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Abstract
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We initiate an investigation into how much the existing theory of (nonselfadjoint)
operator algebras on a Hilbert space generalizes to algebras acting on
-spaces.
In particular we investigate the applicability of the theory of
real positivity, which has recently been useful in the study of
-operator
algebras and Banach algebras, to algebras of bounded operators on
-spaces.
In the process we answer some open questions on real positivity in Banach algebras
from work of Blecher and Ozawa.
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Keywords
$L^p$-operator algebra, accretive, approximate identity,
Banach algebra, Kaplansky density, M-ideal, real positive,
smooth Banach space, strictly convex Banach space, state of
a Banach algebra, unitization
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Mathematical Subject Classification 2010
Primary: 46E30, 46H10, 46H99, 47L10, 47L30
Secondary: 46H35, 47B38, 47B44, 47L75
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Milestones
Received: 3 April 2018
Revised: 18 May 2019
Accepted: 19 May 2019
Published: 4 January 2020
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