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DOI: 10.23671/VNC.2019.21.44595 2-Local Isometries of Non-Commutative Lorentz Spaces
Abstract:
Let \(\mathcal M \) be a von Neumann algebra equipped with a faithful normal finite trace \(\tau\), and let \(S\left( \mathcal{M}, \tau\right)\) be an \(\ast \)-algebra of all \(\tau \)-measurable operators affiliated with \(\mathcal M \). For \(x \in S\left( \mathcal{M}, \tau\right)\) the generalized singular value function \(\mu(x):t\rightarrow \mu(t;x)\), \(t>0\), is defined by the equality \(\mu(t;x)=\inf\{\|xp\|_{\mathcal{M}}:\, p^2=p^*=p \in \mathcal{M}, \, \tau(\mathbf{1}-p)\leq t\}.\) Let \(\psi\) be an increasing concave continuous function on \([0, \infty)\) with \(\psi(0) = 0\), \(\psi(\infty)=\infty\), and let \(\Lambda_\psi(\mathcal M,\tau) = \left \{x \in S\left( \mathcal{M}, \tau\right): \ \| x \|_{\psi} =\int_0^{\infty}\mu(t;x)d\psi(t) < \infty \right \}\) be the non-commutative Lorentz space. A surjective (not necessarily linear) mapping \(V:\, \Lambda_\psi(\mathcal M,\tau) \to \Lambda_\psi(\mathcal M,\tau)\) is called a surjective 2-local isometry, if for any \(x, y \in \Lambda_\psi(\mathcal M,\tau) \) there exists a surjective linear isometry \(V_{x, y}:\, \Lambda_\psi(\mathcal M,\tau) \to \Lambda_\psi(\mathcal M,\tau)\) such that \(V(x) = V_{x, y}(x)\) and \(V(y) = V_{x, y}(y)\). It is proved that in the case when \(\mathcal{M}\) is a factor, every surjective 2-local isometry \(V:\Lambda_\psi(\mathcal M,\tau) \to \Lambda_\psi(\mathcal M,\tau)\) is a linear isometry.
Keywords: measurable operator, Lorentz space, isometry.
Language: English
For citation: Alimov, A. A. and Chilin, V. I. 2-Local Isometries of Non-Commutative Lorentz Spaces,Vladikavkaz Math. J., 2019, vol. 21, no. 4, pp. 5-10. DOI 10.23671/VNC.2019.21.44595
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