  A class of highest weight irreducible representations of the algebra
$U_h(A_\infty)$, the quantum analogue of the completion and central extension
$A_\infty$ of the Lie algebra $gl_\infty$, is constructed. It is considerably
larger than the class of representations known so far. Within each module a
basis is introduced and the transformation relations of the basis under the
action of the Chevalley generators are explicitly given. The verification of
the quantum algebra relations is shown to reduce to a set of nontrivial
$q$-number identities. All representations are restricted in the terminology of
S. Levendorskii and Y. Soibelman (Commun. Math. Phys. {\bf 140}, 399-414
(1991)).
