Linear orthogonality preservers between abelian C*-algebras and JB*-triples
کد مقاله : 1030-SHAA
David Cabezas *، Antonio M Peralta
Department of Mathematical Analysis, Faculty of Sciences, University of Granada, Granada, Spain
چکیده مقاله:
In 1990, Jarosz provided a complete description of all linear orthogonality preserving (non-necessarily continuous) maps between C(K)-spaces as weighted composition operators. The main consequence of this description asserts that every orthogonality preserving linear bijection between C(K)-spaces is automatically continuous and bi-orthogonality preserving. The result by Jarosz, including its interesting consequence, was extended, in similar terms, to the setting of C_0(L)-spaces by Jeang and Wong in 1996. This finishes the study in the case of commutative C*-algebras, but the problem remains open in wider classes of analytical structures.

JB*-triples are complex Banach spaces with a continuous triple product satisfying certain conditions, they were introduced by Kaup in 1983. This class of spaces strictly contains the class of C*-algebras. Namely, if A is a C*-algebra, the triple product {a,b,c}=(ab*c+cb*a)/2 provides a structure of JB*-triple for A. Furthermore, JB*-triples have applications to holomorphic functions theory. Indeed, Kaup showed in that for every bounded symmetric domain D in a complex Banach space, there exists a unique (up to a linear isometry) JB*-triple E whose open unit ball is biholomorphically equivalent to D. Symmetric domains play the role of simply connected domains when extending the Riemann mapping theorem beyond one-dimensional complex Banach spaces.

Until 2022, the form of linear orthogonality preserving mappings between abelian JB*-triples was unknown even under the assumption of continuity. In this talk, we shall present the results by Jarosz and their generalization to the setting of commutative JB*-triples that we have recently achieved.

The study of linear orthogonality preservers between C*-algebras and (of course) JB*-triples remains open in the non-commutative case. In 2008, Burgos, Fernández-Polo, Garcés, Martínez Moreno and Peralta provided a description of linear continuous orthogonality preservers between C*-algebras.
کلیدواژه ها:
Orthogonality preserver, bi-orthogonality preserver, abelian C*-algebra, abelian JB*-triple, automatic continuity
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