Denote by $\phi_{\beta,p}$ the normal state on${\cal B}( \ell ^2 ({\Bbb N}))$
with list of eigenvalues $\lambda_{\beta ,p,k} =p^{-\beta k} (1-p^{-\beta} )$,
where $k \in {\Bbb N}$, $\beta \in (0,1]$and $p$ is a prime number.
For each $\beta $ and each subset ${\cal S}$
of the set ${\cal P}$ of all prime numbers,
denote by $M_{\beta ,{\cal S}}$ the ITPFI factor
(infinite tensor product of factors of type I)$$ \bigotimes\limits_{
p\in {\cal S}} \big( {\cal B}(\ell ^2 ({\Bbb N} )) ,\phi_{\beta,p} \big).$$
We prove that for any $\beta \in (0,1]$ and $\lambda \in [ 0,1]$, there exists
${\cal S}={\cal S}_{\beta,\lambda}$, asubset of ${\cal P}$, such that
$M_{\beta, {\cal S}}$ isof type III$_\lambda$. Moreover,
$M_{\beta ,{\cal P}}$ is the Araki--Woods factor
$R_\infty$ for any $\beta \in (0,1]$. For any
$\beta \in (0.5,1]$ and ${\cal S} \subset {\cal P}$,
we show that $M_{\beta ,{\cal S}}$ is an ITPFI$_2$ factor.
We study the class of Connes' T-groups of type III
factors of the form $M_{\beta,{\cal S}}$, where ${\cal S}
\subset {\cal P}$, which we call $\beta$-representable
subgroups of ${\Bbb R}$ and show that it is
rich, proving for each $\beta \in (0,1]$
that if $H$ is a countable subgroup of ${\Bbb R}$and $\Sigma$ a countable subset
of ${\Bbb R} \setminus H$, there exists a
$\beta$-representable group $\Gamma$ which contains
$H$ and does not intersect $\Sigma$.Moreover, we prove that any
$\beta_0$-representable group is $\beta$-representable
if $1\geq \beta_0 \geq \max (\beta ,0.07)$ and that
the class of $1$-representable groups is closed under
the natural action of ${\Bbb R}^*_+$on the subgroups of ${\Bbb R}$.
Our results show in particular that all hyperfinite
type III$_\lambda$ factors, with $\lambda \in (0,1]$,
and a large class of type III$_0$ ITPFI factors are
isomorphic to group von Neumann algebras associated
with some restricted adelic productsof affine motion groups of $p$-adic lines.
1991 Mathematics Subject Classification:
46L35, 11N05.
