A note on lattice ordered $C^{\ast }$-algebras and Perron--Frobenius theory

Abstract

A classical result of Sherman says that if the space of self-adjoint elements in a $C^{\ast }$-algebra $\mathcal{A}$ is a lattice with respect to its canonical order, then $\mathcal{A}$ is commutative. We give a new proof of this theorem which shows that it is intrinsically connected with the spectral theory of positive operator semigroups. Our methods also show that some important Perron--Frobenius like spectral results fail to hold in any non-commutative $C^{\ast }$-algebra.

7 pages. This is version 3; minor changes compared to version 2

References in corpus (2)

• On the Peripheral Spectrum of Positive Operators
• Growth rates and the peripheral spectrum of positive operators

Cited by in corpus (1)

• On the decoupled Markov group conjecture