Definition
Kronecker factor
The maximal factor of an ergodic transformation whose spectrum is pure point.
For an invertible ergodic system on a standard probability space, its Kronecker factor is the factor associated with the completed sigma-algebra generated by all Koopman eigenfunctions. It is the maximal factor with discrete spectrum.
Why eigenfunctions define a factor
Ergodic eigenfunctions have constant modulus and are bounded after normalization. Their products and conjugates are again eigenfunctions. They therefore generate an invariant function algebra as well as an invariant Hilbert subspace. Its space is the closed span of the eigenfunctions.
Examples
For an ergodic compact abelian rotation, the Kronecker factor is the entire system. For a weakly mixing system it is the one-point factor. In general it captures the persistent frequencies while leaving a continuous spectral component orthogonal to its space.
References
- Asgar Jamneshan and Henrik Kreidler, Ergodic Structure Theory and Applications, ISEM 28 lecture notes. Lecture 6 and the compact-factor construction in Lecture 7.