Isometries of non-commutative Lorentz spaces
Let \(\mathfrak A\) be a finite von Neumann algebra, \(\mu\) be a faithful normal finite trace on \(A\), \(L_1(\mathfrak A,\mu)\) be the Banach space of all \(\mu\)-integrable measurable operators affiliated with \(\mathfrak A\), \(E\) be a subspace in \(L_1(\mathfrak A,\mu)\), \(\| \cdot \| _E\) be a norm on \(E\). The pair \((E,\| \cdot \| _E)\) is said to be a symmetric (respectively a fully symmetric) space on (\(\mathfrak A,\mu)\) if from \(x\in E\), \(y\in L_1(\mathfrak A,\mu)\) and \(\tilde y\le \tilde x\) (respectively \(\int^s_0 \tilde y(t)\,dt\le \int^s_0\tilde x(t)\,dt\), \(s\in (0,\mu(\mathbf{1}))\) it follows that \(y\in E\) and \(\| y\| _E\le \| x\| _E\), where \(\tilde x(t)\) denotes the rearrangement of \(x\). In the paper the general form of a positive linear isometry of a symmetric space onto a fully symmetric space is determined. Also is given a full characterization of all linear isometries of non-commutative Lorentz spaces.
positive linear isometry of a symmetric space onto a fully symmetric space
characterization of all linear isometries of non-commutative Lorentz spaces
scientific article; zbMATH DE number 4054345
scientific article