A characterization and sum decomposition for operator ideals
Authors:
Andreas Blass and Gary Weiss
Journal:
Trans. Amer. Math. Soc. 246 (1978), 407417
MSC:
Primary 47D25; Secondary 03E50
MathSciNet review:
515547
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Abstract: Let be the ring of bounded operators on a separable Hubert space. Assuming the continuum hypothesis, we prove that in every twosided ideal that contains an operator of infinite rank is the sum of two smaller twosided ideals. The proof involves a new combinatorial description of ideals of . This description is also used to deduce some related results about decompositions of ideals. Finally, we discuss the possibility of proving our main theorem under weaker assumptions than the continuum hypothesis and the impossibility of proving it without the axiom of choice.
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 A. Brown, C. Pearcy and N. Salinas, Ideals of compact operators on Hilbert space, Michigan Math. J. 18 (1971), 373384. MR 0291819 (45:909)
 [2]
 J. W. Calkin, Twosided ideals and congruences in the ring of bounded operators on Hilbert space, Ann. of Math. (2) 42 (1941), 839873. MR 0005790 (3:208c)
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 F. Hausdorff, Die Graduierung nach dem Endverlauf, Abh. Sächs. Ges. Wiss. 31 (1909), 295334.
 [4]
 K. Kunen, Ultrafilters and independent sets, Trans. Amer. Math. Soc. 172 (1972), 299306. MR 0314619 (47:3170)
 [5]
 , Some points in , Math. Proc. Cambridge Philos. Soc. 80 (1976), 385398. MR 0427070 (55:106)
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 D. A. Martin and R. Solovay, Internal Cohen extensions, Ann. Math. Logic 2 (1970), 143178. MR 0270904 (42:5787)
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 D. Morris and N. Salinas, Semiprime ideals and irreducible ideals in the ring of bounded operators on Hilbert space, Indiana Univ. Math. J. 23 (1974), 575589. MR 0326414 (48:4758)
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 N. Salinas, Symmetric norm ideals and relative conjugate ideals, Trans. Amer. Math. Soc. 188 (1974), 213240. MR 0336371 (49:1146)
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 R. Solovay, A model of set theory in which every set of reals is Lebesgue measurable, Ann. of Math. (2) 92 (1970), 156. MR 0265151 (42:64)
 [10]
 G. Weiss, Commutators and operator ideals, Thesis, Univ. of Michigan, Ann Arbor, Mich., 1975.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719780515547X
PII:
S 00029947(1978)0515547X
Keywords:
Hubert space,
operator ideals,
compact operators,
continuum hypothesis,
axiom of choice,
Calkin ideal sets
Article copyright:
© Copyright 1978 American Mathematical Society
