A completely positive approach to Ando's theorem
by
Michael Dritschel
University of Newcastle, United Kingdom
Coauthors: Robert Archer (University of Newcastle)

Let T₁ and T₂ be commuting contractions on a Hilbert space H, and let S subset or equal C(T²) be the operator system defined by S = {p(e^i\theta₁, e^i\theta₂)+[`(q(e^i\theta₁, e^i\theta₂))]: p, q are polynomials}. We construct a positive map \phi from S into L(H) with the property that \phi(p) = p(T₁, T₂). Such a map will be completely positive, and extends to a completely positive map of C(T²) into L(H). An application of the Stinespring dilation theorem then gives Ando's theorem that T₁ and T₂ dilate to a pair of commuting unitary operators. We also examine the case of three or more commuting contractions, where Ando's theorem in general does not hold.

Date received: May 28, 2002