I'm reading the paper 'New bounds for the max-$k$-cut and chromatic number of a graph' by van Dam and Sotirov. On page 221, it says:
"It is well known that one can restrict optimization of a SDP problem to feasible points in a matrix $\ast$-algebra that contains the data matrices of that problem as well as the identity matrix, ..."
What is meant precisely with 'data matrices'? Can someone give a definition?
More specifically, what would be the data matrices in the following SDP problem (for the max-$k$-cut problem on a graph)
$\max \quad \frac{1}{2}\text{tr}(LY)$
$\text{s.t.} \quad \text{diag}(Y)=u_n, \enspace kY-J_n \text{ is positive semi-definite, and }\enspace Y\geq 0$,
where $L$ is the Laplacian, $u_n$ is the all-one vector of size $n$ and $J_n$ is the all-one matrix of size $n\times n$?