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]**A. Brown and C. Pearcy,*Multiplicative commutators of operators*, Canad. J. Math.**18**(1966), 737-749. MR**34**#608. MR**0200720 (34:608)****[2]**L. G. Brown,*The determinant invariant for operators with trace class self-commutators*, Proc. Conf. on Operator Theory, Lecture Notes in Math., vol. 345, Springer-Verlag, New York, 1973. MR**0390830 (52:11653)****[3]**-,*Operator algebras and algebraic*-*theory*, Bull. Amer. Math. Soc.**81**(1975), 1119-1121. MR**0383090 (52:3971)****[4]**L. G. Brown, R. G. Douglas and P. A. Fillmore,*Unitary equivalence modulo the compact operators and extensions of*-*algebras*, Proc. Conf. on Operator Theory, Lecture Notes in Math., vol. 345, Springer-Verlag, New York, 1973. MR**0380478 (52:1378)****[5]**-,*Extensions of*-*algebras, operators with compact self-commutators, and*-*homology*, Bull. Amer. Math. Soc.**79**(1973), 973-978. MR**0346540 (49:11265)****[6]**J. W. Helton and R. E. Howe,*Integral operators*:*commutators, traces, index and homology*, Proc. Conf. on Operator Theory, Lecture Notes in Math., vol. 345, Springer-Verlag, New York, 1973. MR**0390829 (52:11652)****[7]**J. Kaminker and C. Schochet,*Steenrod homology and operator algebras*, Bull. Amer. Math. Soc.**81**(1975), 431-434. MR**0450997 (56:9287)****[8]**J. Milnor,*Introduction to algebraic*-*theory*, Ann. of Math. Studies, no. 72, Princeton Univ. Press, Princeton, N. J., 1971. MR**0349811 (50:2304)****[9]**C. Pearcy and D. Topping,*On commutators in ideals of compact operators*, Michigan Math. J.**18**(1971), 247-252. MR**44**#2077. MR**0284853 (44:2077)**

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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