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Differential structure of the Thompson components of selfadjoint operators

Different equivalence relations are defined in the set L(H) s of selfadjoint operators of a Hilbert space H in order to extend a very well known relation in the cone of positive operators. As in the positive case, for a ∈ L(H) s the equivalence class C a admits a differential structure, which is compatible with a complete metric defined on C a . This metric coincides with the Thompson metric when a is positive.