Maps on idempotents

Abstract

Abstract. Let X be an infinite-dimensional real or complex Banach space, B(X) the algebra of all bounded linear operators on X, and P (X) ⊂ B(X) the subset of all idempotents. We characterize bijective maps on P (X) preserving commutativity in both directions. This unifies and extends the characterizations of two types of automorphisms of P (X), with respect to the orthogonality relation and with respect to the usual partial order; the latter have been previously characterized by Ovchinnikov. We also describe bijective orthogonality preserving maps on the set of idempotents of a fixed finite rank. As an application we present a nonlinear extension of the structural result for bijective linear biseparating maps on B(X). Introduction and statement of main results.Let X be a real or complex Banach space and B(X) the algebra of all bounded linear operators on X. Denote by X * the dual of X and by P (X) ⊂ B(X) the subset of all idempotents. Recall that P (X) is a poset with the partial order defined by P ≤ Q ⇔ P Q = QP = P , P, Q ∈ P (X). Orthogonality is another important relation on P (X). Two idempotents P, Q ∈ P (X) are said to be orthogonal if P Q = QP = 0. In this case we write P ⊥ Q. Motivated by some problems in quantum mechanics (see the review MR 95a:46093) Ovchinnikov [16] characterized automorphisms of the poset P (X) in the case that X is a Hilbert space of dimension at least 3. Recall that an automorphism φ of the poset P (X) is a bijective map preserving order in both directions, that is, P ≤ Q if and only if φ(P ) ≤ φ(Q), P, Q ∈ P (X).As far as we know, the automorphisms of P (X) with respect to the orthogonality relation have not been treated in the literature. In fact, the structural result for bijective maps on P (X) preserving orthogonality in both directions follows almost directly from the result of Ovchinnikov. Namely, for a subset S ⊂ P (X) denote by S ⊥ the set of all idempotents from P (X) that are orthogonal to every member of S. In the case S = {P } we simply write P ⊥ = {P } ⊥ . Then, as we shall see, it is easy to verify that for an arbitrary pair of idempotents P, Q ∈ P (X) we have P ≤ Q if and only if

Maps completely preserving idempotents and maps completely preserving square-zero operators

“…In order to prove the main theorem in this section, we need a lemma which is slightly different from the main results in [22], [24] and [26]. The proof is similar and we omit it here.…”

Maps on idempotent operators

“…In [25] we have considered orthogonality preserving maps defined not only on the whole set I(X), but also orthogonality preserving maps defined on the subset I n (X) ⊂ I(X) consisting of all idempotents of rank n. Here, n is a fixed positive integer. It was proved that every bijective map on I n (X) preserving orthogonality in both directions is of a standard form.…”

“…For example, Araujo and Jarosz [1] obtained a characterization of linear biseparating maps (maps preserving zero products) on B(X). Using the above suggested approach a non-linear extension of their result was given in [25].…”

Wigner-Type Theorems for Projections

“…In [14], the author consider the set P (X) of all projections defined on a Banach space X and he proves that Φ : P (X) → P (X) is an order automorphism of P (X) if and only if Φ preserves the orthogonality relation in both directions.…”