Maximal tracial algebras

Abstract

We introduce and study the notion of maximal tracial algebras. We prove several results in a general setting based on dual pairs and multiplier pairs. In a special case that X is a Banach space we determine the abelian subalgebras of \({\mathcal {B}}\left( X\right) \) that are maximal tracial for rank-one tensors. In another special case that \(\mathcal {H\ }\)is a Hilbert space we show that a unital weak-operator closed subalgebra \( {\mathcal {A}}\) of \({\mathcal {B}}\left( {\mathcal {H}}\right) \) is abelian and transitive if and only if it is maximal \(e\otimes e\)-tracial for every unit vector e in \({\mathcal {H}}\). We also make slight connections between our ideas and the Kadison Similarity Problem and also the Connes’ Embedding Problem.