Spreadability for Quantum Stochastic Processes, with an Application to Boolean Commutation Relations

Area of Science:

• Quantum Probability
• Stochastic Processes
• Algebraic Structures

Background:

• Spreadability is crucial for managing quantum stochastic processes.
• Understanding the underlying algebraic structures is key to defining and analyzing spreadability.

Purpose of the Study:

• To analyze the structure of monoids acting on integer indices for quantum stochastic processes.
• To simplify the definition of spreadability by relating it to monoid actions.
• To investigate spreadability in the context of boolean C*-algebras and establish equivalences with other process types.

Main Methods:

• Detailed study of monoids generated by partial shifts and increasing maps on integers.
• Analysis of semidirect product structures of these monoids.
• Application of monoid invariance principles to define and analyze spreadability.
• Investigation using boolean C*-algebras and boolean Fock spaces.
• Adaptation of the Ryll-Nardzewski theorem for the boolean case.

Main Results:

• Three specific monoids acting on integers are identified and shown to be strictly ordered.
• The monoid of strictly increasing maps with finite complement range is a semidirect product.
• Spreadability can be defined via invariance under a simpler monoid action.
• Spreadable, exchangeable, and stationary stochastic processes are shown to be equivalent in the boolean context.

Conclusions:

• The study provides a simplified framework for understanding spreadability in quantum stochastic processes.
• The equivalence of spreadable, exchangeable, and stationary processes in the boolean case offers a unified perspective.
• The findings contribute to the algebraic and probabilistic analysis of quantum systems.