A new proof of the noncommutative Banach-Stone theorem

TL;DR

This paper presents a novel proof of the noncommutative Banach-Stone theorem, offering a new approach to classifying surjective isometries between nonunital C*-algebras and potential extensions to noncommutative Lp spaces.

Contribution

It introduces a fundamentally new proof method for the structure of surjective isometries between nonunital C*-algebras, building on previous work and techniques.

Findings

New proof of the noncommutative Banach-Stone theorem

Extension of classification techniques to noncommutative Lp spaces for 0 < p ≤ 1

Potential for broader applications in noncommutative geometry

Abstract

Surjective isometries between unital C*-algebras were classified in 1951 by Kadison. In 1972 Paterson and Sinclair handled the nonunital case by assuming Kadison's theorem and supplying some supplementary lemmas. Here we combine an observation of Paterson and Sinclair with variations on the methods of Yeadon and the author, producing a fundamentally new proof of the structure of surjective isometries between (nonunital) C*-algebras. In the final section we indicate how our techniques may be applied to classify surjective isometries of noncommutative Lp spaces, extending some recent results of the author to 0 < p leq 1.