An Infinite Transitivity Theorem
Miles Gould
TL;DR
The article extends infinite Kadison transitivity to corona algebras, notably the Calkin algebra, by linking infinite transitivity to countable degree-1 saturation via excision. It introduces and analyzes quantum P-points and P-states, establishing equivalences between saturation properties and order-theoretic filter properties, and proving that P-representations attain infinite transitivity. The work shows that the existence of P-states and infinite transitivity is sensitive to set-theoretic assumptions: CH yields irreducible representations with infinite transitivity (and many P-states), while ZFC alone allows irreducible representations that fail infinite transitivity. A ZFC counterexample using idempotent ultrafilters demonstrates that infinite transitivity is not automatic for all irreducible representations of corona algebras. Overall, the paper clarifies when and how saturation and filter-theoretic conditions guarantee transitivity phenomena in massive C*-algebras and highlights the role of set-theoretic context in these operator-algebraic properties.
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.