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Multiplicative maps that are close to an automorphism on algebras of linear transformations

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Marcoux L., Radjavi H., Sourour A.

Studia Mathematica

2013

tom 214

nr 3

279-296

Let 𝓗 be a complex, separable Hilbert space of finite or infinite dimension, and let ℬ(𝓗) be the algebra of all bounded operators on 𝓗. It is shown that if φ: ℬ(𝓗) → ℬ(𝓗) is a multiplicative map(not assumed linear) and if φ is sufficiently close to a linear automorphism of ℬ(𝓗) in some uniform sense, then it is actually an automorphism; as such, there is an invertible operator S in ℬ(𝓗) such that $φ(A) = S^{-1} AS$ for all A in ℬ(𝓗). When 𝓗 is finite-dimensional, similar results are obtained with the mere assumption that there exists a linear functional f on ℬ(𝓗) so that f ∘ φ is close to f ∘ μ for some automorphism μ of ℬ(𝓗).

Ograniczanie wyników

1 Studia Mathematica

1 Marcoux L.

1 Radjavi H.

1 Sourour A.

1 2013

Wyniki wyszukiwania

Multiplicative maps that are close to an automorphism on algebras of linear transformations