Abstract
We prove that it is relatively consistent with the usual axioms of mathematics that all automorphisms of the Calkin algebra are inner. Together with a 2006 Phillips-Weaver construction of an outer automorphism using the Continuum Hypothesis, this gives a complete solution to a 1977 problem of Brown-Douglas-Fillmore. We also give a simpler and self-contained proof of the Phillips-Weaver result.
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AUTHOR = {Alekseev, M. A. and Glebski{\u\i},
L. Yu. and Gordon, E. I.},
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MRNUMBER = {1708567},
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DOI = {10.1023/A:1012485910692},
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AUTHOR = {Alperin, J. L. and Covington, Jacinta and Macpherson, Dugald},
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BOOKTITLE = {Ordered Groups and Infinite Permutation Groups},
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[new] I. Farah, All automorphisms of all Calkin algebras, 2010.
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author = {Farah, Ilijas},
TITLE = {All automorphisms of all Calkin algebras},
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YEAR={2010},
NOTE={preprint},
} -
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} -
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[PhWe:Calkin]
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VOLUME = {139},
YEAR = {2007},
NUMBER = {1},
PAGES = {185--202},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {46L40 (03E50 46L05)},
MRNUMBER = {2322680},
MRREVIEWER = {Ivan G. Todorov},
DOI = {10.1215/S0012-7094-07-13915-2},
ZBLNUMBER = {05180738},
} -
[Sak:On]
S. Sakai, "On the group isomorphism of unitary groups in $AW$-algebras," Tôhoku Math. J., vol. 7, pp. 87-95, 1955.
@article {Sak:On, MRKEY = {0073139},
AUTHOR = {Sakai, Sh{ô}ichir{ô}},
TITLE = {On the group isomorphism of unitary groups in {$AW$}-algebras},
JOURNAL = {Tôhoku Math. J.},
FJOURNAL = {The Tohoku Mathematical Journal. Second Series},
VOLUME = {7},
YEAR = {1955},
PAGES = {87--95},
ISSN = {0040-8735},
MRCLASS = {46.2X},
MRNUMBER = {0073139},
MRREVIEWER = {J. Feldman},
DOI = {10.2748/tmj/1178245106},
} -
[Sak:Pure] S. Sakai, "Pure states on $C^*$-algebras," in Advances in Quantum Dynamics, Providence, RI: Amer. Math. Soc., 2003, vol. 335, pp. 247-251.
@incollection {Sak:Pure, MRKEY = {2029629},
AUTHOR = {Sakai, Sh{ô}ichir{ô}},
TITLE = {Pure states on {$C\sp *$}-algebras},
BOOKTITLE = {Advances in Quantum Dynamics},
VENUE={{S}outh {H}adley, {MA},
2002},
SERIES = {Contemp. Math.},
VOLUME = {335},
PAGES = {247--251},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2003},
MRCLASS = {46L30 (46L05 46L40)},
MRNUMBER = {2029629},
ZBLNUMBER = {1049.46046},
} -
[Sh:Proper] S. Shelah, Proper Forcing, New York: Springer-Verlag, 1982, vol. 940.
@book {Sh:Proper, MRKEY = {0675955},
AUTHOR = {Shelah, Saharon},
TITLE = {Proper Forcing},
SERIES = {Lecture Notes in Math.},
VOLUME = {940},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1982},
PAGES = {xxix+496},
ISBN = {3-540-11593-5},
MRCLASS = {03-02 (03E35 03E40 03E50)},
MRNUMBER = {0675955},
MRREVIEWER = {F. R. Drake},
ZBLNUMBER = {0495.03035},
} -
[ShSte:PFA]
S. Shelah and J. Stepr=ans, "PFA implies all automorphisms are trivial," Proc. Amer. Math. Soc., vol. 104, iss. 4, pp. 1220-1225, 1988.
@article {ShSte:PFA, MRKEY = {0935111},
AUTHOR = {Shelah, Saharon and Stepr{ā}ns, Juris},
TITLE = {P{FA} implies all automorphisms are trivial},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {104},
YEAR = {1988},
NUMBER = {4},
PAGES = {1220--1225},
ISSN = {0002-9939},
CODEN = {PAMYAR},
MRCLASS = {03E05 (03E35)},
MRNUMBER = {0935111},
MRREVIEWER = {Judith Roitman},
DOI = {10.2307/2047617},
ZBLNUMBER = {0691.03031},
} -
[ShSte:Nontrivial] S. Shelah and J. Stepr=ans, "Nontrivial homeomorphisms of $\beta {\bf N}\setminus{\bf N}$ without the continuum hypothesis," Fund. Math., vol. 132, iss. 2, pp. 135-141, 1989.
@article {ShSte:Nontrivial, MRKEY = {1002627},
AUTHOR = {Shelah, Saharon and Stepr{ā}ns, Juris},
TITLE = {Nontrivial homeomorphisms of {$\beta {\bf N}\setminus{\bf N}$} without the continuum hypothesis},
JOURNAL = {Fund. Math.},
FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},
VOLUME = {132},
YEAR = {1989},
NUMBER = {2},
PAGES = {135--141},
ISSN = {0016-2736},
CODEN = {FUMAAJ},
MRCLASS = {54C05 (54D40)},
MRNUMBER = {1002627},
MRREVIEWER = {Hao Xuan Zhou},
ZBLNUMBER={0693.54003},
} -
[Ste:Size]
J. Stepr=ans, "The autohomeomorphism group of the \v Cech-Stone compactification of the integers," Trans. Amer. Math. Soc., vol. 355, iss. 10, pp. 4223-4240, 2003.
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AUTHOR = {Stepr{ā}ns, Juris},
TITLE = {The autohomeomorphism group of the \v {C}ech-{S}tone compactification of the integers},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {355},
YEAR = {2003},
NUMBER = {10},
PAGES = {4223--4240},
ISSN = {0002-9947},
CODEN = {TAMTAM},
MRCLASS = {03E35 (03E50 54D35)},
MRNUMBER = {1990584},
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DOI = {10.1090/S0002-9947-03-03329-4},
ZBLNUMBER = {1021.03047},
} -
[To:Partition] S. Todorvcević, Partition Problems in Topology, Providence, RI: Amer. Math. Soc., 1989, vol. 84.
@book {To:Partition, MRKEY = {0980949},
AUTHOR = {Todor{\v{c}}evi{ć},
Stevo},
TITLE = {Partition Problems in Topology},
SERIES = {Contemp. Math.},
VOLUME = {84},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1989},
PAGES = {xii+116},
ISBN = {0-8218-5091-1},
MRCLASS = {04-02 (03E05 03E50 54A25)},
MRNUMBER = {0980949},
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ZBLNUMBER = {0659.54001},
} -
[Ve:Applications] B. Velivcković, "Applications of the open coloring axiom," in Set Theory of the Continuum, New York: Springer-Verlag, 1992, vol. 26, pp. 137-154.
@incollection {Ve:Applications, MRKEY = {1233818},
AUTHOR = {Veli{\v{c}}kovi{ć},
Boban},
TITLE = {Applications of the open coloring axiom},
BOOKTITLE = {Set Theory of the Continuum},
VENUE={{B}erkeley, {CA},
1989},
SERIES = {Math. Sci. Res. Inst. Publ.},
VOLUME = {26},
PAGES = {137--154},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1992},
MRCLASS = {03E05 (03E35 03E40 03E65)},
MRNUMBER = {1233818},
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ZBLNUMBER = {0785.03032},
} -
[Ve:OCA]
B. Velivcković, "OCA and automorphisms of ${\mathcal P}(\omega) /\mathrm{fin}$," Top. Appl., vol. 49, pp. 1-12, 1992.
@article{Ve:OCA,
author={Veli{\v{c}}kovi{ć},
Boban},
TITLE={{OCA} and automorphisms of {${\mathcal P}(\omega) /\mathrm{fin}$}},
JOURNAL={Top. Appl.},
VOLUME={49},
YEAR={1992},
PAGES={1--12},
MRNUMBER={1202874},
ZBLNUMBER = {0785.03033},
DOI = {10.1016/0166-8641(93)90127-Y},
} -
[We:Set]
N. Weaver, "Set theory and $C^*$-algebras," Bull. Symbolic Logic, vol. 13, iss. 1, pp. 1-20, 2007.
@article {We:Set, MRKEY = {2300900},
AUTHOR = {Weaver, Nik},
TITLE = {Set theory and {$C\sp *$}-algebras},
JOURNAL = {Bull. Symbolic Logic},
FJOURNAL = {Bulletin of Symbolic Logic},
VOLUME = {13},
YEAR = {2007},
NUMBER = {1},
PAGES = {1--20},
ISSN = {1079-8986},
MRCLASS = {03E75 (03E35 03E50 46L05 46L30)},
MRNUMBER = {2300900},
MRREVIEWER = {Andreas Blass},
ZBLNUMBER = {1127.46043},
DOI = {10.2178/bsl/1174668215},
} -
[Wo:CH2]
H. W. Woodin, "The continuum hypothesis. II," Notices Amer. Math. Soc., vol. 48, iss. 7, pp. 681-690, 2001.
@article {Wo:CH2, MRKEY = {1842471},
AUTHOR = {Woodin, W. Hugh},
TITLE = {The continuum hypothesis. {II}},
JOURNAL = {Notices Amer. Math. Soc.},
FJOURNAL = {Notices of the American Mathematical Society},
VOLUME = {48},
YEAR = {2001},
NUMBER = {7},
PAGES = {681--690},
ISSN = {0002-9920},
CODEN = {AMNOAN},
MRCLASS = {03E50 (03A05 03E05 03E15 03E35)},
MRNUMBER = {1842471},
DOI = {10.2178/bsl/1174668215},
MRREVIEWER = {Yehuda Rav},
ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
ZBLNUMBER = {1047.03041},
}