An equivalent quasinorm for the Lipschitz space of noncommutative martingales (Q2053505)

The noncommutative Lipschitz spaces \(\lambda_0^c(\mathcal{M})\) and \(\lambda_0^r(\mathcal{M})\) associated to a von Neumann algebra \(\mathcal{M}\) with a normal faithful normalized trace \(\tau\) were first introduced by \textit{T. N. Bekjan} et al. [J. Funct. Anal. 258, No.~7, 2483--2505 (2010; Zbl 1198.46047)]. They defined an equivalent quasinorm for the conditioned Hardy space of noncommutative martingales \(h_p(\mathcal{M})\), \(0 < p \leq 2\), and discussed the description of the dual space of \(h_p(\mathcal{M})\), \(0 < p \leq 1\). They asked if one can describe the dual space of \(h_p^c(\mathcal{M})\) as a Lipschitz space for \(0 < p < 1\). The authors of the paper under review answer this question positively by providing an equivalent quasinorm for the Lipschitz spaces \(\lambda_{\beta}^c(\mathcal{M})\) and \(\lambda_{\beta}^r(\mathcal{M})\) for \(\beta \geq 0\). Moreover, they introduce the noncommutative analogue of the classical martingale space \(L_1^2\) defined in [\textit{C. Herz}, Z. Wahrscheinlichkeitstheor. Verw. Geb. 28, 189--205 (1974; Zbl 0269.60040)], and then transfer the equivalence of the spaces \(h_1\) and \(L_1^2\) to the noncommutative martingales. Some equivalent quasinorms for \(h_p^c(\mathcal{M})\) and \(h_p^r(\mathcal{M})\) are also proved for \(2 < p < \infty\).