  Let $A' = \varprojlim_n A'_n$ be the projective limit of the $p$-parts of the
ideal class groups of the $p$ integers in the $\mathbb{Z}_p$-cyclotomic
extension $K_{\infty}/K$ of a CM number field $K$. We prove in this paper that
the $T$-part $(A')^-(T) = \{ 1 \}$ for CM extensions $K/\mathbb{Q}$. This fact
has been conjectured for arbitrary fields $K$ by Kuz'min in 1972 and was proved
by Greenberg in 1973, for abelian extensions $K/\mathbb{Q}$. Federer and Gross
had shown in 1981 that $(A')^-(T) = \{ 1 \}$ is equivalent to the non-vanishing
of the $p$-adic regulator of the $p$-units of $K$.
