Toeplitz operators associated to commuting row contractions

Abstract

Let H be a complex Hilbert space and let {T n } n 1 be a sequence of commuting bounded operators on H such that n 1 T n T * n

Lifting fixed points of completely positive mappings

“…A similar result was proved in [18] in the context of commuting row contractions, and that proof can easily be adapted to the present settings. …”

“…The main result in this section is the following theorem, which extends a similar one for commuting row contractions, proved in [18]. Its proof is very close to the one given in [18].…”

“…It may be viewed as a non-commutative version of similar results for commuting row contractions proved in [17] and [18]. The difference is that the role played in [17] and [18] by a sequence {U j } n j=1 ⊂ B(R) of commuting normal operators with n j=1 U j U * j = I R is now played by a Cuntz isometry. In the case when n j=1 T j T * j = I H , the fact that the mapping ρ appearing below is isometric was proved in [6].…”

“…However, it is not necessarily minimal. For instance, it was proved in [18] that if {T j } n j=1 is a commuting row contraction, then the commutant of the range of the minimal Stinespring dilation of this restriction is an abelian von Neumann algebra.…”

A Class of General Quantum Operations

“…Thus, if either Q V = 0 or Q W = 0, then S(V, W * ) = {0}. By Theorem 5, we have S(T , S * ) = {0}.Remark 7 For the special caseS = T , Q T = Q S = 0 ⇔ S(T , S * ) = {0}[9]. We note that this is not true in general by illustrating the following example.…”