ORDER and ISOMORPHISMS

报 告 人：Ekaterina Turilova 教授
                  俄罗斯喀山联邦大学
主 持 人：黄哲学
日      期：2016 年 11月 2 日
时      间：下午 2:30-4:30
地      点：计软学院623会议室

Path I: Transformations preserving the spectral order

Symmetries of the quantum system are one of the basic topics in quantum structures,   foundations of quantum theory and computational aspects of quantum formalism. They have their beginning in celebrated Wigner Theorem describing quantum transformations  preserving transition probabilities between states of the quantum system. This golden rule is a subject of the research till today, many new generalizations and approaches are appearing and references therein. In its lattice theoretic or quantum logic version Wigner theorem determines transformations of projection lattices that preserve order and orthocomplementation in both directions. In case of Hilbert space logic such symmetries are given by unitary or antiunitary operators, in case of von Neumann projection lattice they are given by Jordan  *-automorphisms. Recently, Wigner type theorems have been considered for spectral order on quantum effects. It turns out that effect algebras of C*-algebras endowed with the spectral order are natural extensions of projection lattices equipped with standard operator order. Transition from projections to effects is interesting on both mathematical and physical side. In quantum formalism it corresponds to replacing sharp observables (projections) with two point spectrum by positive contractions (effects) whose spectrum may be whole unit interval. It has a meaning for quantum measurement.   Spectral order is a natural order on effect operators that organizes them into complete lattice. Introduced by Arveson and Olson,  it plays an important role in  matrix theory  and theory of von Neumann algebras.

Path II: Choquet order of orthogonal measures and abelian subalgebras

An interplay between recent topos theoretic approach and standart convex theoretic approach to quantum theory  is discovered. Combining new results on isomorphisms of the posets of all abelian subalgebras of  von  Neumann algebras with classical Tomita's theorem from state space Choquet theory,  we show that order isomorphism between the sets  of orthogonal measures (resp. finitely supported orthogonal measures) on state spaces endowed  the Choquet order  are given by Jordan *-isomorphims between corresponding operator algebras. It provides new complete  Jordan invariants for σ-finite von Neumann algebras in terms of decompositions of states and shows that one can recover physical system from associated system of convex decompositions (discrete or continuous) of a fixed state.