Maximum modulus principle for 'holomorphic functions' on the quantum matrix ball

We describe the Shilov boundary ideal for a q-analog of the algebra of holomorphic functions on the unit ball in the space of $n\times n$ matrices and show that its $C^*$-envelope is isomorphic to the $C^*$-algebra of continuous functions on the quantum unitary group $U_q(n)$.
27 pages,v.3:accepted for publication in Journal Funct.Anal., crrected som typos, proof of Lemma 10 changed, a reference added, an acknowledgement added