ABSTRACT
CONDITIONING IN GENERAL STRUCTURES
In this chapter we indicate applications and analogs of conditioning
in certain algebras of functions and operators as well as in cones of
(real) function spaces. Some characterizations of averaging projections
on such spaces, noncommutative conditioning, (“free”) independence,
and martingale convergence together with an application to sufficiency
(also in the noncommutative case) are discussed. These are actually
noncommutative analogs of probability theory, but termed noncommu-
tative probability for short. Here conditional operators rather than
measures often play a central role, with “states” of operator algebras
replacing probability measure. The treatment is intended to show the
far reaching impact of Kolmogorov’s idea of conditioning and inde-
pendence in different contexts. Most of the latter is on deterministic
structures and there will be few assertions with positive probability
(not to talk of certainty), but the technical analogs have deep con-
sequences, particularly for operator algebras and quantum mechanics.
Some detailed analysis is included.
