Geodesic
Pullback Diagram of Hilbert Modules Over $H^*$-Algebras
Kragujevac Journal of Mathematics, Tome 39 (2015) no. 1, p. 21

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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.
Classification : 46L08 46L05, 46C50
Keywords: $H^*$-algebra, morphism, Hilbert module, pullback diagram, trace-class 
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     author = {M. Khanehgir and M. Amyari and M. Moradian Khibary},
     title = {Pullback {Diagram} of {Hilbert} {Modules} {Over} $H^*${-Algebras}},
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M. Khanehgir; M. Amyari; M. Moradian Khibary. Pullback Diagram of Hilbert Modules Over $H^*$-Algebras. Kragujevac Journal of Mathematics, Tome 39 (2015) no. 1, p. 21 . http://geodesic.mathdoc.fr/item/KJM_2015_39_1_a2/