The algebra of log-summable functions

The space ${L_0}$ consists of measurable functions $f$ on $[0,1]$ such that $\log (1 + |f(x)|)$ is summable on $[0,1]$, with functions equal almost everywhere identified. The integral defines a quasinorm on ${L_0}$. With this quasinorm, ${L_0}$ becomes a complete quasinormed linear space, the topology of which is not locally bounded. The quasinorm is plurisubharmonic (subharmonic on one-dimensional complex manifolds). ${L_0}$ is closed under multiplication, and multiplication is continuous. Inversion is not continuous, and the group of invertible elements is not open. There are no proper closed maximal ideals. The resolvent ${(\lambda - f)^{ - 1}}$ may exist for all complex $\lambda$, but it cannot be entire.