On Limit theorems in $JW$- algebras

TL;DR

This paper investigates convergence properties and ergodic theorems in $JW$-algebras, focusing on bundle convergence, conditional expectations, and supermartingale convergence.

Contribution

It introduces new ergodic theorems based on bundle convergence and explores conditional expectations in reversible $JW$-algebras, advancing the understanding of their convergence behavior.

Findings

Proved ergodic theorems with respect to bundle convergence.

Established convergence of supermartingales in $JW$-algebras.

Analyzed conditional expectations in reversible $JW$-algebras.

Abstract

In the present paper, we study bundle convergence in $JW$- algebra and prove certain ergodic theorems with respect to such convergence. Moreover, conditional expectations of reversible $JW$-algebras are considered. Using such expectations, the convergence of supermartingales in such is established.