The Schur-Agler class consists of functions over a domain satisfying anappropriate von Neumann inequality. Originally defined over the polydisk, theidea has been extended to general domains in multivariable complex Euclideanspace with matrix polynomial defining function as well as to certainmultivariable noncommutative-operator domains with a noncommutativelinear-pencil defining function. Still more recently there has emerged a freenoncommutative function theory (functions of noncommuting matrix variablesrespecting direct sums and similaritytransformations). The purpose of thepresent paper is to extend the Schur-Agler-class theory to the freenoncommutative functionsetting. This includes the positive-kernel-decompositioncharacterization of the class, transfer-function realization and Pickinterpolation theory. A special class of defining functions is identified forwhich the associated Schur-Agler class coincides with thecontractive-multiplier class on an associated noncommutative reproducing kernelHilbert space; in this case, solution of the Pick interpolation problem is interms of the complete positivity of an associated Pick matrix which isexplicitly determined from the interpolation data.
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