# Invariance of the Kronecker factor

Let $$(X,\mathcal{F},\mu,T)$$ be a measure preserving system and $$U_T$$ Koopman operator on $$L^2(X)$$, i.e. $$U_T f = f\circ T$$. Note that, for the moment, I am not imposing any further assumptions on $$X$$, $$\mathcal{F}$$, $$\mu$$ and $$T$$. Let's define the Kronecker factor of $$(X,\mathcal{F},\mu,T)$$ to be the minimal $$\sigma$$-algebra $$\mathcal{K} \subset \mathcal{F}$$ such that all the eigenfunctions of $$U_T$$ are $$\mathcal{K}$$-measurable.

Is $$\mathcal{K}$$ $$T$$-invariant in general, i.e. $$T^{-1}\mathcal{K} \subseteq \mathcal{K}$$? If not, what are the minimal assumptions that make it $$T$$-invariant? For example, most references prove this is the case if $$X$$ is a compact metric space (and $$\mathcal{F}$$, $$\mu$$ and $$T$$ corresponding to that structure), but I am wondering if this assumption is really necessary and if it can be replaced by a weaker one.

Strictly speaking, the definition of a factor of a measure preserving system (m.p.s.) is as follows.

Definition: Let $$(X,\mathcal{B},\mu,T)$$ and $$(Y,\mathcal{C},\nu,S)$$ be m.p.s. Then, a set function $$\phi: X'\to Y'$$ is called a factor map if it is surjective, measure preserving in the sense that $$\mu(\phi^{-1}A)=\nu(A)$$, for any $$A \in \mathcal{C}$$, and satisfies the equality $$S\circ \phi (x) = \phi \circ T(x),\ \text{for}\ x \in X',$$ where $$X'$$ and $$Y'$$ are full measure subsets of $$X$$ and $$Y$$ respectively, with $$TX' \subseteq X'$$ and $$SY' \subseteq Y'$$. We say that the system $$(Y,\mathcal{C},\nu,S)$$ is a factor of $$(X,\mathcal{B},\mu,T)$$, if such a factor map exists and in this case we also say that $$(X,\mathcal{B},\mu,T)$$ is an extension of $$(Y,\mathcal{C},\nu,S)$$.

''Equivalent'' definitions: There are two more equivalent ways of seeing factors of m.p.s. one through $$T$$-invariant sub-$$\sigma$$-algebras and one through $$T$$-invariant algebras of bounded measurable functions.

Let $$(X,\mathcal{B},\mu,T)$$ be a m.p.s. and $$(Y,\mathcal{C},\nu,S)$$ some factor of it, with factor map $$\phi: X'\to Y'$$. Then, it is easy to see that $$\phi^{-1}(\mathcal{C})$$ is a $$T$$-invariant sub-$$\sigma$$-algebra of $$\mathcal{B}$$ (modulo sets of measure zero). For example, if $$A \in \phi^{-1}(\mathcal{C})$$, then $$A=\phi^{-1}(C)$$, for some $$C\in \mathcal{C}$$ and so $$T^{-1}A=(\phi \circ T)^{-1}(C)=(S \circ \phi)^{-1}(C)=\phi^{-1}(S^{-1}(C)) \in \phi^{-1}(\mathcal{C}).$$
A partial converse is also true. Precisely, if $$(X,\mathcal{B},\mu,T)$$ is some m.p.s. and $$\mathcal{A} \subset \mathcal{B}$$ is a $$T$$-invariant sub-$$\sigma$$-algebra then, there exists a system $$(Y,\mathcal{C},\nu,S)$$ and a factor map $$\phi: X \to Y$$, such that $$\mathcal{A}= \phi^{-1}(\mathcal{C})$$.

We can also describe a correspondence between invariant $$\sigma$$-algebras and invariant algebras of functions. Indeed, if $$(X,\mathcal{B},\mu,T)$$ is a m.p.s. and $$\mathcal{F}$$ is a $$T$$-invariant algebra of $$L^{\infty}(\mu)$$, that is, $$T\mathcal{F} \subset \mathcal{F}$$ and for any $$f,g \in \mathcal{F}$$ and $$c\in \mathbb{C}$$, we have that $$f\cdot g, f+cg \in \mathcal{F}$$, the set $$\mathcal{A}:=\{B\in \mathcal{B}: B=f^{-1}(D),\ \text{for some}\ f\in \mathcal{F}\ \text{and}\ D\ \text{a complex Borel set}\}$$ is a $$T$$-invariant sub-$$\sigma$$-algebra of $$\mathcal{B}$$. On the other hand, given $$\mathcal{A} \subset \mathcal{B}$$ some $$T$$-invariant sub-$$\sigma$$-algebra of $$\mathcal{B}$$, the set $$\mathcal{F}:=\{f\in L^{\infty}(\mu): f\ \text{is measurable with respect to}\ \mathcal{A}\}$$ is a $$T$$-invariant sub-algebra of $$L^{\infty}(\mu)$$.

Conclusion: So, to conclude answering your question, we can define the Kronecker factor through functions; i.e. as the closed linear subspace of $$L^2(\mu)$$ $$\mathcal{K}_T:=\overline{\text{span}\{f\in L^{\infty}(\mu):f\ \text{is an eigenfunction of}\ T\}}^{L^2(\mu)},$$ where, a function $$f\in L^{\infty}(\mu)$$ is an eigenfunction of $$T$$ if there exists $$a\in \mathbb{R}$$, so that $$Tf=e^{2\pi ia}f$$ and $$e^{2\pi ia}$$ is the respective eigenvalue, we can define it as the minimal $$T$$-invariant sub-$$\sigma$$-algebra of $$\mathcal{B}$$ with respect to which all eigenfunctions are measurable, or finally, we can define it as the maximal factor of the system $$(X,\mathcal{B},\mu,T)$$ that is isomorphic to a rotation on a compact abelian group.

• If $X$ is not a probability space, then the constants are not integrable. Pretty easy to show that in that case, the Kronecker factor is empty as well (see Aaronson's book).
– Asaf
Jun 19 at 21:06