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Let $H$ be a Hilbert space and $E$ a Banach space. We set up a theory of stochastic integration of $ℒ(H,E)$-valued functions with respect to $H$-cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter $0<\beta <1$. For $0<\beta <\frac{1}{2}$ we show that a function $\Phi :(0,T)\to ℒ(H,E)$ is stochastically integrable with respect to an $H$-cylindrical Liouville fractional Brownian motion if and only if it is stochastically integrable with respect to an $H$-cylindrical fractional Brownian motion. We apply our results to stochastic evolution equations...