Spectral radii for subsets of Hilbert $C^*$-modules and spectral properties of positive maps

TL;DR

This paper extends spectral radius concepts to Hilbert $C^*$-modules, providing new characterizations, approximation results, and a constructive approach to the Perron-Frobenius theorem for positive maps in finite-dimensional $C^*$-algebras.

Contribution

It introduces spectral radius notions for Hilbert $C^*$-bimodules, proves a Rota-Strang type characterization, and offers a constructive Perron-Frobenius approach for positive maps.

Findings

Established a Rota-Strang type characterization for joint spectral radius.

Proved an approximation result linking joint and outer spectral radii.

Provided a constructive treatment of the Perron-Frobenius theorem for positive maps.

Abstract

The notions of joint and outer spectral radii are extended to the setting of Hilbert $C^*$-bimodules. A Rota-Strang type characterisation is proved for the joint spectral radius. In this general setting, an approximation result for the joint spectral radius in terms of the outer spectral radius has been established. This work leads to a new proof of the Wielandt-Friedland's formula for the spectral radius of positive maps. Following an idea of J. E. Pascoe, a positive map called the maximal part has been associated to any positive map with non-zero spectral radius, on finite dimensional $C^*$-algebras. This provides a constructive treatment of the Perron-Frobenius theorem. It is seen that the maximal part of a completely positive map has a very simple structure and it is irreducible if and only if the original map is irreducible. It is observed that algebras generated by tuples of matrices can be determined and their dimensions can be computed by realizing them as linear span of Choi-Kraus coefficients of some easily computable completely positive maps.