We show that there are no automorphic Banach spaces of the form C(K) with K continuous image of Valdivia compact except the spaces c0(Γ). Nevertheless, when K is an Eberlein compact of finite height such that C(K) is not isomorphic to c0(Γ), all isomorphism between subspaces of C(K) of size less than אω extend to automorphisms of C(K).

We study Lp-theory of second order elliptic divergence type operators with measurable coefficients. To this end, we introduce a new method of constructing positive C0-semigroups on Lp associated with sesquilinear (not necessarily sectorial) forms in L2. A precise condition ensuring that the elliptic operator is associated with a quasi-contractive C0-semigroup on Lp is established.

Relying on the recent work of Liu-Sz\'ekelyhidi we give a weak asymptotic estimate for Bergman kernels polarized K\"ahler manifolds with Ricci lower bound and Sobolev constant upper bound. We will also simple proof partial $C^0$ along (generalized) K\"ahler-Ricci flow Fano manifolds.