This article is a continuation of [1], to which the reader is referred for the definition and properties of the JC -tensor product of two JC -algebras. Our standard references for nuclear and postliminal C * -algebras are [ 2 , 3 , 4 , 5 , 6 , 7 ] . We extend the notion of nuclearity to JC -algebras and prove that postliminal JC -algebras are nuclear. In contrast with the situation which occurs for C * -algebras, the JC -tensor product of two postliminal JC -algebras turns out, in general, to be non-postliminal and can even be anitliminal.