Vrftsjtryk

Let $X$ be a compact metric space and $mathcal B$ be its Borel $sigma$-algebra.
Let $mu$ be a Borel probability measure on $X$ and $T:Xto X$ me an invertible measure preserving transformation.
Let $U_T:L^2(X, mu)to L^2(X, mu)$ be the associated Koopman operator.
Let $mathcal A$ be the smallest sub-$sigma$-algebra on $X$ with respect to which all the eigenfunctions of $U_T$ are measurable.
Theorem 6.10 in Einsiedler and Ward's Ergodic Theory with a View Towards Number Theory [EW] states the following:

Theorem. The factor of $(X, mathcal B, mu, T)$ corresponding to $mathcal A$ is the largest factor of $(X, mathcal B, mu, T)$ which is omorphic to a rotation on some compact abelian group.

I do not follow why are we allowed to use the definite article "the" when asking for a factor corresponding to the sub-$sigma$-algebra $mathcal A$.

I ask this question because the following theorem [EW, Theorem 6.5] only guarantees 'a' factor.

Theorem. Let $(X, mathcal B, mu, T)$ be a measure preserving system, where $X$ is a compact metric space and $mathcal B$ is its Borel $sigma$-algebra.
Let $mathcal A$ be a $T$-invariant sub-$sigma$-algebra on $mathcal B$.
Then there is a measure preserving system $(Y, mathcal B_Y, nu, S)$, where $Y$ is a compact metric space with Borel $sigma$-algebra $mathcal B_Y$, and a factor map $phi:Xto Y$ with $mathcal A=phi^{-1}(mathcal B_Y)pmod mu$.

Is is perhaps true that any two factors corresponding to a given sub-$sigma$-algerba $mathcal A$ are isomorphic?

P.S. I have changed the wording of the theorems I have taken from [EW]. Also, in [EW] the theorems are stated for Borel probability spaces, which are more general than compact metric spaces.

Lemma $5.25$ from EW gives a map between two such factors. I think you can prove some quasi-uniqueness statement about two maps arising from this lemma.