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 $T:X\to X$ 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 isoomorphic 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 $\phi:X\to Y$ 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.