  In this paper, we analyze problems involving matrix variables for which we
use a noncommutative algebra setting. To be more specific, we use a class of
functions (called NC analytic functions) defined by power series in
noncommuting variables and evaluate these functions on sets of matrices of all
dimensions; we call such situations dimension-free.
  In an earlier paper we characterized NC analytic maps that send
dimension-free matrix balls to dimension-free matrix balls and carry the
boundary to the boundary; such maps we call "NC ball maps". In this paper we
turn to a more general dimension-free ball B_L, called a "pencil ball",
associated with a homogeneous linear pencil L(x):= A_1 x_1 + ... + A_m x_m,
where A_j are complex matrices. For an m-tuple X of square matrices of the same
size, define L(X):=\sum A_j \otimes X_j and let B_L denote the set of all such
tuples X satisfying ||L(X)||<1.
  We study the generalization of NC ball maps to these pencil balls B_L, and
call them "pencil ball maps". We show that every B_L has a minimal dimensional
(in a certain sense) defining pencil L'. Up to normalization, a pencil ball map
is the direct sum of L' with an NC analytic map of the pencil ball into the
ball. That is, pencil ball maps are simple, in contrast to the classical result
of D'Angelo on such analytic maps in C^m. To prove our main theorem, this paper
uses the results of our previous paper mentioned above plus entirely different
techniques, namely, those of completely contractive maps.
