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.


1 Answer 1


The short answer is yes.

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.

  • $\begingroup$ 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). $\endgroup$
    – Asaf
    Jun 19 at 21:06

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