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

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On ring-theoretic (in)finiteness of Banach algebras of operators on Banach spaces. / Laustsen, Niels Jakob.

In: Glasgow Mathematical Journal, Vol. 45, No. 1, 31.01.2003, p. 11-19.

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Laustsen, Niels Jakob. / On ring-theoretic (in)finiteness of Banach algebras of operators on Banach spaces. In: Glasgow Mathematical Journal. 2003 ; Vol. 45, No. 1. pp. 11-19.

Bibtex

@article{5d93bda11bb74a3dbf7fe3bfafcb80d7,
title = "On ring-theoretic (in)finiteness of Banach algebras of operators on Banach spaces",
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.",
author = "Laustsen, {Niels Jakob}",
year = "2003",
month = jan,
day = "31",
doi = "10.1017/S0017089502008947",
language = "English",
volume = "45",
pages = "11--19",
journal = "Glasgow Mathematical Journal",
issn = "0017-0895",
publisher = "Cambridge University Press",
number = "1",

}

RIS

TY - JOUR

T1 - On ring-theoretic (in)finiteness of Banach algebras of operators on Banach spaces

AU - Laustsen, Niels Jakob

PY - 2003/1/31

Y1 - 2003/1/31

N2 - 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.

AB - 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.

U2 - 10.1017/S0017089502008947

DO - 10.1017/S0017089502008947

M3 - Journal article

AN - SCOPUS:0037263066

VL - 45

SP - 11

EP - 19

JO - Glasgow Mathematical Journal

JF - Glasgow Mathematical Journal

SN - 0017-0895

IS - 1

ER -