On the extension of isometries between the unit spheres of a C*-algebra and 𝐵(𝐻)

Abstract

Abstract. Given two complex Hilbert spaces H and K, let S(B(H)) and S(B(K)) denote the unit spheres of the C * -algebras B(H) and B(K) of all bounded linear operators on H and K, respectively. We prove that every surjective isometry f : S(B(K)) → S(B(H)) admits an extension to a surjective complex linear or conjugate linear isometry T : B(K) → B(H).This provides a positive answer to Tingley's problem in the setting of B(H) spaces.

Tingley's problem through the facial structure of an atomic JBW⁎-triple

“…In our main result we prove that every surjective isometry f : S(E) → S(B) admits a unique extension to a surjective real linear isometry T : E → B (see Theorem 2.9). This theorem extends the main conclusion in [15] to the setting of atomic JBW * -triples. In [15], we strived for arguments essentially based on standard techniques of C * -algebras, Geometry and Functional Analysis.…”

“…The spaces in this class enjoy a unique geometry which makes more interesting the study of certain geometric problems in a wider setting. This paper is devoted to extend the recent results in [15] to the context of atomic JBW * -triples (i.e. ℓ ∞ -sums of Cartan factors).…”

“…In a recent contribution we establish that every surjective isometry between the unit spheres of two B(H)-spaces extends uniquely to a surjective complex linear or conjugate linear surjective isometry between the corresponding spaces (see [15]). This result constitutes a positive answer to Tingley's isometric extension problem [31] in the setting of B(H)-spaces and atomic von Neumann algebras.…”

“…Solutions to Tingley's problem for compact operators, compact C * -algebras and weakly compact JB * -triples have been previously obtained in [26,14]. For additional information on the historic background and the state of the art of Tingley's problem the reader is referred to the introduction of [15] and to the monograph [32].…”

“…Solutions to Tingley's problem for compact operators, compact C * -algebras and weakly compact JB * -triples have been previously obtained in [26,14]. For additional information on the historic background and the state of the art of Tingley's problem the reader is referred to the introduction of [15] and to the monograph [32].Problems in C * -algebras, von Neumann algebras and operator algebras are often considered in the context of Banach Jordan algebras and Jordan triple systems. Such studies widen the scope and often introduce new ideas and techniques not present in the associative case.…”