝN.

The process of Fronsdal inducing is discussed as a method of quantizing physical systems, and is seen to reduce to a technique for obtaining representations of *-algebras.

It is asked whether this above process is at all successful, and whether it is at all related to the representation inducing methods of Fell. The two inducing techniques are shown to be equivalent for algebras satisfying three specified properties, and it is also shown that all the strictly irreducible representations of such algebras can be found by these methods.

Examples are produced whereby it is shown that algebras of the type required by the theory are of physical interest.