#### Vol. 303, No. 2, 2019

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$L^p$-operator algebras with approximate identities, I

### David P. Blecher and N. Christopher Phillips

Vol. 303 (2019), No. 2, 401–457
##### Abstract

We initiate an investigation into how much the existing theory of (nonselfadjoint) operator algebras on a Hilbert space generalizes to algebras acting on ${L}^{p}$-spaces. In particular we investigate the applicability of the theory of real positivity, which has recently been useful in the study of ${L}^{2}$-operator algebras and Banach algebras, to algebras of bounded operators on ${L}^{p}$-spaces. In the process we answer some open questions on real positivity in Banach algebras from work of Blecher and Ozawa.

##### 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
##### Mathematical Subject Classification 2010
Primary: 46E30, 46H10, 46H99, 47L10, 47L30
Secondary: 46H35, 47B38, 47B44, 47L75