Pro-C⁎-algebras associated to tensor products of pro-C⁎-correspondences

In this paper, we show that given two pro-C⁎-correspondences (X,A,φX) and (Y,B,φY) with positively invariant kernels which are ideal-compatible and Katsura non-degenerate, the pro-C⁎-algebra OX⊗Y associated to the minimal tensor product pro-C⁎-correspondence (X⊗Y,A⊗B,φX⊗Y) is isomorphic with a pro-C⁎-subalgebra of OX⊗OY, the minimal tensor product of the pro-C⁎-algebras OX and OY, which is generated by the elements x⊗y with the property that αz(x)⊗y=x⊗βz(y) for all z∈T, the unit circle, where α and β are the gauge actions on OX and OY.