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arXiv · 2512.07549

An Infinite Transitivity Theorem

Miles Gould
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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.

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Miles Gould. 2025-12-08. An Infinite Transitivity Theorem. https://arxiv.org/abs/2512.07549

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