Complete invariant fuzzy metrics on groups

We study some questions related to complete fuzzy metric topological groups (in the sense of Kramosil and Michael). Invariant fuzzy metrics are characterized and we prove that if (G,M,⁎) is a fuzzy metric group such that (M,⁎) is invariant, then a fuzzy metric completion (G˜,M˜,⁎) of (G,M,⁎) is a fuzzy metric group and (M˜,⁎) is invariant. Moreover if (M˜,⁎) is an invariant complete metric on (G,τ), then every compatible left invariant (or right invariant) fuzzy metric on G is complete. We also study the so-called three spaces problem: given a fuzzy metric group (G,M,⁎) and a closed normal subgroup N, if one has information about two complete fuzzy metric groups (G,M,⁎), (N,M,⁎) and (G/N,M,⁎), what can be said about the third one. Our results apply to fuzzy Banach spaces.