A Kadison–Kastler row metric and intermediate subalgebras

Abstract

In this paper we introduce a row version of Kadison and Kastler's metric on the set of C*-subalgebras of B(H). By showing C*-algebras have row length (in the sense of Pisier) of at most 2 we show that the row metric is equivalent to the original Kadison Kastler metric. Ino and Watatani have recently proved that in certain circumstances sufficiently close intermediate C*-algebras occur as small unitary perturbations. By adjusting their arguments to work with the row metric we are able to obtain universal constants independent of inclusions.

Perturbations of crossed product C*-algebras by amenable groups

“…The author and Watatani [12] showed that for an inclusion of simple C * -algebras C ⊆ D with finite index in the sense of Watatani [23], sufficiently close intermediate C * -subalgebras are unitarily equivalent. The implementing unitary can be chosen close to the identity and in the relative commutant algebra C ′ ∩ D. Our estimates depend on the inclusion C ⊆ D, since we use the finite basis for C ⊆ D. Dickson obtained uniform estimates independent of all inclusions in [10]. To get this, Dickson showed that row metric is equivalent to the Kadison-Kastler metric.…”