Tensor product of generalized sample spaces

Robin H. Lock

R. F. Streater, Aubrey Wulfsohn

Patricia Frazer Lock

A manual of operations is a model for a collection of physical experiments, or a collection of (perhaps overlapping) sample spaces. To understand this theory, it is essential to understand the relations between manuals. Hence, in this dissertation, we study the morphisms (and, more specifically, the interpretations) of manuals. This is followed by an investigation of the categorical properties of the tensor product of two manuals. We define biinterpretations and prove a universal mapping theorem. We then prove that the tensor product is associative, and define, using an inductive limit, an infinite tensor product. In the final chapter, we develop a language and theory that is broad enough to encompass much of the work which has been done under the general title of "quantum logic."

Matthias P. Kl�y, Charles H. Randall, David J. Foulis

Kalyanapuram Rangachari Parthasarathy

Suppose (Ω i , F i ), 1 ≤ i ≤ n are sample spaces describing the elementary outcomes and events concerning n different statistical systems in classical probability. To integrate them into a unified picture under the umbrella of a single sample space one takes their cartesian product (Ω, F) where Ω = Ω1 x … x Ω n , F = F1 x … x F n , the smallest σ-algebra containing all rectangles of the form F1 x F1 x … F n , F j ∈ F j for each j. Now we wish to search for an analogue of this description in quantum probability when we have n systems where the events concerning the j-th system are described by the set P(ℋ j ) of all projections in a Hubert space ℋ j j = 1, 2,…, n. Such an attempt leads us to consider tensor products of Hilbert spaces. We shall present a somewhat statistically oriented approach to the definition of tensor products which is at the same time coordinate free in character. To this end we introduce the notion of a positive definite kernel.

Stanley Gudder