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A note on lattice ordered ‐algebras and Perron–Frobenius theory
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AbstractA classical result of Sherman says that if the space of self‐adjoint elements in a ‐algebra is a lattice with respect to its canonical order, then 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 ‐algebra.
Title: A note on lattice ordered ‐algebras and Perron–Frobenius theory
Description:
AbstractA classical result of Sherman says that if the space of self‐adjoint elements in a ‐algebra is a lattice with respect to its canonical order, then 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 ‐algebra.
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