[Szőkefalvi-Nagy's theorem](https://en.wikipedia.org/wiki/Sz.-Nagy%27s_dilation_theorem) says the following: if $A$ is a contraction on a Hilbert space $H$, then there exists a unitary $U$ on a Hilbert space $H'\supset H$ for which $A^n=P\_HU^nP\_H$ for all $0\le n$. ($P\_H$ denotes projection onto $H$.)


If $H$ is finite-dimensional and we only need this to be true for $0\le n\le N$ then this can be modified such that $H'$ is also finite-dimensional (cf. [Levy and Shalit - Dilation theory in finite dimensions: the possible, the impossible and the unknown](https://arxiv.org/abs/1012.4514)).


Ando's theorem (see e.g. Levy–Shalit) says the following: if $A$ and $B$ are commuting contractions on $H$, then there exist commuting unitaries $U$ and $V$ on $H'\supset H$ for which $A^nB^m=P\_HU^nV^mP\_H$ for all $0\le n,m$.


The construction in the proof is naturally infinite-dimensional even if $H$ is finite-dimensional so a natural question (Problem C in Levy–Shalit) is: if we only need this to be true for $0\le n+m\le N$ then does an analogous modification work to make $H'$ finite-dimensional?


Levy–Shalit make this out to be wide-open, as of 2011. **My question here is if there are any recent developments in the years since then on this or related problems.**



Status of finite-dimensional Ando's theorem
In their 2013 paper "[Unitary $N$-dilations for tuples of commuting matrices](https://doi.org/10.1090/S0002-9939-2012-11714-9)" McCarthy and Shalit showed that there does indeed exist an $N$-dilation to a finite-dimensional space (see Theorem 1.2 in the paper).


By the way, I found this paper by taking a look at the recent survey article "[Dilation theory: a guided tour](https://zbmath.org/?q=an%3A7393048)" (2021) by Shalit ([link to arXiv](https://arxiv.org/abs/2002.05596)).


Since you're apparently interested in dilation theory, you might also be interested to know that Orr Shalit has a blog [Noncommutative analysis](https://noncommutativeanalysis.wordpress.com/) where he sometimes writes about dilation theory. For instance, the survey article that I mentioned above is announced in the blog post [A survey (another one!) on dilation theory](https://noncommutativeanalysis.wordpress.com/2020/02/14/a-survey-another-one-on-dilation-theory/).


