Lifting fixed points of completely positive mappings

Let 1 ⩽ n ⩽ ∞ , and let { T j } j = 1 n ⊂ B ( H ) be a row contraction on some Hilbert space H . Let F ( T ) be the space of all X ∈ B ( H ) such that ∑ j = 1 n T j X T j ∗ = X . We show that, if non-zero, this space is completely isometric to the commutant of the Cuntz part of the minimal isometric dilation of { T j } j = 1 n .