Convergence in the dual of a \(\sigma\)-complete \(C^*\)-algebra (Q596702)

Let \(B\) be a monotone \(\sigma\)-complete C*-algebra, that is, a C*-algebra which possesses a unit element and so that each upper bounded, monotone increasing sequence of self-adjoint elements of \(B\) has a least upper bound. In particular, every von Neumann algebra is a monotone \(\sigma\)-complete C*-algebra. It is also known that the class of monotone \(\sigma\)-complete C*-algebras is strictly larger than the class of von Neumann algebras. In the paper under review, the authors provide sufficient conditions to assure that a sequence in the dual of a monotone \(\sigma\)-complete C*-algebra is weakly convergent. Concretely, they prove that if \((\mu_n)\) is a sequence in the dual of a monotone \(\sigma\)-complete C*-algebra \(B\) such that \((\mu_n (p))\) converges for each projection \(p\in B,\) then \((\mu_n)\) converges weakly to a bounded functional on \(B\). This interesting result extends the non-commutative version of the Dieudonné theorem proved by \textit{J. K. Brooks, K. Saitô} and \textit{J. D. M. Wright} [J. Math. Anal. Appl. 276, No.~1, 160-167 (2002; Zbl 1018.46031)], where the hypotheses are essentially stronger because, in the latter, it is assumed that the sequence \((\mu_n)\) converges on each range projection in \(B^{**}\). In a second section, the authors study the absolute continuity of bounded linear functionals on a monotone \(\sigma\)-complete C*-algebra.