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In: Advances in Mathematics, Vol. 452, 109747, 31.08.2024.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Choi, Y, Gardella, E & Thiel, H 2024, 'Rigidity results for L p -operator algebras and applications', *Advances in Mathematics*, vol. 452, 109747. https://doi.org/10.1016/j.aim.2024.109747

Choi, Y., Gardella, E., & Thiel, H. (2024). Rigidity results for L p -operator algebras and applications. *Advances in Mathematics*, *452*, Article 109747. https://doi.org/10.1016/j.aim.2024.109747

Choi Y, Gardella E, Thiel H. Rigidity results for L p -operator algebras and applications. Advances in Mathematics. 2024 Aug 31;452:109747. Epub 2024 Jul 1. doi: 10.1016/j.aim.2024.109747

@article{eb289289cf544822aa4ecdbd703e3e37,

title = "Rigidity results for L p -operator algebras and applications",

abstract = "For p ∈ [ 1 , ∞ ) , we show that every unital L p -operator algebra contains a unique maximal C ⁎ -subalgebra, which is always abelian if p ≠ 2 . Using this, we canonically associate to every unital L p -operator algebra A an {\'e}tale groupoid G A , which in many cases of interest is a complete invariant for A. By identifying this groupoid for large classes of examples, we obtain a number of rigidity results that display a stark contrast with the case p = 2 ; the most striking one being that of crossed products by topologically free actions. Our rigidity results give answers to questions concerning the existence of isomorphisms between different algebras. Among others, we show that for the L p -analog O 2 p of the Cuntz algebra, there is no isometric isomorphism between O 2 p and O 2 p ⊗ p O 2 p , when p ≠ 2 . In particular, we deduce that there is no L p -version of Kirchberg's absorption theorem, and that there is no K-theoretic classification of purely infinite simple amenable L p -operator algebras for p ≠ 2 . Our methods also allow us to recover a folklore fact in the case of C*-algebras ( p = 2 ), namely that no isomorphism O 2 ≅ O 2 ⊗ O 2 preserves the canonical Cartan subalgebras.",

author = "Yemon Choi and Eusebio Gardella and Hannes Thiel",

year = "2024",

month = aug,

day = "31",

doi = "10.1016/j.aim.2024.109747",

language = "English",

volume = "452",

journal = "Advances in Mathematics",

issn = "0001-8708",

publisher = "Academic Press Inc.",

}

TY - JOUR

T1 - Rigidity results for L p -operator algebras and applications

AU - Choi, Yemon

AU - Gardella, Eusebio

AU - Thiel, Hannes

PY - 2024/8/31

Y1 - 2024/8/31

N2 - For p ∈ [ 1 , ∞ ) , we show that every unital L p -operator algebra contains a unique maximal C ⁎ -subalgebra, which is always abelian if p ≠ 2 . Using this, we canonically associate to every unital L p -operator algebra A an étale groupoid G A , which in many cases of interest is a complete invariant for A. By identifying this groupoid for large classes of examples, we obtain a number of rigidity results that display a stark contrast with the case p = 2 ; the most striking one being that of crossed products by topologically free actions. Our rigidity results give answers to questions concerning the existence of isomorphisms between different algebras. Among others, we show that for the L p -analog O 2 p of the Cuntz algebra, there is no isometric isomorphism between O 2 p and O 2 p ⊗ p O 2 p , when p ≠ 2 . In particular, we deduce that there is no L p -version of Kirchberg's absorption theorem, and that there is no K-theoretic classification of purely infinite simple amenable L p -operator algebras for p ≠ 2 . Our methods also allow us to recover a folklore fact in the case of C*-algebras ( p = 2 ), namely that no isomorphism O 2 ≅ O 2 ⊗ O 2 preserves the canonical Cartan subalgebras.

AB - For p ∈ [ 1 , ∞ ) , we show that every unital L p -operator algebra contains a unique maximal C ⁎ -subalgebra, which is always abelian if p ≠ 2 . Using this, we canonically associate to every unital L p -operator algebra A an étale groupoid G A , which in many cases of interest is a complete invariant for A. By identifying this groupoid for large classes of examples, we obtain a number of rigidity results that display a stark contrast with the case p = 2 ; the most striking one being that of crossed products by topologically free actions. Our rigidity results give answers to questions concerning the existence of isomorphisms between different algebras. Among others, we show that for the L p -analog O 2 p of the Cuntz algebra, there is no isometric isomorphism between O 2 p and O 2 p ⊗ p O 2 p , when p ≠ 2 . In particular, we deduce that there is no L p -version of Kirchberg's absorption theorem, and that there is no K-theoretic classification of purely infinite simple amenable L p -operator algebras for p ≠ 2 . Our methods also allow us to recover a folklore fact in the case of C*-algebras ( p = 2 ), namely that no isomorphism O 2 ≅ O 2 ⊗ O 2 preserves the canonical Cartan subalgebras.

U2 - 10.1016/j.aim.2024.109747

DO - 10.1016/j.aim.2024.109747

M3 - Journal article

VL - 452

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

M1 - 109747

ER -