On ring-theoretic (in)finiteness of Banach algebras of operators on Banach spaces

Laustsen, Niels Jakob (2003) On ring-theoretic (in)finiteness of Banach algebras of operators on Banach spaces. Glasgow Mathematical Journal, 45 (1). pp. 11-19. ISSN 0017-0895

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Abstract

Let script B sign (x) denote the Banach algebra of all bounded linear operators on a Banach space x. We show that script B sign(x) is finite if and only if no proper, complemented subspace of x is isomorphic to x, and we show that script B sign(x) is properly infinite if and only if x contains a complemented subspace isomorphic to x ⊕ x. We apply these characterizations to find Banach spaces x1, x2 and x3 such that script B sign(x1) is finite, script B sign(x2) is infinite, but not properly infinite, and script B sign(x3) is properly infinite. Moreover, we prove that every unital, properly infinite ring has a continued bisection of the identity, and we give examples of Banach spaces η1 and η2 such that script B sign(η1 and script B sign(η2) are infinite without being properly infinite, script B sign(η1) has a continued bisection of the identity, and script B sign(η2) has no continued bisection of the identity. Finally, we exhibit a unital C*-algebra which is finite and has a continued bisection of the identity.

Item Type:
Journal Article
Journal or Publication Title:
Glasgow Mathematical Journal
Uncontrolled Keywords:
/dk/atira/pure/subjectarea/asjc/2600
Subjects:
ID Code:
160251
Deposited By:
Deposited On:
05 Oct 2021 15:05
Refereed?:
Yes
Published?:
Published
Last Modified:
19 Nov 2021 11:59