Let $\alpha$ be a fixed complex number, and let $\Omega$ be a simply connected region in complex plane $\mathbb C$ that is starlike with respect to $\alpha\in\Omega$. We define some Banach space of analytic functions on $\Omega$ and prove that it is a Banach algebra with respect to the $\alpha$-Duhamel product defined by \[ \big(f\circledast_{lpha}g\big)(z):=\frac{d}{dz}ıtimits_{lpha}^zf(z+lpha-t)g(t)\,dt. \] We prove that its maximal ideal space consists of the homomorphism $h_{\alpha}$ defined by $h_{\alpha}(f)=f(\alpha)$. Further, we characterize the lattice of invariant subspaces of the integration operator $J_{\alpha}f(z)=\int_{\alpha}^zf(t)\,dt$. Moreover, we describe in terms of $\alpha$-Duhamel operators the extended eigenvectors of $J_{\alpha}$.