  We introduce the notion of regular operator mappings of several variables
generalising the notion of spectral function. This setting is convenient for
studying maps more general than what can be obtained from the functional
calculus, and it allows for Jensen type inequalities and multivariate
non-commutative perspectives.
  As a main application of the theory we consider geometric means of k operator
variables extending the geometric mean of k commuting operators and the
geometric mean of two arbitrary positive definite matrices. We propose
different types of updating conditions that seems natural in many applications
and prove that each of these conditions, together with a few other natural
axioms, uniquely defines the geometric mean for any number of operator
variables. The means defined in this way are given by explicit formulas and are
computationally tractable.
