Noncommutative Mathematics for Quantum Systems

172 Noncommutative Mathematics for Quantum Systems

nlim→∞

1

n

n− 1

∑

k= 0

‖αk(x)y−yαk(x)‖ 2 =0.

The last result is also proved in [AET].

2.6.6 Final remarks

The above notes of course touch only a few aspects of the theory of
noncommutative – or ‘not necessarily commutative’ – dynamical
systems. This theory in a sense ‘contains’ the classical theory, but,
as we had a chance to see, extending even basic concepts or
theorems related to ‘standard’ dynamical systems to the
noncommutative setting requires usually completely new tools,
mostly coming from the theory of operator algebras (but also from
group theory, matrix analysis, combinatorics, and so on), and often
also forces additional assumptions. Recent decades have seen
similar developments of noncommutative probability and
noncommutative geometry.