Banach spaces whose algebras of operators have a large group of unitary elements
2008 ◽
Vol 144
(1)
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pp. 97-108
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Keyword(s):
AbstractWe prove that a complex Banach space X is a Hilbert space if (and only if) the Banach algebra $\mathcal L (X)$ (of all bounded linear operator on X) is unitary and there exists a conjugate-linear algebra involution • on $\mathcal L (X)$ satisfying T• = T−1 for every surjective linear isometry T on X. Appropriate variants for real spaces of the result just quoted are also proven. Moreover, we show that a real Banach space X is a Hilbert space if and only if it is a real JB*-triple and $\mathcal L (X)$ is $w_{op}'$-unitary, where $w'_{op}$ stands for the dual weak-operator topology.
2010 ◽
Vol 10
(2)
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pp. 325-348
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Keyword(s):
1977 ◽
Vol 18
(1)
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pp. 13-15
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1969 ◽
Vol 21
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pp. 592-594
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1997 ◽
Vol 56
(2)
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pp. 303-318
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Keyword(s):
2020 ◽
Vol 65
(4)
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pp. 585-597
Keyword(s):
2012 ◽
Vol 34
(1)
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pp. 132-152
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2003 ◽
Vol 2003
(30)
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pp. 1899-1909
1968 ◽
Vol 9
(2)
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pp. 106-110
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