Envelopes in Banach spaces (Q6613288)

This is an outstanding paper, recipient of the 2025 Banach Journal of Mathematical Analysis prize, full of new ideas and results on topics that nowadays we can call classical Banach space theory; in this case, envelopes of Banach spaces. There are various notions of envelopes in the paper, ranging from the \emph{minimal} envelope of a subspace \(A\) of a Banach space \(X\) (the smallest \(1\)-complemented subspace of \(X\) containing \(A\)), that might or might not exist, to the \emph{abstract} envelope of a subset of a Banach space. Among all, the notions that attracted our attention the most are the \emph{S-algebraic envelope} of a subspace \(Y\) of a Banach space \(X\), in which \(S\) is a semigroup of operators acting on \(X\): it is the subspace formed by all points of \(X\) that are fixed by all operators in \(S\) that fix the points of \(Y\), and the \emph{Korovkin envelope} of a subset \(A\) of \(X\): the elements \(x\) that, whenever a net of operators of \(S\) pointwisely converges on \(A\) to the identity, they also converge to \(x\) on \(X\). Why this notion? Because the choice of \(S\) as the group Isom(X) of isometries of \(X\) leads to the notion of \emph{full} subspace \(Y\) of \(X\): one whose isometric envelope is the whole \(X\). This will be one more notion because the authors obtain (among a plethora of results) in Proposition~3.17: for \(1\leq p\leq q<\infty \) with \(p\) not even there are full copies of \(L_q\) in \(L_p\).\N\NOther results deal with envelopes in Fraïssé or Gurarij spaces as well.