Resumen de Banach algebras with large groups of unitary elements

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

• We study unitary Banach algebras, as defined by M. L. Hansen and R. V. Kadison in 1996, as well as some related concepts like maximal or uniquely maximal Banach algebras. We show that a norm-unital Banach algebra is uniquely maximal if and only if it is unitary and has minimality of the equivalent norm. We prove that every unitary semisimple commutative complex Banach algebra has a conjugate-linear involution mapping each unitary element to its inverse, and that, endowed with such an involution, becomes a hermitian *-algebra. The possibility of removing the requirement of commutativity in the above statement is also considered. The paper concludes by translating to real algebras some results previously known in the complex case. In particular, we show that every maximal semisimple finite-dimensional real Banach algebra is isometrically isomorphic to a real C*-algebra.