A remark on the slice map problem (Q1326632)

The slice map problem is to find all pairs \(({\mathcal S},{\mathcal T})\) of \(\sigma\)-weakly closed subspaces \({\mathcal S}\subset {\mathcal B}({\mathcal H}_ 1)\), \({\mathcal T}\subset {\mathcal B}({\mathcal H}_ 2)\) for two Hilbert spaces \({\mathcal H}_ 1\) and \({\mathcal H}_ 2\) such that \({\mathcal S}\overline\otimes{\mathcal T}= F({\mathcal S},{\mathcal T})\), where \(F({\mathcal S},{\mathcal T})\) denotes the Fubini product. \(\mathcal S\) is said to have the property \(S_ \sigma\) if \({\mathcal S}\overline\otimes{\mathcal T}= F({\mathcal S},{\mathcal T})\) for each \(\sigma\)-weakly closed subspace \(\mathcal T\) of a von Neumann algebra. \textit{J. Kraus} has shown that a \(\sigma\)-weakly closed subalgebra \({\mathcal A}\) of \({\mathcal B}({\mathcal H})\) generated by finite rank operators has the property \(S_ \sigma\) [J. Lond. Math. Soc. II. Ser. 28, 350-358 (1983; Zbl 0525.46034)]. In the present paper an example is given, which shows that the property of \(\mathcal A\) to contain an identity is necessary in this result.