L^p operator algebras with approximate identities I

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 the first author and Ozawa.

Rigidity results for $L^p$-operator algebras and applications

“…Definition 3.4. A topological groupoid is a topological space G endowed with a distinguished subset G (2) ⊆ G × G of composable arrows and continuous operations…”

“…In these works, an L p -operator algebra is obtained from combinatorial or dynamical data, and properties of the underlying data (such as hereditary saturation of a graph, or minimality of an action) are related to properties of the algebra (such as simplicity). More recent works have approached the study of L p -operator algebras in a more abstract and systematic way [15,17,2], showing that there is an interesting theory waiting to be unveiled.…”

A modern look at algebras of operators onLp-spaces

“…It is clear that (5) ⇔ (4) ⇔ (3) ⇔ (2) and that (5) ⇔ (1). We show that (1) implies (2) and that (2) implies (5).…”

“…In view of Proposition 4.20, one can restrict the attention to group algebras F p (G) and F p λ (G) for Hölder exponents p in [1,2]. The remaining question is whether the algebras one gets for different values in [1,2] are really different. This is indeed the case: Theorem 4.22.…”

“…More recent works have approached the study of L p -operator algebras in a more abstract and systematic way [15,18,2], showing that there is an interesting theory waiting to be unveiled, of which only very little is currently known.…”