When are $L_p$ spaces homologically trivial?

In this talk, we will provide necessary and sufficient conditions for the projectivity, injectivity, and flatness of $L_p$ spaces in metric topological and relative Banach homology. To be precise, we will consider $L_p(S,\mu)$ spaces for $1<p<+\infty$, with finite Borel regular measures $\mu$ on a locally compact space $S$. These spaces will be regarded as Banach modules over the algebra $C_0(S)$, consisting of continuous functions vanishing at infinity. Most of the talk will be dedicated to the finite-dimensional case, where we will prove various inequalities regarding operator norms. Our main tools for this discussion will be convex optimization and Lyapunov's inequality.