Lifting invertibles in von Neumann algebras

Author:
Robert R. Rogers

Journal:
Proc. Amer. Math. Soc. **113** (1991), 381-388

MSC:
Primary 46L10

DOI:
https://doi.org/10.1090/S0002-9939-1991-1087469-1

MathSciNet review:
1087469

Full-text PDF

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 well-known 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**, https://doi.org/10.1007/BF01350663**[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**, https://doi.org/10.1512/iumj.1977.26.26035**[3]**Catherine L. Olsen,*Index theory in von Neumann algebras*, Mem. Amer. Math. Soc.**47**(1984), no. 294, iv+71. MR**727597**, https://doi.org/10.1090/memo/0294**[4]**Masamichi Takesaki,*Theory of operator algebras. I*, Springer-Verlag, New York-Heidelberg, 1979. MR**548728****[5]**Jun Tomiyama,*Generalized dimension function for 𝑊*-algebras of infinite type*, Tôhoku Math. J. (2)**10**(1958), 121–129. MR**0102762**, https://doi.org/10.2748/tmj/1178244706**[6]**W. Wils,*Two-sided ideals in 𝑊*-algebras*, J. Reine Angew. Math.**244**(1970), 55–68. MR**0273427**, https://doi.org/10.1515/crll.1970.244.55

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
46L10

Retrieve articles in all journals with MSC: 46L10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1991-1087469-1

Keywords:
von Neumann Algebra,
Fredholm,
generalized dimension function,
generalized index

Article copyright:
© Copyright 1991
American Mathematical Society