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BANACH SPACES WHOSE ALGEBRAS OF OPERATORS ARE UNITARY: A HOLOMORPHIC APPROACH

Published online by Cambridge University Press:  20 March 2003

JULIO BECERRA GUERRERO
Affiliation:
Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071-Granada, Spain juliobg@ugr.es
ANGEL RODRÍGUEZ-PALACIOS
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071-Granada, Spain apalacio@goliat.ugr.es
GEOFFREY V. WOOD
Affiliation:
Department of Mathematics, University of Wales Swansea, Swansea SA2 8PP G.V.Wood@swansea.ac.uk
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Abstract

An element u of a norm-unital Banach algebra A is said to be unitary if u is invertible in A and satisfies $\Vert u\Vert =\Vert u^{-1}\Vert =1$. The norm-unital Banach algebra A is called unitary if the convex hull of the set of its unitary elements is norm-dense in the closed unit ball of A. If X is a complex Hilbert space, then the algebra $\BL(X)$ of all bounded linear operators on X is unitary by the Russo–Dye theorem. The question of whether this property characterizes complex Hilbert spaces among complex Banach spaces seems to be open. Some partial affirmative answers to this question are proved here. In particular, a complex Banach space X is a Hilbert space if (and only if) $\BL(X)$ is unitary and, for Y equal to $X,$ $X^*$ or $X^{**},$ there exists a biholomorphic automorphism of the open unit ball of Y that cannot be extended to a surjective linear isometry on Y.

Type
NOTES AND PAPERS
Copyright
© The London Mathematical Society 2003

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Footnotes

Partially supported by Junta de Andalucía grant FQM 0199 and Acción Integrada HB1999-0052.
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BANACH SPACES WHOSE ALGEBRAS OF OPERATORS ARE UNITARY: A HOLOMORPHIC APPROACH
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