Maps preserving peripheral spectrum of Jordan semi-triple products of operators

Abstract

Let A and B be (not necessarily unital or closed) standard operator algebras on complex Banach spaces X and Y, respectively. For a bounded linear operator A on X, the peripheral spectrum σ π ( A ) of A is the set σ π ( A ) = { z ∈ σ ( A ) : | z | = max ω ∈ σ ( A ) | ω | } , where σ ( A ) denotes the spectrum of A. Assume that Φ : A → B is a map the range of which contains all operators of rank at most two. It is shown that the map Φ satisfies the condition that σ π ( BAB ) = σ π ( Φ ( B ) Φ ( A ) Φ ( B ) ) for all A , B ∈ A if and only if there exists a scalar λ ∈ C with λ 3 = 1 and either there exists an invertible operator T ∈ B ( X , Y ) such that Φ ( A ) = λ TAT - 1 for every A ∈ A ; or there exists an invertible operator T ∈ B ( X ∗ , Y ) such that Φ ( A ) = λ TA ∗ T - 1 for every A ∈ A . If X = H and Y = K are complex Hilbert spaces, the maps preserving the peripheral spectrum of the Jordan skew semi-triple product BA ∗ B are also characterized. Such maps are of the form A ↦ UAU ∗ or A ↦ UA t U ∗ , where U ∈ B ( H , K ) is a unitary operator, A t denotes the transpose of A in an arbitrary but fixed orthonormal basis of H.