  The problem of the description of finite factor representations of the
infinite-dimensional unitary group, investigated by Voiculescu in 1976, is
equivalent to the description of all totally positive Toeplitz matrices.
Vershik-Kerov showed that this problem is also equivalent to the description of
the simplex of central (i.e. possessing a certain Gibbs property) measures on
paths in the Gelfand-Tsetlin graph. We study a quantum version of the latter
problem. We introduce a notion of a q-centrality and describe the simplex of
all q-central measures on paths in the Gelfand-Tsetlin graph. Conjecturally,
q-central measurets are related to representations of the quantized universal
enveloping algebra U_\epsilon(gl_\infty). We also define a class of q-Toeplitz
matrices and show that every extreme q-central measure corresponds to a
q-Toeplitz matrix with non-negative minors. Finally, our results can be viewed
as a classification theorem for certain Gibbs measures on rhombus tilings of
the halfplane.
  We use a class of q-interpolation polynomials related to Schur functions. One
of the key ingredients of our proofs is the binomial formula for these
polynomials proved by Okounkov.
