Trivial endomorphisms of the Calkin algebra

Vaccaro, Andrea

doi:10.1007/s11856-021-2284-0

Cited by 2 publications

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“…This is a consequence of [51, Theorem B] and the straightforward fact that in this situation, there are no topologically trivial isomorphisms. The same axioms imply that the Calkin algebra has no outer automorphisms ([21, 22, §17.2–17.8]) and even that each one of its endomorphisms is unitarily equivalent to the amplifications by a matrix algebra ([50]). However, the Continuum Hypothesis implies that the Calkin algebra has outer automorphisms ([44], see also [22, §17.1]).…”

Section: Discussionmentioning

confidence: 99%

The Calkin algebra, Kazhdan's property (T), and strongly self‐absorbing C∗$\mathrm{C}^*$‐algebras

Farah

2023

Proceedings of London Math Soc

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It is well known that the relative commutant of every separable nuclear ‐subalgebra of the Calkin algebra has a unital copy of Cuntz algebra . We prove that the Calkin algebra has a separable ‐subalgebra whose relative commutant has no simple, unital, and noncommutative ‐subalgebra. On the other hand, the corona of every stable, separable ‐algebra that tensorially absorbs the Jiang–Su algebra has the property that the relative commutant of every separable ‐subalgebra contains a unital copy of . Analogous result holds for other strongly self‐absorbing ‐algebras. As an application, the Calkin algebra is not isomorphic to the corona of the stabilization of the Cuntz algebra , any other Kirchberg algebra, or even the corona of the stabilization of any unital, ‐stable ‐algebra.

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