Math. Proc. Camb. Phil. Soc. (2008), 144, 97 c  2008 Cambridge Philosophical Society doi:10.1017/S0305004107000655 Printed in the United Kingdom First published online 17 January 2008 97 Banach spaces whose algebras of operators have a large group of unitary elements BY JULIO BECERRA GUERRERO Departamento de Matem ´ atica Aplicada, Universidad de Granada, Facultad de Ciencias, 18071-Granada, Spain. e-mail: juliobg@ugr.es MAR ´ IA BURGOS AND EL AMIN KAIDI Departamento de ´ Algebra y An´ alisis Matem´ atico, Universidad de Almer´ ıa, Facultad de Ciencias Experimentales, 04120-Almer´ ıa, Spain. e-mail: mburgos@ual.es;elamin@ual.es AND ´ ANGEL RODR ´ IGUEZ PALACIOS Departamento de An ´ alisis Matem´ atico, Universidad de Granada, Facultad de Ciencias, 18071-Granada, Spain. e-mail: apalacio@ugr.es (Received 12 September 2006; revised 15 December 2006) Abstract We prove that a complex Banach space X is a Hilbert space if (and only if) the Banach algebra L( X ) (of all bounded linear operator on X ) is unitary and there exists a conjugate- linear algebra involution • on L( X ) satisfying T • = T −1 for every surjective linear iso- metry 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 L( X ) is w ′ op -unitary, where w ′ op stands for the dual weak-operator topology. 1. Introduction Unitary elements of a norm-unital normed (associative) algebra A are defined as those invertible elements u of A satisfying ‖u ‖=‖u −1 ‖= 1. By a unitary normed algebra we mean a norm-unital normed algebra A such that the convex hull of the set of its unitary elements is norm-dense in the closed unit ball of A. In the sequel we will denote by U A the set of unitary elements of A. Relevant examples of unitary Banach algebras are all unital (complex) C ∗ -algebras and the real or complex discrete group algebras ℓ 1 (G) for every group G. The reader is referred to [1, 2, 7, 9, 10, 12, 25] for a full view of the theory of unitary normed algebras. We remark that unital C ∗ -algebras and discrete group algebras, as well as those unitary Banach algebras which are finite-dimensional or commutative and semisimple, satisfy Property (S ): (S ) There exists an algebra involution on the algebra, which is linear in the real case and conjugate-linear in the complex one, and maps each unitary element to its inverse.