The aim of this paper is to start a systematic investigation of the existence of angular limits and angular derivatives of holomorphic maps of infinite dimensional Siegel domains in $J^{*}$-algebras. Since $J^{*}$-algebras are natural generalizations of $C^{*}$-algebras, $B^{*}$-algebras, $J{C}^{*}$-algebras, ternary algebras and complex Hilbert spaces, various significant results follow. Examples are given.

We study the reflexivity of the automorphism (and the isometry) group of the Banach algebras $L_{\infty }\left(\mu \right)$ for various measures μ. We prove that if μ is a non-atomic σ-finite measure, then the automorphism group (or the isometry group) of $L_{\infty }\left(\mu \right)$ is [algebraically] reflexive if and only if $L_{\infty }\left(\mu \right)$ is *-isomorphic to $L_{\infty }[0,1]$. For purely atomic measures, we show that the group of automorphisms (or isometries) of ${\ell }_{\infty }\left(\Gamma \right)$ is reflexive if and only if Γ has non-measurable cardinal. So, for most “practical” purposes, the automorphism group...

The aim of this paper is to prove the statement announced in the title which can be reformulated in the following way. Let H be a separable infinite-dimensional Hilbert space and let Φ: B(H) → B(H) be a continuous linear mapping with the property that for every A ∈ B(H) there exists a sequence $\left({\Phi }_n\right)$ of automorphisms of B(H) (depending on A) such that $\Phi \left(A\right)=li{m}_n{\Phi }_n\left(A\right)$. Then Φ is an automorphism. Moreover, a similar statement holds for the set of all surjective isometries of B(H).

Let X be an infinite-dimensional Banach space, and let ϕ be a surjective linear map on B(X) with ϕ(I) = I. If ϕ preserves injective operators in both directions then ϕ is an automorphism of the algebra B(X). If X is a Hilbert space, then ϕ is an automorphism of B(X) if and only if it preserves surjective operators in both directions.