Random matrices and \(K\)-theory for exact \(C^*\)-algebras
A \(C^\ast\)-algebra \(\mathcal A\) is said to be exact, after \textit{E. Kirchberg} [Invent. Math. 112, No. 3, 449-489 (1993; Zbl 0803.46071)], if for every \(C^\ast\)-algebra \(\mathcal B\) and each closed ideal \(\mathcal I\) of \(\mathcal B\) the sequence \(0\to {\mathcal A}\underset{\text{min}}\otimes {\mathcal I} {\mathcal A}\underset{\text{min}}\otimes {\mathcal B} {\mathcal A}\underset{\text{min}}\otimes {\mathcal B}/{\mathcal I} 0\) is exact. For example, every nuclear \(C^\ast\)-algebra is exact, while the full \(C^\ast\)-algebra of a noncommutative free group is not. Let now \(\mathcal H\) and \(\mathcal K\) be Hilbert spaces and let \(a_1,\cdots,a_r\) be elements of \({\mathcal B}({\mathcal H},{\mathcal K})\) such that the products \(a_i^\ast a_j\) are contained in some exact \(C^\ast\)-subalgebra \(\mathcal A\) of \({\mathcal B}({\mathcal H})\). Let \((\Omega, {\mathcal F}, {\mathcal P})\) be a probability space, and let for every natural \(n\), \(Y_1^{(n)},\cdots, Y_r^{(n)}\) be random \(n\times n\)-matrices on \(\Omega\) such that the real and complex parts of all their entries form independent Gaussian variables (that is, \(2rn^2\) of them altogether) with mean \(0\) and variance \(\frac 1 {2n}\). The main result of the paper gives asymptotic (as \(n\to\infty\)) lower and upper bounds on the spectra of the operators \(S^\ast_n(\omega) S_n(\omega)\) and \(S_n(\omega)S^\ast_n(\omega)\), where \(S_n=\sum_{i=1}^r a_i\otimes Y^{(n)}_i\); these bounds are valid for almost all \(\omega\in\Omega\), and it is noted that the subsequent paper by the second named author [``Mixed moments of Voiculescu's Gaussian random matrices'', preprint, Odense University (1999)] establishes the (asymptotic) tightness of the bounds. The bounds are no longer valid if the \(C^\ast\)-subalgebra generated by \(a_i^\ast a_j\) is not exact, as an example shows. As an application of the bounds derived, new proofs are given to the following two results: (1) every stably finite exact unital \(C^\ast\)-algebra has a tracial state; (2) if \(\mathcal A\) is an exact unital \(C^\ast\)-algebra, then every state \(\psi\) on \(K_0({\mathcal A})\) is given by a tracial state \(\tau\) on \(\mathcal A\) in the sense that \(\psi([p])=(\tau\otimes \text{Tr}_m)(p)\) for all \(m\in{\mathbb N}\) and all projections \(p\) in \(M_m({\mathcal A})\).
full \(C^*\)-algebra of noncommutative free group
scientific article; zbMATH DE number 1332117