Causality in Schwinger’s Picture of Quantum Mechanics

This paper begins the study of the relation between causality and quantum mechanics, taking advantage of the groupoidal description of quantum mechanical systems inspired by Schwinger’s picture of quantum mechanics. After identifying causal structures on groupoids with a particular class of subcategories, called causal categories accordingly, it will be shown that causal structures can be recovered from a particular class of non-selfadjoint class of algebras, known as triangular operator algebras, contained in the von Neumann algebra of the groupoid of the quantum system. As a consequence of this, Sorkin’s incidence theorem will be proved and some illustrative examples will be discussed.

Some noncommutative subsequential weighted individual ergodic theorems

This paper is devoted to studying individual ergodic theorems for subsequential weighted ergodic averages on the noncommutative [Formula: see text]-spaces associated to a semifinite von Neumann algebra [Formula: see text]. In particular, we establish the convergence of these averages along sequences with density one and certain types of block sequences with positive lower density, and we extend known results along uniform sequences in the sense of Brunel and Keane.

Some aspects of quantum sufficiency for finite and full von Neumann algebras

Some features of the notion of sufficiency in quantum statistics are investigated. Three kinds of this notion are considered: plain sufficiency (called simply: sufficiency), strong sufficiency and Umegaki’s sufficiency. It is shown that for a finite von Neumann algebra with a faithful family of normal states the minimal sufficient von Neumann subalgebra is sufficient in Umegaki’s sense. Moreover, a proper version of the factorization theorem of Jenčová and Petz is obtained. The structure of the minimal sufficient subalgebra is described in the case of pure states on the full algebra of all bounded linear operators on a Hilbert space.

Completely order bounded maps on non-commutative $${\varvec{L_p}}$$-spaces

We define norms on $$L_p({\mathcal {M}}) \otimes M_n$$ L p ( M ) ⊗ M n where $${\mathcal {M}}$$ M is a von Neumann algebra and $$M_n$$ M n is the space of complex $$n \times n$$ n × n matrices. We show that a linear map $$T: L_p({\mathcal {M}}) \rightarrow L_q({\mathcal {N}})$$ T : L p ( M ) → L q ( N ) is decomposable if $${\mathcal {N}}$$ N is an injective von Neumann algebra, the maps $$T \otimes Id_{M_n}$$ T ⊗ I d M n have a common upper bound with respect to our defined norms, and $$p = \infty $$ p = ∞ or $$q = 1$$ q = 1 . For $$2p< q < \infty $$ 2 p < q < ∞ we give an example of a map $$T$$ T with uniformly bounded maps $$T \otimes Id_{M_n}$$ T ⊗ I d M n which is not decomposable.

The norm attainment problem for functions of projections

The paper is concerned with the problem of identifying the norm attaining operators in the von Neumann algebra generated by two orthogonal projections on a Hilbert space. Every skew projection on that Hilbert space is contained in such an algebra and hence the results of the paper also describe functions of skew projections and their adjoints that attain the norm.

A quantum route to the classical Lagrangian formalism

Using the recently developed groupoidal description of Schwinger’s picture of Quantum Mechanics, a new approach to Dirac’s fundamental question on the role of the Lagrangian in Quantum Mechanics is provided. It is shown that a function [Formula: see text] on the groupoid of configurations (or kinematical groupoid) of a quantum system determines a state on the von Neumann algebra of the histories of the system. This function, which we call q-Lagrangian, can be described in terms of a new function [Formula: see text] on the Lie algebroid of the theory. When the kinematical groupoid is the pair groupoid of a smooth manifold M, the quadratic expansion of [Formula: see text] will reproduce the standard Lagrangians on TM used to describe the classical dynamics of particles.