An Infinite Transitivity Theorem

TL;DR

This paper extends an infinite transitivity theorem to massive C*-algebras, including the Calkin algebra, linking saturation properties to order-theoretic features of quantum filters and exploring their set-theoretic independence.

Contribution

It introduces an infinite transitivity theorem for massive C*-algebras, connecting saturation of pure states with quantum P-points, and examines the existence and limitations of such states.

Findings

Transitivity extends to massive C*-algebras including the Calkin algebra.

Saturation of pure states is equivalent to being a quantum P-point.

Existence of quantum P-points is independent of ZFC, but can be shown under certain set-theoretic assumptions.

Abstract

In this note, we promote an infinite Kadison transitivity theorem on massive $C^*$-algebras, including the Calkin algebra. This transitivity stems from the analog of countable degree-1 saturation on pure states which is inherited from these algebras via excision. We show this saturation to be equivalent to several order-theoretic properties on the quantum filter associated to the state, in particular the property of being a quantum P-point. While we show their existence is independent from ZFC, under basic set theoretic assumptions, we produce a plethora of these states. Finally, we find an irreducible representation of the Calkin algebra which fails infinite transitivity.