Linear Orthogonality Preservers of Standard Operator Algebras

Abstract

In 2003, Araujo and Jarosz showed that every bijective linear map θ : A → B between unital standard operator algebras preserving zero products in two ways is a scalar multiple of an inner automorphism. Later in 2007, Zhao and Hou showed that similar results hold if both A, B are unital standard algebras on Hilbert spaces and θ preserves range or domain orthogonality. In particular, such maps are automatically bounded. In this paper, we will study linear orthogonality preservers in a unified way. We will show that every surjective linear map between standard operator algebras preserving range/domain orthogonality carries a standard form, and is thus automatically bounded.

Linear Maps Which are Anti-derivable at Zero

“…Let us simply observe that a linear map T between Banach algebras is a homomorphism at zero if and only if it preserves zero products (i.e., ab = 0 implies T (a)T (b) = 0). We find in this way a natural link with the results on zero products preservers (see, for example, [1,2,8,10,28,29,32,33,[47][48][49][50][51] for additional details and results). Burgos, Cabello-Sánchez and the third author of this note explore in [6] those linear maps between C * -algebras which are * -homomorphisms at certain points of the domain, for example, at the unit element or at zero.…”

Linear disjointness preservers of W*-algebras

“…On the other hand, it is shown in [2,28] that every surjective linear map θ : A → B between two standard operator algebras preserving zero products, or range/domain orthogonality in two directions is basically an inner automorphism, and thus it is automatically bounded as well. Recall that standard operator algebras are those containing all finite rank operators.…”

Operators preserving the strong Birkhoff–James orthogonality on B(H)

“…Let us note that the proof of the part (b) is a modification of the proof of Theorem 1 from [21], but we include all the details for the sake of completeness. (a) Let A ∈ B(H) be a rank-one operator.…”

“…[6,8,9,15,16,19,21,22]). The notion of orthogonality in a normed linear space can be introduced in many ways.…”

“…It was shown in [23] (see also [21]) that every surjective linear (conjugate linear) operator Φ : B(H) → B(H) satisfying (1.1), where ∼ is the orthogonality relation ⊥, is of the form Φ(X) = UXB (Φ(X) = U (X * ) t B) for some unitary U ∈ B(H) and invertible B ∈ B(H). A characterization of surjective (conjugate) linear operators Φ satisfying (1.1), where ∼ is the orthogonality relation ⊥ B , was given in [22].…”