The formulation of Doob’s inequality for non-commutative martingales in the theory of non-commutative probability encounters the following difficulty: for an increasing sequence \(( E_n)\), \(n \in \mathbb{N}\), of conditional expectations and a positive operator \(x\) in \(L_p\), the quantity \(\sup_n E_n(x)\) (or \(\sup_n |E_n(x) |\)) lies in neither \(L_p\) nor the class of unbounded operators at all. By adopting G. Pisier’s non-commutative vector-valued \(L_p\)-space \(L_p( N; \ell_{\infty})\): cf. [“Non-commutative vector valued \(L_p\)-spaces and completely \(p\)-summing maps”, Astérisque 247 (1998; Zbl 0937.46056)], the author overcomes the difficulty and succeeds in gaining the right formulation of Doob’s inequality that should be taken. However, Pisier’s definition is restricted only to hyperfinite von Neumann algebras \(\widetilde{N}\), namely, von Neumann algebras with a \(\sigma\)-weakly dense net of finite dimensional subalgebras. All these obstacles automatically disappear for the so-called dual version (DDp) of Doob’s inequality: that is, under the general setting with von Neumann algebras \(N\), for \(1 \leq p < \infty\) and a sequence \((x_n)\), \(n \in \mathbb{N}\), of positive elements in a non-commutative \(L_p\) space \(L_p(N)\), there exists a constant \(c_p\) depending only on \(p\) such that \[ \|\sum_n E_n (x_n) \|_p \leq c_p \|\sum_n x_n \|_p. \tag{DDp} \] This inequality is originally due to D. L. Burkholder, B. J. Davis and R. F. Gundy inequality in the commutative case [Proc. VIth Berkeley Symp. Math. Stat. Prob., Univ., Calif., 1970/1971, Vol. II, 223-240 (1972; Zbl 0253.60056)]. By resorting to duality argument, the author derives from (DDp) a version of Doob’s maximal inequality for non-commutative martingales with parameter range \(1 < p \leq \infty\). Indeed, (DDp) implies that \(T((x_n))\) \(=\) \(\sum_n E_n (x_n)\) defines a continuous linear map between \(L_p(l_1)\) and \(L_p\). Moreover, the norm \(\|T^* \|\) yields the best constant in Doob’s inequality for the conjugate index \(p' :=p/(p-1)\). In fact, it follows that \[ \left\|\sup_n |E_n (x) |\right\|_{p'} = \|T^*(x) \|_{L_{p'}(l_{\infty})} \leq \|T \|\cdot \|x \|_{p'} = c_p \|x \|_{p'}. \]
More precisely, although (DDp) admits an entirely elementary proof in the commutative case, its proof still works in the non-commutative case for \(p=2\), too. This fact gives the author a nice proper starting point towards the final aim of derivation of a non-commutative version of Doob’s inequality. Next the complex interpolation method is used to extend (DDp) to the interval \(1 \leq p \leq 2\), and suitable norms are introduced to make the above duality argument work even in the non-commutative case. Then the author establishes the dual version (DDp) in the range \(2 \leq p < \infty\), by using the duality arguments which rely on Pisier-Xu’s version of Stein’s inequality for non-commutative martingales [cf. G. Pisier and Q. Xu, Commun. Math. Phys. 189, No. 3, 667-698 (1997; Zbl 0898.46056)] in combination with techniques from Hilbert \(C^*\)-modules. By duality again, the author obtains the non-commutative Doob inequality in the more delicate range \(1 < p \leq 2\). The core of his arguments consists in the new connection between Hilbert \(C^*\)-modules and non-commutative \(L_p\) spaces (used before). Lastly, applications of Doob’s inequality in terms of submartingales and Doob decomposition are presented as well.