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The mathematics of classical probability theory was subsumed into classical measure theory by Kolmogorov in 1933. Quantum theory as nonclassical probability theory was incorporated into the beginnings of noncommutative measure theory by von Neumann in the early thirties, as well. To precisely this end, von Neumann initiated the study of what are now called von Neumann algebras and, with Murray, made a first classification of such algebras into three types. The nonrelativistic quantum theory of systems with finitely many degrees of freedom deals exclusively with type I algebras. However, for the description of further quantum systems, the other types of von Neumann algebras are indispensable. The paper reviews quantum probability theory in terms of general von Neumann algebras, stressing the similarity of the conceptual structure of classical and noncommutative probability theories and emphasizing the correspondence between the classical and quantum concepts, though also indicating the nonclassical nature of quantum probabilistic predictions. In addition, differences between the probability theories in the type I, II and III settings are explained. A brief description is given of quantum systems for which probability theory based on type I algebras is known to be insufficient. These illustrate the physical significance of the previously mentioned differences.

It is shown that, given any finite set of pairs of random events in a Boolean algebra which are correlated with respect to a fixed probability measure on the algebra, the algebra can be extended in such a way that the extension contains events that can be regarded as common causes of the correlations in the sense of Reichenbach's definition of common cause. It is shown, further, that, given any quantum probability space and any set of commuting events in it which are correlated with respect to a fixed quantum state, the quantum probability space can be extended in such a way that the extension contains common causes of all the selected correlations, where common cause is again taken in the sense of Reichenbach's definition. It is argued that these results very strongly restrict the possible ways of disproving Reichenbach's Common Cause Principle.

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The notion of common cause closedness of a classical, Kolmogorovian probability space with respect to a causal independence relation between the random events is defined, and propositions are presented that characterize common cause closedness for specific probability spaces. It is proved in particular that no probability space with a finite number of random events can contain common causes of all the correlations it predicts; however, it is demonstrated that probability spaces even with a finite number of random events can be common cause closed with respect to a causal independence relation that is stronger than logical independence. Furthermore it is shown that infinite, atomless probability spaces are always common cause closed in the strongest possible sense. Open problems concerning common cause closedness are formulated and the results are interpreted from the perspective of Reichenbach's Common Cause Principle. The problem and informal review of resultsLet T be a theory formal part of which contains classical probability theory (S, p), where S is a Boolean algebra of sets representing random events (with Boolean operations ∪, ∩, ⊥) and where p is a probability measure possessing the standard properties. Typically, T predicts correlations between certain elements of S: if A, B ∈ S, then the quantity According to a classical tradition in philosophy of science, articulated especially by H. Reichenbach [16] and more recently by W. Salmon [19], correlations are always results of causal relations. This is the content of what became called Reichenbach's Common Cause Principle (RCCP). The principle asserts that if Corr(A, B) > 0 then either the events A, B stand in a direct causal relation responsible for the correlation, or there exists a third event C causally affecting both A and B, and it is this third event, the so-called (Reichenbachian) common cause, which brings about the correlation by being related to A, B in a specific way (see Definition 2). So formulated, Reichenbach's Common Cause Principle is a metaphysical claim about the causal structure of the World, its status has been investigated extensively in the literature, especially by Butterfield [2]; Cartwright [3]; Placek [9,10]; Salmon [17,18,19]; Sober [20,21,22] Van Fraassen [29,30,31]. Corr(A, B) . = p(A ∩ B) − p(A)p(B)Assuming that RCCP is valid, one is led to the question of whether our theories predicting probabilistic correlations can be causally rich enough to contain also the causes of the 1 correlations. The aim of the present paper is to formulate precisely and to investigate this question.According to RCCP, causal richness of a theory T would manifest in T 's being causally closed in the sense of being capable to explain the correlations by containing a common cause of every correlation between causally independent events A, B. More explicitly, if T is a theory containing probability theory (S, p) and if R ind (A, B) is a causal independence relation on S, then we call T common cause closed with respect to R ind , if for every p...