Noncommutative transforms and free pluriharmonic functions

In this paper, we study free pluriharmonic functions on noncommutative balls [Bn(H)]γ[B(H)n]γ, γ>0γ>0, and their boundary behavior. These functions have the formf(X1,…,Xn)=∑k=1∞∑|α|=kbαXα∗+a0I+∑k=1∞∑|α|=kaαXα,aα,bα∈C, where the convergence of the series is in the operator norm topology for any (X1,…,Xn)∈[Bn(H)]γ(X1,…,Xn)∈[B(H)n]γ, and B(H)B(H) denotes the algebra of all bounded linear operators on a Hilbert space HH. The main tools used in this study are certain noncommutative transforms which are introduced in the present paper and which generalize the classical transforms of Berezin, Poisson, Fantappiè, Herglotz, and Cayley. Several classical results from complex analysis have free analogues in our noncommutative multivariable setting.