On reduced amalgamated free products of \(C^\ast\)-algebras and the MF property (Q2866933)

Matricial field \(C^*\)-algebras or MF algebras were introduced by \textit{B. Blackadar} and \textit{E. Kirchberg} in [Math. Ann. 307, No. 3, 343--380 (1997; Zbl 0874.46036)] in connection with the study of finite-dimensional approximations of \(C^*\)-algebras. Among other definitions, a \(C^*\)-algebra is said to be an MF algebra if it embeds into the quotient \(C^*\)-algebra \(\left(\prod_k M_{n_k}(\mathbb{C})\right)/\left(\bigoplus_k M_{n_k}(\mathbb{C})\right)\) for some sequence \((n_k)_k\) of positive integers, see [\textit{B. Blackadar}, Operator algebras. Theory of \(C^*\)-algebras and von Neumann algebras. Berlin: Springer (2006; Zbl 1092.46003)].NEWLINENEWLINE\textit{U. Haagerup} and \textit{S. Thorbjørnsen} proved in [Ann. Math. (2) 162, No. 2, 711--775 (2005; Zbl 1103.46032)] that reduced \(C^*\)algebras of free groups are MF algebras.NEWLINENEWLINEThe central result of the paper under review states that the reduced \(C^*\)-algebra of an amalgamated free product of abelian groups is an MF algebra. In particular, it is the case of reduced \(C^*\)-algebras of torus knot groups.NEWLINENEWLINEThe proof partly relies on results of \textit{D. Hadwin} et al. [J. Oper. Theory 68, No. 1, 275--302 (2012; Zbl 1274.46116)]. As a by-product of the method, the author obtains a tensor product factorization for the von Neumann algebras of amalgamated free products of abelian groups and other new examples of reduced group \(C^*\)-algebras that are MF algebras.NEWLINENEWLINEThe main result is also used to obtain a characterization of amalgamated free products of abelian groups for which the Brown-Douglas-Fillmore semigroup of the reduced \(C^*\)-algebra is a group.