I will consider the space $ M_{n}(\mathbb{C})$ of $ n\times n$ complex matrices for simplicity, although many of the results below are true in more general Banach spaces.

I have encountered an interesting representation of the famous Grothendieck inequality recently. Basically, the usual representation of the inequality states that there are two (not relevant here) norms $ \|X\|_{(1)}, \|X\|_{(2)}$ such that the relation $ \|X\|_{(2)} \leq \|X\|_{(1)} \leq k \|X\|_{(2)}$ holds for any $ X \in M_{n}(\mathbb{C})$ and the (dimension-independent) constant $ k$ is called Grothendieck’s constant. Now, in the book Summing and Nuclear Norms in Banach Space Theory by G. J. O. Jameson (Cambridge, 1987), the author essentially takes the dual norms of the norms involved in Grothendieck’s inequality, giving $ \frac{1}{k} \|X\|^*_{(2)} \leq \|X\|^*_{(1)} \leq \|X\|^*_{(2)}$ . The dual norms can be given as

\begin{align} \|X\|^*_{(1)} &= \min_{U,V \in M_{n}(\mathbb{C})} \left \{ \|U\|_{2\to\infty} \|V\|_{2\to\infty} : X = U V^H \right\}\ \|X\|^*_{(2)} &= \min_{\{x_i\}, \{y_i\} \in \mathbb{C}^n} \left \{ \sum_{i,j} \|x_i \|_\infty \|y_j\|_\infty :X = \sum_{i,j} x_i y^H_j \right\} \end{align} with $ \|U\|_{2\to\infty}$ being an operator norm that in this case is the maximum $ \ell_2$ norm of the rows of a matrix $ U$ , and $ \|x\|_\infty$ denoting the $ \ell_\infty$ norm in $ \mathbb{C}^n$ . Interestingly, for a positive semidefinite matrix $ P$ the computation of $ \|P\|^*_{(1)}$ reduces to computing the largest magnitude of the entries of $ P$ , that is, $ \|P\|^*_{(1)} = \max_{i,j} |P_{ij}| = \max_{i} |P_{ii}|$ .

This motivates my question: is it true that for positive semidefinite matrices $ \|X\|^*_{(1)} = \|X\|^*_{(2)}$ ? If not, is there a non-trivial class of matrices for which it does hold true (e.g. positive semidefinite with further restrictions)?

Put in another way, when is it true that a positive semidefinite matrix $ P$ with largest entry magniutde $ |P_{\max}|$ can be expressed as a convex combination $ P = \sum_{i,j} c_{ij} x_i y^H_j$ with each $ \|x_i\|_{\infty} \leq \sqrt{|P_{\max}|}$ , $ \|y_j\|_{\infty} \leq \sqrt{|P_{\max}|}$ and $ \sum_{i,j} c_{ij} = 1$ ?

I have been experimenting with different sets of positive semidefinite matrices (with fixed trace, fixed operator norm…) but ultimately I was not able to prove the result for any non-trivial $ P$ . Any advice about this problem would be appreciated.