Free pluriharmonic majorants and commutant lifting

In this paper we initiate the study of sub-pluriharmonic curves and free pluriharmonic majorants on the noncommutative open ball B(H)n 1 := (X1, . . . , Xn) ∈ B(H)n: X1X∗1 +···+XnX∗n 1/2 < 1 , where B(H) is the algebra of all bounded linear operators on a Hilbert space H. Several classical results from complex analysis have analogues in this noncommutative multivariable setting. We present basic properties for sub-pluriharmonic curves, characterize the class of sub-pluriharmonic curves that admit free pluriharmonic majorants and find, in this case, the least free pluriharmonic majorants. We show that, for any free holomorphic function Θ on [B(H)n]1, themap ϕ : [0, 1)→C∗(R1, . . . , Rn), ϕ(r) := Θ(rR1, . . . , rRn)∗Θ(rR1, . . . , rRn), is a sub-pluriharmonic curve in the Cuntz–Toeplitz algebra generated by the right creation operators R1, . . . , Rn on the full Fock space with n generators. We prove that Θ is in the noncommutative Hardy space H2 ball if and only if ϕ has a free pluriharmonic majorant. In this case, we find Herglotz–Riesz and Poisson type representations for the least pluriharmonic majorant of ϕ. Moreover, we obtain a characterization of the unit ball of H2 ball and provide a parametrization and concrete representations for all free pluriharmonic majorants of ϕ, when Θ is in the unit ball of H2 ball. In the second part of this paper, we introduce a generalized noncommutative commutant lifting (GNCL) problem which extends, to our noncommutative multivariable setting, several lifting problems including the classical Sz.-Nagy–Foia¸s commutant lifting problem and the extensions obtained by Treil–Volberg, Foia¸s– Frazho–Kaashoek, and Biswas–Foia¸s–Frazho, as well as their multivariable noncommutative versions. Wesolve the GNCL problem and, using the results regarding sub-pluriharmonic curves and free pluriharmonic majorants on noncommutative balls, we provide a complete description of all solutions. In particular, we obtain a concrete Schur type description of all solutions in the noncommutative commutant lifting theorem. © 2008 Elsevier Inc. All rights reserved