Hermitian elements in projective tensor products (Q803892)

Let A and B be unital Banach algebras such that the canonical map A\({\tilde \otimes}_{\pi}B\to A{\tilde \otimes}_{\epsilon}B\) from the projective into the injective tensor product is injective (this is true for example if A or B has the approximation property); \textit{S. Kaijser} and \textit{A. M. Sinclair} [Math. Scand. 55, 161-187 (1984; Zbl 0546.46047)] showed that then every hermitian element u in A\({\tilde \otimes}_{\pi}B\) is of the form \(u=x\otimes 1+1\otimes y\) with hermitian \(x\in A\) and \(y\in B\). The author proves that x and y can be even chosen such that \(\| u\|_{A{\tilde \otimes}_{\pi}B}=\| x\| +\| y\|.\) They extend this result to the tensor product of n algebras and, as an application, show that for every hermitian operator T on B(H) there are hermitian operators x,y\(\in B(H)\) such that \(T=L_ x+R_ y\) (left and right multiplication on B(H)) and \(\| T\| =\| x\| +\| y\|\).