On centralizers of standard operator algebras with involution

 

Maja Fošner¹, Benjamin Marcen¹ and Nejc Širovnik²

¹Faculty of Logistics, University of Maribor, Mariborska cesta 7 3000 Celje Slovenia, maja.fosner@fl.uni-mb.si, benjamin.marcen@fl.uni-mb.si

²Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160 2000 Maribor Slovenia. nejc.sirovnik@uni-mb.si

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ABSTRACT

The purpose of this paper is to prove the following result. Let be a complex Hilbert space, let () be the algebra of all bounded linear operators on and let () () be a standard operator algebra, which is closed under the adjoint operation. Let : () () be a linear mapping satisfying the relation 2(*) = ()* + *() for all (). In this case is of the form () = λ for all (), where λ is some fixed complex number.

Keywords and Phrases: ring, ring with involution, prime ring, semiprime ring, Banach space, Hilbert space, standard operator algebra, H*-algebra, left (right) centralizer, two-sided centralizer.

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RESUMEN

El propósito de este artículo es probar el siguiente resultado. Sea un espacio de Hilbert complejo, sea () el álgebra de todos los operadores lineales acotados sobre y sea () () la álgebra de operadores clásica, la cual es cerrada bajo la operación adjunto. Sea : ()() una aplicación lineal satisfaciendo la relación 2(*) = ()* + *() para todo (). En este caso, es de la forma () = λ para todo (), donde λ es un número complejo fijo.

2010 AMS Mathematics Subject Classication: 16N60, 46B99, 39B42.

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Received: April 2013 / Accepted: September 2013.