Johansson, Jimmy2016-12-062016-12-062016-12-06http://hdl.handle.net/2077/49978We introduce Arveson's generalization of the Shilov boundary to the noncommutative case and give a proof based on the work of Hamana of the existence of the Shilov boundary ideal. Moreover, we describe the Shilov boundary for a noncommutative analog of the algebra of holomorphic functions on the unit polydisk Dn and for a q-analog of the algebra of holomorphic functions on the unit ball in the space of symmetric complex 2 x 2 matrices.text