Almost commuting self-adjoint matrices: the real and self-dual cases (Q2822031)

A result of \textit{H.-X. Lin} [in: Operator algebras and their applications. Providence, RI: American Mathematical Society. 193--233 (1997; Zbl 0881.46042)] asserts, roughly speaking that, given two self-adjoint contractive matrices with complex entries which almost commute, one can find two commuting self-adjoint matrices close to them. The aim of this paper is to prove a version of this result for matrices with real entries. Specifically, the authors show that any pair of almost commuting self-adjoint, symmetric matrices is close, in a uniform way, to a pair of commuting self-adjoint ones. To obtain such results, as well as some related ones, the authors develop a theory of semiprojectivity for real \(C^*\)-algebras.