We prove that for every closed locally convex subspace E of $L_0$ and for any continuous linear operator T from $L_0$ to $L_0/E$ there is a continuous linear operator S from $L_0$ to $L_0$ such that T = QS where Q is the quotient map from $L_0$ to $L_0/E$.

Let A be a unital strict Banach algebra, and let K + be the one-point compactification of a discrete topological space K. Denote by the weak tensor product of the algebra A and C(K +), the algebra of continuous functions on K +. We prove that if K has sufficiently large cardinality (depending on A), then the strict global dimension is equal to .

We study the structure of Lipschitz algebras under the notions of approximate biflatness and Johnson pseudo-contractibility. We show that for a compact metric space $X$, the Lipschitz algebras ${\mathrm{Lip}}_{\alpha }\left(X\right)$ and ${\mathrm{lip}}_{\alpha }\left(X\right)$ are approximately biflat if and only if $X$ is finite, provided that $0<\alpha <1$. We give a necessary and sufficient condition that a vector-valued Lipschitz algebras is Johnson pseudo-contractible. We also show that some triangular Banach algebras are not approximately biflat.