Unital Banach algebras not isomorphic to Calkin algebras of separable Banach spaces

Horváth, Bence; Kania, Tomasz

doi:10.1090/proc/15589

Cited by 4 publications

(2 citation statements)

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“…With regards to the question of what unital algebras can be realized as Calkin algebras of a Banach space (see, e.g., Tarbard's PhD thesis [39, page 134] in 2012), there is a lot of progress to be made. For example, there does not exist a known property of unital Banach algebras that precludes them from being Calking algebras (some progress was made in [20] which is discussed further below). The appearance of the Argyros-Haydon construction led to the description of a plethora of explicit Calkin algebras, both finite and infinite dimensional.…”

Section: Open Problemsmentioning

confidence: 99%

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Separable Spaces of Continuous Functions as Calkin Algebras

Motakis¹

2021

Preprint

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Section: Open Problemsmentioning

confidence: 99%

“…Horváth and Kania proved in [20] that for any cardinal λ there exists a C(K) space of density 2 λ that is not the Calkin algebra of any space of density λ. Of course, this does not mean that C(K) is not the Calkin algebra of a space with larger density.…”

Section: Open Problemsmentioning

confidence: 99%

Separable Spaces of Continuous Functions as Calkin Algebras

Motakis¹

2021

Preprint

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Separable spaces of continuous functions as Calkin algebras

Motakis¹

2023

J. Amer. Math. Soc.

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It is proved that for every compact metric space K K there exists a Banach space X X whose Calkin algebra L ( X ) / K ( X ) \mathcal {L}(X)/\mathcal {K}(X) is homomorphically isometric to C ( K ) C(K) . This is achieved by appropriately modifying the Bourgain-Delbaen L ∞ \mathscr {L}_\infty -space of Argyros and Haydon in such a manner that sufficiently many diagonal operators on this space are bounded.

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The existence of UFO implies projectively universal morphisms

Balcerzak

Kania

2023

Proc. Amer. Math. Soc.

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Let C \mathcal C be a concrete category. We prove that if C \mathcal {C} admits a universally free object F \mathsf F , then there is a projectively universal morphism u : F → F u\colon \mathsf F\to \mathsf F , i.e., a morphism u u such that for any B ∈ C B\in \mathcal {C} and τ ∈ Mor ⁡ ( B ) \tau \in \operatorname {Mor}(B) there exists an epimorphism π ∈ Mor ⁡ ( F , B ) \pi \in \operatorname {Mor}(\mathsf F, B) such that π τ = u π \pi \tau = u \pi . This builds upon and extends various ideas by Darji and Matheron [Proc. Amer. Math. Soc. 145 (2017), pp. 251–265] who proved such a result for the category of separable Banach spaces with contractive operators as well as certain classes of dynamical systems on compact metric spaces. Specialising from our abstract setting, we conclude that the result applies to various categories of Banach spaces/lattices/algebras, C*-algebras, etc.

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Unital Banach algebras not isomorphic to Calkin algebras of separable Banach spaces

Cited by 4 publications

References 17 publications

Separable Spaces of Continuous Functions as Calkin Algebras

Separable Spaces of Continuous Functions as Calkin Algebras

Separable spaces of continuous functions as Calkin algebras

The existence of UFO implies projectively universal morphisms

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