We determine when there is a unique conditional expectation from a semifinite von Neumann algebra onto a singly-generated maximal abelian *-subalgebra. Our work extends the results of Kadison and Singer via new methods, notably the observation that a unique conditional expectation onto a singly-generated maximal abelian *-subalgebra must be normal.

A subclass of dynamical semigroups induced by the interaction of a quantum system with an environment is introduced. Such semigroups lead to the selection of a stable subalgebra of effective observables. The structure of this subalgebra is completely determined. ∗Supported by the KBN research grant No 2 P03B 086 16

Necessary and sufficient conditions will be found for quadruples of lattices to be isomorphic to lattices of weak, relative, strong subalgebras and initial segments, respectively, of one partial unary algebra. To this purpose we will start with a characterization of pairs of lattices that are weak and strong subalgebra lattices of one partial unary algebra, respectively. Next, we will describe ...

There is shown that there exists a complete, atomless, σ-centered Boolean algebra, which does not contain any regular countable subalgebra if and only if there exist a nowhere dense ultrafilter. Therefore the existence of such algebras is undecidable in ZFC. A subalgebra B of a Boolean algebra A is called regular whenever for every X ⊆ B, supB X = 1 implies supA X = 1; see e.g. Heindorf and Sha...

To calculate the entropy of a subalgebra or of a channel with respect to a state, one has to solve an intriguing optimalization problem. The latter is also the key part in the entanglement of formation concept, in which case the subalgebra is a subfactor. I consider some general properties, valid for these definitions in finite dimensions, and apply them to a maximal commutative subalgebra of a...

To calculate the entropy of a subalgebra or of a channel with respect to a state, one has to solve an intriguing optimalization problem. The latter is also the key part in the entanglement of formation concept, in which case the subalgebra is a subfactor. I consider some general properties, valid for these definitions in finite dimensions, and apply them to a maximal commutative subalgebra of a...

We analyze the most commonly used method for shuffling cards. The main result is a simple expression for the chance of any arrangement after any number of shuffles. This is used to give sharp bounds on the approach to randomness: $ log, n + 0 shuffles are necessary and sufficient to mix up n cards. Key ingredients are the analysis of a card trick and the determination of the idempotents of a na...

We find a remarkable subalgebra of higher symmetries of the elliptic Euler-Darboux equation. To this aim we map such equation into its hyperbolic analogue already studied by Shemarulin. Taking into consideration how symmetries and recursion operators transform by this complex contact transformation, we explicitly give the structure of this Lie algebra and prove that it is finitely generated. Fu...

Preex iteration a x is added to Minimal Process Algebra (MPA), which is a subalgebra of BPA equivalent to Milner's basic CCS. We present a-nite equational axiomatization for MPA , and prove that this axiomatization is complete with respect to strong bisimulation equivalence. To obtain this result, we set up a term rewriting system, based on the axioms, and show that bisimilar terms have the sam...

Prefix iteration a∗x is added to Minimal Process Algebra (MPAδ), which is a subalgebra of BPAδ equivalent to Milner’s basic CCS. We present a finite equational axiomatization for MPA∗ δ , and prove that this axiomatization is complete with respect to strong bisimulation equivalence. To obtain this result, we set up a term rewriting system, based on the axioms, and show that bisimilar terms have...