Orthogonally Additive and Orthogonality Preserving Holomorphic Mappings between C*-Algebras

Abstract

We study holomorphic maps between C * -algebras A and B. When f : BA(0, ̺) −→ B is a holomorphic mapping whose Taylor series at zero is uniformly converging in some open unit ball U = BA(0, δ) and we assume that f is orthogonality preserving on Asa ∩ U , orthogonally additive on U and f (U ) contains an invertible element in B, then there exist a sequence (hn) in B * * and Jordan * -homomorphisms Θ, Θ :Θ(a n )hn, uniformly in a ∈ U . When B is abelian the hypothesis of B being unital and f (U ) ∩ inv(B) = ∅ can be relaxed to get the same statement.2010 MSC: Primary 46G20, 46L05; Secondary 46L51, 46E15, 46E50.

A note on 2-local representations of C^∗-algebras

“…Every homomorphism between C * -algebras preserves zero products and every * -homomorphism is orthogonality preserving. Orthogonality preserving bounded linear maps between C * -algebras have been completely described in [6, Theorem 17] (see [7] and [18] for completeness).…”

Orthogonally additive holomorphic maps between C$^{*}$-algebras

“…See Theorem 3.8 and Example 3.9 for details. In a very interesting recent paper [21], the authors there consider orthogonally additive holomorphic maps H between general C*-algebras, which preserve doubly orthogonality, i.e., In this case, with the extra assumption that the range of H contains an invertible element, a corresponding result to Theorem 3.11 is established in [21] through the technique of JB*-algebras. Nevertheless, we will see in Example 3.9 that one cannot directly make use of this new result to study zero product preserving holomorphic maps.…”

“…See Theorem 3.8 and Example 3.9 for details. In a very interesting recent paper [21], the authors there consider orthogonally additive holomorphic maps H between general C*-algebras, which preserve doubly orthogonality, i.e.,…”

“…In this case, with the extra assumption that the range of H contains an invertible element, a corresponding result to Theorem 3.11 is established in [21] through the technique of JB*-algebras. Nevertheless, we will see in Example 3.9 that one cannot directly make use of this new result to study zero product preserving holomorphic maps.…”

Orthogonally additive polynomials on the algebras of approximable operators

“…. , g k } satisfy the requirements (3) and (4). Let M be the subalgebra of F (X) generated by the set y i ⊗g j : i, j ∈ {1, .…”