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Commutativity Preserving Maps of Factors

Published online by Cambridge University Press:  20 November 2018

C. Robert Miers*
Affiliation:
University of Victoria, Victoria, British Columbia
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By a von Neumann algebra M we mean a weakly closed, self-adjoint algebra of operators on a Hilbert space which contains I, the identity operator. A factor is a von Neumann algebra whose centre consists of scalar multiples of I.

In all that follows ϕ:M → N will be a one to one, *-linear map from the von Neumann factor M onto the von Neumann algebra N such that both ϕ and ϕ−1 preserve commutativity. Our main result states that if M is not of type I2 then where is an isomorphism or an antiisomorphism, c is a non-zero scalar, and λ is a *-linear map from M into ZN, the centre of N.

Our interest in this problem was aroused by several recent results. In [1], Choi, Jafarian, and Radjavi proved that if S is the real linear space of n × n matrices over any algebraically closed field, n ≧ 3, and ψ a linear operator on S which preserves commuting pairs of matrices, then either ψ(S) is commutative or there exists a unitary matrix U such that

for all A in S. They proved an analogous result for the collection of all bounded self-adjoint operators on an infinite dimensional Hilbert space when ψ is one to one. Subsequently, Omladic [7] proved that if ψ:L(X)L(X) is a bijective linear operator preserving commuting pairs of operators where X is a non-trivial Banach space, then

where U is a bounded invertible operator on X and A′ is the adjoint of A.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Choi, M. D., Jafarian, A. A. and Radjavi, H., Linear maps preserving commutativity, Linear Algebra and Its Applications. 87 (1987), 227242.Google Scholar
2. Dixmier, J., Les algèbres d'operateurs dans l'espace Hilbertien, Cahiers Scientifiques. 25 (Gauthier-Villars, Paris, 1969).Google Scholar
3. Dye, H. A., On the geometry of projections in certain operator algebras, Ann. of Math.. 61 (1955), 7389.Google Scholar
4. Fack, T. and Harpe, P. de la, Sommes de commutateurs dans les algèbres de von Neumann finies continues, Ann. Inst. Fourier. 30 (1980), 4973.Google Scholar
5. Kadison, R. V., Normalcy in operator algebras, Duke Math. J.. 29 (1962), 459464.Google Scholar
6. Miers, C. R., Lie isomorphisms of factors, Trans. Amer. Math. Soc.. 147 (1970), 5563.Google Scholar
7. Omladic, M., On operators preserving commutativity, J. Func. Anal.. 66 (1986), 105122.Google Scholar
8. Pearcy, C. and Topping, D., Sums of small numbers of idempotents, Mich. Math. J.. 14 (1967), 453465.Google Scholar
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