  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 known so far representations. Within each module a basis is
introduced and the transformation relations of the basis under the action of
the Chevalley generators are explicitly written. The verification of the
quantum algebra relations to be satisfied is shown to reduce to a set of
nontrivial $q$-number identities. All our representations are restricted in the
terminology of S. Levendorskii and Y. Soibelman (Commun. Math. Phys. 140,
399-414 (1991)).
