Journal of the
Korean Mathematical Society JKMS

In this paper we study the structure of closed weakly dense ideals in Privalov spaces $N^p$ $(1 < p < \infty)$ of holomorphic functions on the disk $\mathbb D:|z| < 1$. The space $N^p$ with the topology given by Stoll's metric [21] becomes an $F$-algebra. N. Mochizuki [16] proved that a closed ideal in $N^p$ is a principal ideal generated by an inner function. Consequently, a closed subspace $E$ of $N^p$ is invariant under multiplication by $z$ if and only if it has the form $IN^p$ for some inner function $I$. We prove that if ${\mathcal M}$ is a closed ideal in $N^p$ that is dense in the weak topology of $N^p$, then ${\mathcal M}$ is generated by a singular inner function. On the other hand, if $S_{\mu}$ is a singular inner function whose associated singular measure $\mu$ has the modulus of continuity $O(t^{(p-1)/p})$, then we prove that the ideal $S_{\mu}N^p$ is weakly dense in $N^p$. Consequently, for such singular inner function $S_{\mu}$, the quotient space $N^p/S_{\mu}N^p$ is an $F$-space with trivial dual, and hence $N^p$ does not have the separation property.

Keywords: Privalov space $N^p$, $F$-algebra, weakly dense ideal, singular inner function, topological dual