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The Quarterly Journal of Mathematics Advance Access originally published online on April 18, 2007
The Quarterly Journal of Mathematics 2007 58(2):203-220; doi:10.1093/qmath/ham004
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© 2007. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

BANACH ALGEBRAS WITH LARGE GROUPS OF UNITARY ELEMENTS

Julio Becerra Guerrero1 {dagger}, Angel Rodríguez-Palacios1,{ddagger}, María Burgos2 § and El Amin Kaidi2 ¶

1 Universidad de Granada, Facultad de Ciencias, Departamento de Análisis Matemático, 18071 Granada, Spain
2 Universidad de Almería, Facultad de Ciencias Experimentales, Departamento de Álgebra y Análisis Matemático, 04120 Almería, Spain

{ddagger} Corresponding author. E-mail: apalacio{at}ugr.es

Received 25 June 2006;
  Abstract

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.

{dagger} E-mail: juliobg{at}ugr.es

§ E-mail: mburgos{at}ual.es

E-mail: elamin{at}ual.es


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