We prove that a pure state on a ȁlgebra or a JB algebra is a unique extension of some pure state on a singly generated subalgebra if and only if its left kernel has a countable approximative unit. In particular, any pure state on a separable JB algebra is uniquely determined by some singly generated subalgebra. By contrast, only normal pure states on JBW algebras are determined by singly generated subalgebras, which provides a new characterization of normal pure states. As an application we contribute to the extension problem and strengthen the hitherto known results on independence of operator algebras arising in the quantum field theory.
Here we present some interesting aspects of interplay between quantum theory and functional analysis. We show that some of the deepest problems in both disciplines are unexpectedly equivalent. Starting from a historical point of view we comment on the following topics: connection between the structure of von Neumann algebras and the structure of quantum theories, construction of quantum integral and logic of quantum systems, Bell inequalities in light of Grothendieck inequality and operator space theory. Finally, we discuss bohrification and topos approach to quantum theory., Martin Bohata, Jan Hamhalter., and Obsahuje bibliografii