Lifting invertibles in von Neumann algebras
Author:
Robert R. Rogers
Journal:
Proc. Amer. Math. Soc. 113 (1991), 381388
MSC:
Primary 46L10
MathSciNet review:
1087469
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Given , the algebra of bounded operators on a separable Hilbert space , and , the ideal of compact operators, it is a wellknown fact that in is a Fredholm operator if and only if is invertible in where is the canonical quotient map. A natural question arises: When can a Fredholm operator be perturbed by a compact operator to obtain an invertible operator? Equivalently, when does the invertible element lift to an invertible operator? The answer is well known: may be perturbed by a compact operator to obtain an invertible operator if and only if the Fredholm Index of is 0. In this case, the perturbation may be made as small in norm as we wish. Using the generalized Fredholm index for a von Neumann algebra developed by C. L. Olsen [3], the following generalization is obtained: let be a von Neumann algebra with norm closed ideal and canonical quotient map . Let be such that is invertible. Then there exists such that is invertible if and only if .
 [1]
Manfred
Breuer, Fredholm theories in von Neumann algebras. I, Math.
Ann. 178 (1968), 243–254. MR 0234294
(38 #2611)
 [2]
Victor
Kaftal, On the theory of compact operators in von Neumann algebras.
I, Indiana Univ. Math. J. 26 (1977), no. 3,
447–457. MR 0463970
(57 #3908)
 [3]
Catherine
L. Olsen, Index theory in von Neumann algebras, Mem. Amer.
Math. Soc. 47 (1984), no. 294, iv+71. MR 727597
(85k:46064), http://dx.doi.org/10.1090/memo/0294
 [4]
Masamichi
Takesaki, Theory of operator algebras. I, SpringerVerlag, New
YorkHeidelberg, 1979. MR 548728
(81e:46038)
 [5]
Jun
Tomiyama, Generalized dimension function for 𝑊*algebras of
infinite type, Tôhoku Math. J. (2) 10 (1958),
121–129. MR 0102762
(21 #1548)
 [6]
W.
Wils, Twosided ideals in 𝑊*algebras, J. Reine Angew.
Math. 244 (1970), 55–68. MR 0273427
(42 #8306)
 [1]
 M. Breuer, Fredholm theories in von Neumann algebras I, Math. Ann. 178 (1968), 243254. MR 0234294 (38:2611)
 [2]
 V. Kaftal, On the theory of compact operators in von Neumann algebras I, Indiana Univ. Math. J. 26 (1977), 447457. MR 0463970 (57:3908)
 [3]
 C. L. Olsen, Index theory in von Neumann algebras, Mem. Amer. Math. Soc., no. 294, Amer. Math. Soc., Providence, RI, 1984. MR 727597 (85k:46064)
 [4]
 M. Takesaki, Theory of operator algebras. I, SpringerVerlag, New York, 1979. MR 548728 (81e:46038)
 [5]
 J. Tomiyama, Generalized dimension function for algebras of infinite type, Tokoku Math. J. 10 (1958), 121129. MR 0102762 (21:1548)
 [6]
 W. Wils, Twosided ideals in algebras, J. Reine Angew. Math. 2424 (1970), 5568. MR 0273427 (42:8306)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
46L10
Retrieve articles in all journals
with MSC:
46L10
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199110874691
PII:
S 00029939(1991)10874691
Keywords:
von Neumann Algebra,
Fredholm,
generalized dimension function,
generalized index
Article copyright:
© Copyright 1991
American Mathematical Society
