Strong convergence for tensor-embedded unitarily invariant random matrices
Determine whether strong convergence in operator norm holds for tensor-embedded families Y_1, ..., Y_L constructed from unitarily invariant d^2×d^2 random matrices X_1, ..., X_L via embeddings Y_j := X_j^{(0j)} ⊗ (I_d)^{⊗([L]\{j})} ∈ M_d^{⊗(L+1)}. Specifically, establish whether Y_1, ..., Y_L converge strongly to a free semicircular system s_1, ..., s_L as d → ∞ in the case X_1 = ··· = X_L, and more generally under mild unitarily invariant assumptions on the family (X_1, ..., X_L).
References
Can one say anything about the strong convergence (i.e., convergence of operator norms) of these families? For example, suppose each X_j is a (normalized) GUE matrix of size d2 and Y_j:=X_j{(0j)}\otimes (I_d){\otimes [L]\setminus {j}\in M{d}{\otimes(L+1)}. It has recently been shown that Y_1,\ldots, Y_L are asymptotically strongly free as d\to \infty if X_1,\ldots, X_L are independent. On the other hand, considers the case X_1=X_2=\cdots=X_L and shows that as d\to \infty, E[|Y_1+\cdots +Y_L|_{\infty}]\to 2\sqrt{L}= |s_1+\cdots +s_L| where {s_1,\ldots, s_L} is a free semicircular system in a C*-probability space. A natural question is whether the strong convergence Y_1,\ldots, Y_L\to s_1,\ldots, s_L still holds in this case and, more generally, whether it extends to the case when the family {X_1,\ldots, X_L} is unitarily invariant with reasonably nice conditions.