A remark on non-commutative \(L^{p}\)-spaces (Q6579307)

Let \(M\) be a von Neumann algebra. Choose a faithful semifinite normal weight \(\varphi\) on \(M\). Let \(\widetilde{M}:=M\rtimes_{\sigma^\varphi}\mathbb{R}\) be the crossed product of \(M\) by the modular group \(\sigma^\varphi\) of \(\varphi\), and denote by \(\tau\) the canonical faithful normal semifinite trace on \(M\) (i.e., the trace used for the Haagerup construction). The purpose of the paper is to link the algebraic tensor product \(L^p(M_1)\otimes_{alg} L^p(M_2)\) of Haagerup \(L^p\)-spaces \(L^p(M_1)\) and \(L^p(M_2)\) (for \(0<p<\infty\)) with the Haagerup \(L^p\)-space \(L^p(M_1\bar{\otimes}M_2)\) of the von Neumann algebra tensor product of \(M_1\) and \(M_2\). Here, the algebraic tensor product of \(L^p(M_1)\) and \(L^p(M_2)\) is understood as the linear span of `simple tensors' \(x_1\bar\otimes x_2\), where \(x_i\), \(i=1,2\), are treated as \(\tau_i\)-measurable operators affiliated with resp. \(\widetilde{M_i}\), and \(x_1\bar\otimes x_2\) is the closure of the algebraic tensor product of \(x_1\) and \(x_2\). \N\NThe main result states that for any pair \((x_1, x_2)\in L^p(M_1)\times L^p(M_2)\), the element \(x_1\bar\otimes x_2\) yields, in fact, an element of \(L^p(M_1\bar{\otimes}M_2)\), which satisfies additionally \(\|x_1\bar\otimes x_2\|_p=\|x_1\|_p\|x_2\|_p\). Moreover, the map \((x_1,x_2)\mapsto x_1\bar\otimes x_2\) extends to a natural bilinear mapping of \(L^p(M_1)\otimes_{alg} L^p(M_2)\) into \(L^p(M_1\bar{\otimes}M_2)\), and the image of \(L^p(M_1)\otimes_{alg} L^p(M_2)\) in \(L^p(M_1\bar{\otimes}M_2)\) under the mapping is dense whenever both \(M_i\) are \(\sigma\)-finite and \(p\geq 1\). The authors show how this result can be applied in Quantum Information Theory.