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For a finite noncommutative martingale $x=(x_k)_{1\\leq k\\leq n} \\subseteq L_1(\\M)$ adapted to $(\\M_k)_{k\\geq 1}$ and $0<\\alpha<1$, the fractional integral of $x$ of order $\\alpha$ is defined by setting: $$I^\\alpha x = \\sum_{k=1}^n \\zeta_k^{\\alpha} dx_k$$ for an appropriate sequence of scalars $(\\zeta_k)_{k\\geq 1}$. 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We investigate abstract fractional integrals associated to the filtration $(\\M_k)_{k\\geq 1}$. For a finite noncommutative martingale $x=(x_k)_{1\\leq k\\leq n} \\subseteq L_1(\\M)$ adapted to $(\\M_k)_{k\\geq 1}$ and $0<\\alpha<1$, the fractional integral of $x$ of order $\\alpha$ is defined by setting: $$I^\\alpha x = \\sum_{k=1}^n \\zeta_k^{\\alpha} dx_k$$ for an appropriate sequence of scalars $(\\zeta_k)_{k\\geq 1}$. 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