This question will probably appeal to people in operator systems theory as it is very much related. However, I'm interested in down-to-earth concrete systems with finite dimensional Hilbert space representations. Please suggest the adequate tags if you think I'm missing some.

Let $V\subset\mathcal B(\mathcal H)$ be a subspace of Hermitian operators in finite-dimensional Hilbert space $\mathcal H$, containing $1$. Let $\{X_i,Y_j\}$, (with $1\leq i\leq r$ and $1\leq j\leq d$) be a basis of $V$ and let $\{\tilde X_i,\tilde Y_j\}$ be the dual basis w.r.t the Hilbert-Schmidt inner product \begin{align} \mathrm{tr}[X_i \tilde X_{i'}]=\delta_{ii'}, \quad \mathrm{tr}[Y_j \tilde Y_{j'}]=\delta_{jj'},\quad \mathrm{tr}[X_i \tilde Y_{j}]=0, \quad \mathrm{tr}[Y_i \tilde X_{j}]=0. \end{align} Note that defining $\alpha_i=\mathrm{tr}[\tilde X_i]$ and $\beta_j=\mathrm{tr}[\tilde Y_j]$ we have \begin{align} 1=\sum_{i=1}^r \alpha_i X_i+\sum_{j=1}^d \beta_j Y_j, \end{align} however not all $\alpha_i$'s are zero, meaning that $1\notin\mathrm{span}\{Y_j\}$.

Regard $\mathbb{R}^n\otimes\mathcal B(\mathcal H)$ as the algebra of $n$-tuples of elements from $\mathcal B(\mathcal H)$, as in operator systems theory. Let $\mathcal C\subset \mathbb{R}^r\otimes\mathcal B(\mathcal H)$ be the pointed cone defined by

\begin{align} \mathcal C=\left\{Z=\{Z_i\}\Big|\exists \{K_j\}\in\mathbb{R}^d\otimes\mathcal B(\mathcal H)\mathrm{~s.t.~} \sum_{i=1}^r X_i\otimes Z_i+\sum_{j=1}^d Y_j\otimes K_j\geq0\right\}. \end{align}

Question: Under what conditions does $\{\tilde X_i^\top\}\in\mathcal C$?

Note: If $1\in\mathrm{span}\{Y_j\}$ the answer is trivial: Always. This is however, not my case.


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.