Resumen de Banach spaces whose algebras of operators have a large group of unitary elements

Julio Becerra Guerrero , María Burgos , El Amin Kaidi Lhachmi , Ángel Rodríguez Palacios

• We prove that a complex Banach space X is a Hilbert space if (and only if) the Banach algebra (of all bounded linear operator on X) is unitary and there exists a conjugate-linear algebra involution ¿ on satisfying T¿ = T-1 for every surjective linear isometry T on X. Appropriate variants for real spaces of the result just quoted are also proven. Moreover, we show that a real Banach space X is a Hilbert space if and only if it is a real JB*-triple and is -unitary, where stands for the dual weak-operator topology