Banach and Fr\'{e}chet groups, and the derivation and automorphism lengths of C*-algebras

We generalize the notions of Banach space and Fréchet space from the context of topological vector spaces to arbitrary topological groups.

We then extend to Banach and Fréchet groups previous results of Saint-Raymond for Banach and Fréchet spaces.

We also consider groups with a TSI Polish cover, proving a general comparison criterion for their phantom length.

This is applied, together with a dichotomy theorem of Shani, to the study of the derivation length and automorphism length of separable C*-algebras.

Building of previous works of Elliott and Akemann--Pedersen, we obtain a general dichotomy theorem for derivations and automorphisms of C*-algebras.

We deduce that the derivation length and automorphism length of a separable unital C*-algebras cannot attain certain values, and are in fact (almost) equal to each other for a large class of C*-algebras.