Separable spaces of continuous functions as Calkin algebras
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- by Pavlos Motakis
- J. Amer. Math. Soc. 37 (2024), 1-37
- DOI: https://doi.org/10.1090/jams/1024
- Published electronically: July 10, 2023
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Abstract:
It is proved that for every compact metric space $K$ there exists a Banach space $X$ whose Calkin algebra $\mathcal {L}(X)/\mathcal {K}(X)$ is homomorphically isometric to $C(K)$. This is achieved by appropriately modifying the Bourgain-Delbaen $\mathscr {L}_\infty$-space of Argyros and Haydon in such a manner that sufficiently many diagonal operators on this space are bounded.References
- S. A. Argyros and I. Deliyanni, Examples of asymptotic $l_1$ Banach spaces, Trans. Amer. Math. Soc. 349 (1997), no. 3, 973–995. MR 1390965, DOI 10.1090/S0002-9947-97-01774-1
- Spiros A. Argyros, Irene Deliyanni, and Andreas G. Tolias, Hereditarily indecomposable Banach algebras of diagonal operators, Israel J. Math. 181 (2011), 65–110. MR 2773038, DOI 10.1007/s11856-011-0004-x
- Spiros A. Argyros, Ioannis Gasparis, and Pavlos Motakis, On the structure of separable $\mathcal {L}_\infty$-spaces, Mathematika 62 (2016), no. 3, 685–700. MR 3521348, DOI 10.1112/S0025579315000492
- Spiros A. Argyros and Richard G. Haydon, A hereditarily indecomposable $\scr L_\infty$-space that solves the scalar-plus-compact problem, Acta Math. 206 (2011), no. 1, 1–54. MR 2784662, DOI 10.1007/s11511-011-0058-y
- Spiros A. Argyros and Pavlos Motakis, A reflexive hereditarily indecomposable space with the hereditary invariant subspace property, Proc. Lond. Math. Soc. (3) 108 (2014), no. 6, 1381–1416. MR 3218313, DOI 10.1112/plms/pdt062
- Spiros A. Argyros and Pavlos Motakis, The scalar-plus-compact property in spaces without reflexive subspaces, Trans. Amer. Math. Soc. 371 (2019), no. 3, 1887–1924. MR 3894038, DOI 10.1090/tran/7353
- Spiros A. Argyros and Pavlos Motakis, On the complete separation of asymptotic structures in Banach spaces, Adv. Math. 362 (2020), 106962, 51. MR 4050585, DOI 10.1016/j.aim.2019.106962
- Kevin Beanland and Lon Mitchell, Operators on the $\scr L_\infty$-spaces of Bourgain and Delbaen, Quaest. Math. 33 (2010), no. 4, 443–448. MR 2755547, DOI 10.2989/16073606.2010.541614
- J. Bourgain and F. Delbaen, A class of special ${\cal L}_{\infty }$ spaces, Acta Math. 145 (1980), no. 3-4, 155–176. MR 590288, DOI 10.1007/BF02414188
- Jean Bourgain and Gilles Pisier, A construction of ${\scr L}_{\infty }$-spaces and related Banach spaces, Bol. Soc. Brasil. Mat. 14 (1983), no. 2, 109–123. MR 756904, DOI 10.1007/BF02584862
- L. G. Brown, R. G. Douglas, and P. A. Fillmore, Extensions of $C^*$-algebras and $K$-homology, Ann. of Math. (2) 105 (1977), no. 2, 265–324. MR 458196, DOI 10.2307/1970999
- J. W. Calkin, Two-sided ideals and congruences in the ring of bounded operators in Hilbert space, Ann. of Math. (2) 42 (1941), 839–873. MR 5790, DOI 10.2307/1968771
- S. R. Caradus, W. E. Pfaffenberger, and Bertram Yood, Calkin algebras and algebras of operators on Banach spaces, Lecture Notes in Pure and Applied Mathematics, Vol. 9, Marcel Dekker, Inc., New York, 1974. MR 415345
- Irene Deliyanni and Antonis Manoussakis, Asymptotic $l_p$ hereditarily indecomposable Banach spaces, Illinois J. Math. 51 (2007), no. 3, 767–803. MR 2379722
- Ilijas Farah, All automorphisms of the Calkin algebra are inner, Ann. of Math. (2) 173 (2011), no. 2, 619–661. MR 2776359, DOI 10.4007/annals.2011.173.2.1
- W. T. Gowers, A solution to Banach’s hyperplane problem, Bull. London Math. Soc. 26 (1994), no. 6, 523–530. MR 1315601, DOI 10.1112/blms/26.6.523
- W. T. Gowers, A new dichotomy for Banach spaces, Geom. Funct. Anal. 6 (1996), no. 6, 1083–1093. MR 1421876, DOI 10.1007/BF02246998
- W. T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), no. 4, 851–874. MR 1201238, DOI 10.1090/S0894-0347-1993-1201238-0
- W. T. Gowers and B. Maurey, Banach spaces with small spaces of operators, Math. Ann. 307 (1997), no. 4, 543–568. MR 1464131, DOI 10.1007/s002080050050
- Carl Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 3, 91–123 (English, with French summary). MR 355482, DOI 10.5802/aif.473
- Bence Horváth and Tomasz Kania, Unital Banach algebras not isomorphic to Calkin algebras of separable Banach spaces, Proc. Amer. Math. Soc. 149 (2021), no. 11, 4781–4787. MR 4310103, DOI 10.1090/proc/15589
- Shizuo Kakutani, A generalization of Brouwer’s fixed point theorem, Duke Math. J. 8 (1941), 457–459. MR 4776
- Tomasz Kania, Piotr Koszmider, and Niels Jakob Laustsen, Banach spaces whose algebra of bounded operators has the integers as their $K_0$-group, J. Math. Anal. Appl. 428 (2015), no. 1, 282–294. MR 3326988, DOI 10.1016/j.jmaa.2015.03.021
- Tomasz Kania and Niels Jakob Laustsen, Ideal structure of the algebra of bounded operators acting on a Banach space, Indiana Univ. Math. J. 66 (2017), no. 3, 1019–1043. MR 3663335, DOI 10.1512/iumj.2017.66.6037
- Piotr Koszmider, Banach spaces of continuous functions with few operators, Math. Ann. 330 (2004), no. 1, 151–183. MR 2091683, DOI 10.1007/s00208-004-0544-z
- Niels Jakob Laustsen, $K$-theory for algebras of operators on Banach spaces, J. London Math. Soc. (2) 59 (1999), no. 2, 715–728. MR 1709676, DOI 10.1112/S0024610799007206
- Niels Jakob Laustsen, $K$-theory for the Banach algebra of operators on James’s quasi-reflexive Banach spaces, $K$-Theory 23 (2001), no. 2, 115–127. MR 1857077, DOI 10.1023/A:1017573608843
- J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in $L_{p}$-spaces and their applications, Studia Math. 29 (1968), 275–326. MR 231188, DOI 10.4064/sm-29-3-275-326
- Jordi Lopez-Abad, A Bourgain-Pisier construction for general Banach spaces, J. Funct. Anal. 265 (2013), no. 7, 1423–1441. MR 3073260, DOI 10.1016/j.jfa.2013.05.039
- Antonis Manoussakis, Anna Pelczar-Barwacz, and MichałŚwiȩtek, An unconditionally saturated Banach space with the scalar-plus-compact property, J. Funct. Anal. 272 (2017), no. 12, 4944–4983. MR 3639519, DOI 10.1016/j.jfa.2017.01.017
- B. Maurey and H. P. Rosenthal, Normalized weakly null sequence with no unconditional subsequence, Studia Math. 61 (1977), no. 1, 77–98. MR 438091, DOI 10.4064/sm-61-1-77-98
- P. Motakis and N. C. Phillips, New examples of $K$-groups of algebras of operators on Banach spaces, In preparation.
- Pavlos Motakis, Daniele Puglisi, and Andreas Tolias, Algebras of diagonal operators of the form scalar-plus-compact are Calkin algebras, Michigan Math. J. 69 (2020), no. 1, 97–152. MR 4071347, DOI 10.1307/mmj/1574845272
- Pavlos Motakis, Daniele Puglisi, and Despoina Zisimopoulou, A hierarchy of Banach spaces with $C(K)$ Calkin algebras, Indiana Univ. Math. J. 65 (2016), no. 1, 39–67. MR 3466455, DOI 10.1512/iumj.2016.65.5756
- E. Odell and Th. Schlumprecht, On the richness of the set of $p$’s in Krivine’s theorem, Geometric aspects of functional analysis (Israel, 1992–1994) Oper. Theory Adv. Appl., vol. 77, Birkhäuser, Basel, 1995, pp. 177–198. MR 1353459
- E. Odell and Th. Schlumprecht, A Banach space block finitely universal for monotone bases, Trans. Amer. Math. Soc. 352 (2000), no. 4, 1859–1888. MR 1637094, DOI 10.1090/S0002-9947-99-02425-3
- N. C. Phillips, Analogs of Cuntz algebras on $L^p$ spaces, 2012, arXiv:1201.4196.
- N. C. Phillips, Crossed products of $l^p$ operator algebras and the $k$-theory of Cuntz algebras on $l^p$ spaces, 2013, arXiv:1309.6406.
- N. C. Phillips, Simplicity of UHF and Cuntz algebras on $L^p$ spaces, 2013, arXiv:1309.0115.
- N. Christopher Phillips and Nik Weaver, The Calkin algebra has outer automorphisms, Duke Math. J. 139 (2007), no. 1, 185–202. MR 2322680, DOI 10.1215/S0012-7094-07-13915-2
- Grzegorz Plebanek, A construction of a Banach space $C(K)$ with few operators, Topology Appl. 143 (2004), no. 1-3, 217–239. MR 2081013, DOI 10.1016/j.topol.2004.03.001
- M. Rørdam, F. Larsen, and N. Laustsen, An introduction to $K$-theory for $C^*$-algebras, London Mathematical Society Student Texts, vol. 49, Cambridge University Press, Cambridge, 2000. MR 1783408
- Thomas Schlumprecht, An arbitrarily distortable Banach space, Israel J. Math. 76 (1991), no. 1-2, 81–95. MR 1177333, DOI 10.1007/BF02782845
- Matthew Tarbard, Hereditarily indecomposable, separable $\scr L_\infty$ Banach spaces with $\ell _1$ dual having few but not very few operators, J. Lond. Math. Soc. (2) 85 (2012), no. 3, 737–764. MR 2927806, DOI 10.1112/jlms/jdr066
- Matthew Tarbard, Operators on banach spaces of Bourgain-Delbaen type, ProQuest LLC, Ann Arbor, MI, 2013. Thesis (D.Phil.)–University of Oxford (United Kingdom). MR 3271750
- Bertram Yood, Difference algebras of linear transformations on a Banach space, Pacific J. Math. 4 (1954), 615–636. MR 68117, DOI 10.2140/pjm.1954.4.615
- D. Zisimopoulou, Bourgain-Delbaen $\mathcal {L}^{\infty }$-sums of Banach spaces, 2014, arXiv:1402.6564.
- András Zsák, A Banach space whose operator algebra has $K_0$-group $\Bbb Q$, Proc. London Math. Soc. (3) 84 (2002), no. 3, 747–768. MR 1888430, DOI 10.1112/S0024611502013448
Bibliographic Information
- Pavlos Motakis
- Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario M3J 1P3, Canada
- MR Author ID: 1037097
- Email: pmotakis@yorku.ca
- Received by editor(s): October 20, 2021
- Published electronically: July 10, 2023
- Additional Notes: The author was supported by NSERC Grant RGPIN-2021-03639.
- © Copyright 2023 American Mathematical Society
- Journal: J. Amer. Math. Soc. 37 (2024), 1-37
- MSC (2020): Primary 46B07, 46B25, 46B28, 46J10
- DOI: https://doi.org/10.1090/jams/1024
- MathSciNet review: 4654606