of the compact operators is zero

Authors:
L. G. Brown and Claude Schochet

Journal:
Proc. Amer. Math. Soc. **59** (1976), 119-122

MSC:
Primary 47B05; Secondary 58G15

DOI:
https://doi.org/10.1090/S0002-9939-1976-0412863-0

MathSciNet review:
0412863

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that of the compact operators is zero. This theorem has the following operator-theoretic formulation: *any invertible operator of the form* (*identity*) (*compact*) *is the product of* (*at most eight*) *multiplicative commutators* , *where each* *is of the form* (*identity*) (*compact*). The proof uses results of L. G. Brown, R. G. Douglas, and P. A. Fillmore on essentially normal operators and a theorem of A. Brown and C. Pearcy on multiplicative commutators.

**[1]**Arlen Brown and Carl Pearcy,*Multiplicative commutators of operators*, Canad. J. Math.**18**(1966), 737–749. MR**0200720**, https://doi.org/10.4153/CJM-1966-074-1**[2]**Lawrence G. Brown,*The determinant invariant for operators with trace class self commutators*, Proceedings of a Conference on Operator Theory (Dalhousie Univ., Halifax, N.S., 1973) Springer, Berlin, 1973, pp. 210–228. Lecture Notes in Math., Vol. 345. MR**0390830****[3]**Lawrence G. Brown,*Operator algebras and algebraic 𝐾-theory*, Bull. Amer. Math. Soc.**81**(1975), no. 6, 1119–1121. MR**0383090**, https://doi.org/10.1090/S0002-9904-1975-13943-7**[4]**L. G. Brown, R. G. Douglas, and P. A. Fillmore,*Unitary equivalence modulo the compact operators and extensions of 𝐶*-algebras*, Proceedings of a Conference on Operator Theory (Dalhousie Univ., Halifax, N.S., 1973) Springer, Berlin, 1973, pp. 58–128. Lecture Notes in Math., Vol. 345. MR**0380478****[5]**L. G. Brown, R. G. Douglas, and P. A. Fillmore,*Extensions of 𝐶*-algebras, operators with compact self-commutators, and 𝐾-homology*, Bull. Amer. Math. Soc.**79**(1973), 973–978. MR**0346540**, https://doi.org/10.1090/S0002-9904-1973-13284-7**[6]**J. William Helton and Roger E. Howe,*Integral operators: commutators, traces, index and homology*, Proceedings of a Conference Operator Theory (Dalhousie Univ., Halifax, N.S., 1973) Springer, Berlin, 1973, pp. 141–209. Lecture Notes in Math., Vol. 345. MR**0390829****[7]**Jerome Kaminker and Claude Schochet,*Steenrod homology and operator algebras*, Bull. Amer. Math. Soc.**81**(1975), no. 2, 431–434. MR**0450997**, https://doi.org/10.1090/S0002-9904-1975-13775-X**[8]**John Milnor,*Introduction to algebraic 𝐾-theory*, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. Annals of Mathematics Studies, No. 72. MR**0349811****[9]**Carl Pearcy and David Topping,*On commutators in ideals of compact operators*, Michigan Math. J.**18**(1971), 247–252. MR**0284853**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1976-0412863-0

Keywords:
Compact operator,
essentially normal operator,
algebraic -theory,
extensions of -algebras

Article copyright:
© Copyright 1976
American Mathematical Society