Pullback Diagram of Hilbert Modules Over $H^*$-Algebras

M. Khanehgir, M. Amyari, M. Moradian Khibary

In this paper, we generalize the construction of a pullback diagram in the framework of Hilbert modules over $H^*$-algebras. More precisely we prove that if a commutative diagram of Hilbert $H^*$-modules and morphisms $$\begin{CD} X_1 @>\Phi_1>> Y_1 @VV\Psi_1V @VV\Psi_2V X_2 @>\Phi_2>>Y_2 \end{CD}$$ is pullback and $\Psi_2$ is a surjection, then (i) $\Psi_1$ is a surjection and (ii)$\ker \Phi_1\cap \ker \Psi_1=\{0\}$. Conversely, if (i) and (ii) hold, $\psi_1(\tau(A_1))$ is $\tau_{A_2}$-closed and $\Psi_2$ is injective, then the above diagram is pullback.