Let \(T\) be a bounded linear operator from a Banach space \(A\) into a Banach space \(X\). Consider the second adjoint \(T''\) of \(T\), and remember that \(T''\) maps the bidual \(A''\) into \(X''\). Then a useful characterization of weakly compact operators states that \(T\) is weakly compact if and only if \(T''\) maps \(A''\) into \(X\). Consider next a bounded multilinear map \(\Gamma : A_1\times A_2\times \dots\times A_r\to X\), where \(A_j\) (\(j=1,2,\dots ,r\)) are Banach spaces. If the \(A_j\) are \(C^*\)-algebras, then there exists a unique bounded multilinear map \(\Gamma'' : A''_1\times A''_2\times \dots\times A''_r\to X''\) that is separately weak* to weak* continuous and extends \(\Gamma\). In order to generalize the above characterization of weak compactness to multilinear maps, it is known: if \(\Gamma\) is weakly compact, then \(\Gamma''\) takes its values in \(X\). But for \(r\geq 2\), the converse is not true in general. However, if the algebras \(A_j\) (\(j=1,2,\dots ,r\)) are commutative, then \(\Gamma''\) takes its values in \(X\) if and only if \(\Gamma\) is completely continuous [I. Villanueva, Proc. Am. Math. Soc. 128, 793–801 (1999; Zbl 0946.46026)].
The paper under review is concerned with the generalization of the above result to the case that some of the algebras \(A_j\) are non-commutative. Along these lines, it is shown that \(\Gamma''\) takes its values in \(X\) if and only if \(\Gamma\) is quasi completely continuous (a property introduced in the paper under review). Finally, this result is applied to obtaining a multilinear generalization of the normal-singular decomposition of a bounded linear operator on a von Neumann algebra.