# Multiplicative mappings at unit operator on B(H) 1

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```					   Scientia Magna
Vol. 5 (2009), No. 4, 57-64

Multiplicative mappings at unit
operator on B(H) 1
Jinping Jia † , Wansheng He     ‡
and Fandi Zhang

†‡       College of Mathematics and Statistics, Tianshui Normal University, Tianshui,
Gansu, 741001, China
E-mail: jjping2006@163.com

Abstract Let A be a subalgebra of B(H). We say that a linear mapping ϕ from A into
itself is a multiplicative mapping at Z(Z ∈ A) if ϕ(ST ) = ϕ(S)ϕ(T ) for any S, T ∈ A with
ST = Z. Let H be an inﬁnite dimensional complex Hilbert space, and let ϕ be a surjective
linear map on B(H). In this paper, we prove that if ϕ is a multiplicative mapping at I and
continuous in the weak operator topology, then ϕ is an automorphism. We also prove that if
ϕ is a weak continuous multiplicative mapping at any invertible operator with ϕ(I) = I then
ϕ is an automorphism.
Keywords Operator algebra, multiplicative mappings at unit operator, automorphism.

§1. Introduction and preliminaries
Let H be an inﬁnite dimensional complex Hilbert space. We denote by B(H) the algebra
of all bounded linear operators on H. The purpose of this paper is to show the following
theorem.
Theorem 1.1. Let ϕ : B(H) → B(H) be a continuous linear surjective mapping in the
weak operator topology. Then the following statements are equivalent:
(1) ϕ is a multiplicative mapping at I from B(H) into itself, i.e.

ϕ(ST ) = ϕ(S)ϕ(T )      (S, T ∈ B(H), ST = I).

(2) ϕ is an automorphism, i.e. there exists an invertible operator A ∈ B(H) such that

ϕ(T ) = AT A−1      (T ∈ B(H)).

Characterizing linear maps on operator algebras is one of the most active and fertile re-
search topics in the theory of operator algebras during the past one hundred years. Recently,
some authors have paid their attention to the study of automorphisms and derivations. Many
profound results have been obtained in these domain
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Description: Let A be a subalgebra of B(H). We say that a linear mapping [straight phi] from A into itself is a multiplicative mapping at Z(Z ∈ A) if [straight phi](ST) = [straight phi](S)[straight phi](T) for any S, T ∈ A with ST = Z. Let H be an infinite dimensional complex Hilbert space, and let [straight phi] be a surjective linear map on B(H). In this paper, we prove that if [straight phi] is a multiplicative mapping at I and continuous in the weak operator topology, then [straight phi] is an automorphism. We also prove that if [straight phi] is a weak continuous multiplicative mapping at any invertible operator with [straight phi](I) = I then [straight phi] is an automorphism. [PUBLICATION ABSTRACT]
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