Bilocal *-automorphisms of B(h)

In this note we show that the bilocal *-automorphisms of the C∗-algebra B(H) of all bounded linear operators acting on a complex infinite dimensional separable Hilbert space H are precisely the unital algebra *-endomorphisms of B(H). The study of local derivations of operator algebras has been initiated by Kadison [5] and Larson and Sourour [8]. A linear map ∆ on an algebra is called a local derivation if for each point in the algebra the value of ∆ at that point coincides with the value of a derivation at the point (the derivation may vary from point to point). In [5] and [8] results were presented which show that in certain settings local derivations are in fact derivations. As for automorphisms of operator algebras, the first hint to the concept of local automorphisms appeared in [7] (see Some concluding remarks (5) in [7]). The definition is straightforward: a local automorphism of a given Banach algebra A is a linear map φ : A → A with the property that for every x ∈ A there exists an (algebra) automorphism φx of A such that φ(x) = φx(x). In the paper [8], Larson and Sourour proved that if X is an infinite dimensional Banach space, then every surjective local automorphism of the algebra B(X) of all bounded linear operators on X is an automorphism. In [1], Brešar and Šemrl showed that in the case of an infinite dimensional separable Hilbert space H the assumption of surjectivity can be relaxed, i.e., every local automorphism of B(H) is an automorphism. Since then considerable attention has been paid to those concepts, and several results have been obtained and published concerning local derivations and local automorphisms. For some more details see Introduction 0.5 and Chapter 3 in [9]. In the paper [12] Zhu and Xiong introduced the interesting notion of bilocal derivations. If X is a Banach space and A is a subalgebra of B(X), then the linear map Θ : A → A is called a bilocal derivation if for every T ∈ A and x ∈ X there exists a derivation δT,x : A → A such that Θ(T )x = δT,x(T )x. One of the main results in [12] tells us that if A is a Banach subalgebra of B(X) which contains all finite rank operators and the identity, 2010 Mathematics Subject Classification. Primary: 47B49. Secondary: 15A86, 47L10.