Real ideals of compact operators of complex factors
Let \(M\) be a \(W^*\)-algebra. A real subspace \(I\) of \(M\) is called a real ideal of \(M\) if \(I\cdot M\subset I+iI\). In the present paper, the authors investigate the real ideal of compact operators in complex \(W^*\)-algebras and give a description (up to isomorphisms) of real two-sided ideals in complex \(W^*\)-factors. In particular, they obtain real analogues of Halpern-Kaftal's theorem [\textit{H. Halpern} and \textit{V. Kaftal}, Math. Ann. 273, 251--270 (1986; Zbl 0591.46053)]. A concept of relative weak \((RW)_r\) convergence in a real Hilbert space is introduced. The classical Hilbert characterization of compactness of operators is extended to the compact operators in semi-finite real \(W^*\)-algebras.
scientific article; zbMATH DE number 6347426