Noncommutative dynamics and orbit spaces

Core idea

The function algebra and its state encode classical probability. Crossed products retain orbit motion; the rotation algebra links this construction to foliations and noncommutative differentiation.

Prerequisite doorway

• C*-algebra
• Von Neumann algebra
• State on a C*-algebra
• Normal state
• Tracial state
• GNS construction
• Center of a von Neumann algebra
• Von Neumann factor
• Von Neumann crossed product
• Gelfand duality
• Unitary group $U(n)$

Reading path

• Noncommutative probability space
• State-preserving operator-algebra dynamical system
• Fixed-point algebra of an action
• Ergodic action on an operator algebra
• Classical dynamics as a state-preserving function-algebra system
• GNS implementation of state-preserving dynamics
• Mean ergodic theorem for a finite tracial system
• Algebraic crossed product by an automorphism
• Group–measure-space construction
• Essentially free action
• Maximal abelian von Neumann subalgebra
• Ergodicity and factoriality of a free crossed product
• Orbit space of an irrational rotation
• Noncommutative two-torus
• Derivation of an associative algebra
• Canonical derivations of the noncommutative torus
• Smooth foliation
• First-return map
• Kronecker foliation of the two-torus
• Smooth noncommutative torus
• Ergodicity of unitary conjugation on matrices

Return to Ergodic theory.