Modular Johnson–Lindenstrauss flattening conjecture

Let $\mathcal{A}$ be a unital C*-algebra, let $0<\varepsilon<1$, and let $M,N\in\mathbb{N}$ and $\mathbf{x}_1,\dots,\mathbf{x}_M\in\mathcal{A}^N$. Modular Johnson–Lindenstrauss flattening conjecture. There is a universal constant $C>0$, possibly depending on $\mathcal{A}$, such that for every natural number $m$ satisfying

$$m>\frac{C}{\varepsilon^2}\log M,$$

there exists a matrix $M\in\mathbb{M}_{m\times N}(\mathcal{A})$ for which

$$(1-\varepsilon)\langle\mathbf{x}_j-\mathbf{x}_k,\mathbf{x}_j-\mathbf{x}_k\rangle\leq\langle M(\mathbf{x}_j-\mathbf{x}_k),M(\mathbf{x}_j-\mathbf{x}_k)\rangle\leq(1-\varepsilon)\langle\mathbf{x}_j-\mathbf{x}_k,\mathbf{x}_j-\mathbf{x}_k\rangle,$$

for all $1\leq j,k\leq M$. This is the C*-algebraic analogue of the Johnson–Lindenstrauss flattening lemma. The paper presents it as a particular case of a broader problem about the optimal function controlling the target dimension for arbitrary unital C*-algebras; the displayed upper bound is reproduced exactly as stated, including its apparent use of $1-\varepsilon$ rather than $1+\varepsilon$.