Envelopes in Banach spaces

We define the notion of isometric envelope of a subspace in a Banach space, and relate it to a) the mean ergodic projection on the space of fixed points of a semigroup of contractions, b) results on Korovkin sets from the 70's, and c) extension properties of linear isometric embeddings. We use this concept to address the recent conjecture that the Gurarij space and the spaces $L_p$, $p \notin 2\mathbb N+4$ are the only separable Approximately Ultrahomogeneous Banach spaces (a certain multidimensional transitivity of the action of the linear isometry group). The similar conjecture for Fra\"iss\'e Banach spaces (a strenghtening of the Approximately Homogeneous Property) is also considered. We characterize the Hilbert space as the only separable reflexive space in which any closed subspace coincides with its envelope. We compute some envelopes in the case of Lebesgue spaces, showing that the reflexive $L_p$-spaces are the only reflexive rearrangement invariant spaces on $[0,1]$ for which all $1$-complemented subspaces are envelopes. We also identify the isometrically unique "full" quotient space of $L_p$ by a Hilbertian subspace, for appropriate values of $p$, as well as the associated topological group embedding of the unitary group into the isometry group of $L_p$.