Lifting strong commutants of unbounded subnormal operators

Various theorems on lifting strong commutants of unbounded subnormal (as well as formally subnormal) operators are proved. It is shown that the strong symmetric commutant of a closed symmetric operatorS lifts to the strong commutant of some tight selfadjoint extension ofS. Strong symmetric commutants of orthogonal sums of subnormal operators are investigated. Examples of (unbounded) irreducible subnormals, pure subnormals with rich strong symmetric commutants and cyclic subnormals with highly nontrivial strong commutants are discussed.This work was supported by the KBN grant # 2P03A 041 10.     

Sufficiency and completeness for classes of probability distributions generated by a single probability distribution

Let (Ω,A, P₀) denote a probability space andA some sub-σ-algebra ofA. Moreover,P _{P 0} stands for the set consisting of all probability distributionsP _{P 0} of the typeP _{P 0}(A) = ∫E _{P 0} (P _{A |B)dP,A ∈A, whereP is a probability distribution onA satisfyingP |B ≪P 0} |B. It is shown thatB is sufficient and (even totally) complete forP _{P 0}. This results is illustrated by examples. Finally, it is shown thatP _{P 0} ∈P _{P 0} is an extremal point ofP _{P 0} if and only ifP |B is {0, 1}-valued. Dedicated to the memory of Professor K. Behnen     

The possibility of reconciling quantum mechanics with classical probability theory

We describe a scheme for constructing quantum mechanics in which a quantum system is considered as a collection of open classical subsystems. This allows using the formal classical logic and classical probability theory in quantum mechanics. Our approach nevertheless allows completely reproducing the standard mathematical formalism of quantum mechanics and identifying its applicability limits. We especially attend to the quantum state reduction problem. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 3, pp. 457–472, December, 2006.     

Measurement and probability in quantum mechanics

Measurement-theoretical foundations of quantum probabilities are investigated in the form of measurement statistics and a statistical ensemble interpretation of quantum mechanics.     

Brownian bridge in quantum probability

The Classical Brownian Bridge is constructed in Symmetric Fock space over an appropriate base Hilbert space. While the representation of the classical Ito-Wiener integral with respect to the increments of the Brownian bridge implements the unitary isomorphism between the Fock space and the (classical) L² space of the Brownian bridge (as is the case with the standard Brownian motion (SBM)), the quantum Ito-integrals with respect to the associated creation and annihilation bridge processes give different left-and right-integrals. This essentially displays the feature that the Brownian Bridge is not a process of independent increments.     

Approach theory meets probability theory

In this paper we reconsider the basic topological and metric structures on spaces of probability measures and random variables, such as e.g. the weak topology and the total variation metric, replacing them with more intrinsic and richer approach structures. We comprehensibly investigate the relationships among, and basic facts about these structures, and prove that fundamental results, such as e.g. the portmanteau theorem and Prokhorov?s theorem, can be recaptured in a considerably stronger form in the new setting.     

Unbounded probability theory compatible with the probability theory of numbers

We study the case in which the values of random variables increase without bound.     

A transference method in quantum probability

Working with a rather general notion of independence, we provide a transference method which allows to compare the p-norm of sums of independent copies with the p-norm of sums of free copies. Our main technique is to construct explicit operator space Lp embeddings preserving independence to reduce the problem to L₁, where some recent results by the first-named author can be used. We find applications on noncommutative Khintchine/Rosenthal type inequalities and on noncommutative Lp embedding theory.     

Discrete problems in probability theory

A short survey is given of some directions in probability theory that have developed most intensively in recent years. Separable statistics and criteria for the verification of statistical hypotheses based on them, various schemes for distributing particles among cells, and problems connected with estimating the unknown size of a finite collection are considered.Translated from Itogi Nauki i Tekhniki, Seriya Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 22, pp. 3–60, 1984.     

On unbounded probability theory

The value of the empirical expectation coincides with that of the mean energy of an ideal Bose gas for one particle. The exact mathematical identity for these quantities makes it possible to carry over the concept of temperature corresponding to the mean energy to an unboundedly increasing sequence of random values for a new unbounded probability theory and for a generalization of Kolmogorov complexity theory. The notion of spectral gap, which was introduced in superconductivity theory, is carried over to unbounded probability theory.     

Belief and probability: A general theory of probability cores

This paper considers varieties of probabilism capable of distilling paradox-free qualitative doxastic notions (e.g., full belief, expectation, and plain belief) from a notion of probability taken as a primitive. We show that core systems, collections of nested propositions expressible in the underlying algebra, can play a crucial role in these derivations. We demonstrate how the notion of a probability core can be naturally generalized to high probability, giving rise to what we call a high probability core, a notion that when formulated in terms of classical monadic probability coincides with the notion of stability proposed by Hannes Leitgeb [32]. Our work continues by one of us in collaboration with Rohit Parikh [7]. In turn, the latter work was inspired by the seminal work of Bas van Fraassen [46]. We argue that the adoption of dyadic probability as a primitive (as articulated by van Fraassen [46]) admits a smoother connection with the standard theory of probability cores as well as a better model in which to situate doxastic notions like full belief. We also illustrate how the basic structure underlying a system of cores naturally leads to alternative probabilistic acceptance rules, like the so-called ratio rule initially proposed by Isaac Levi [34].Core systems in their various guises are ubiquitous in many areas of formal epistemology (e.g., belief revision, the semantics of conditionals, modal logic, etc.). We argue that core systems can also play a natural and important role in Bayesian epistemology and decision theory. In fact, the final part of the article shows that probabilistic core systems are naturally derivable from basic decision-theoretic axioms which incorporate only qualitative aspects of core systems; that the qualitative aspects of core systems alone can be naturally integrated in the articulation of coherence of primitive conditional probability; and that the guiding idea behind the primary qualitative features of a core system gives rise to the formulation of lexicographic decision rules.     

Exchangeable stochastic processes and symmetric states in quantum probability

Annali di Matematica Pura ed Applicata (1923 -) - We analyze general aspects of exchangeable quantum stochastic processes, as well as some concrete cases relevant for several applications to...     

Rotations and quantum theory

Summary A classical theory of rotations is presented in a simple way that can easily be taken over to quantum theory. The application to the theory of elementary particles is indicated. To Enrico Bompiani on his scientific Jubilee     

Cumulants in noncommutative probability theory II

We continue the investigation of noncommutative cumulants. In this paper various characterizations of generalized Gaussian random variables are proved. Supported by the European Network No. HPRN-CT-2000-00116 and the Austrian Science Fund (FWF) Project No. R2-MAT.     

A new concept in probability theory

A new concept called “Disparity” between two distributions is introduced, which could be useful in the study of stochastic processes. It includes the concept of “Activity” introduced earlier by the author.