engleski

Extensions of Hilbert C*-modules

The paper begins a study of extensions of Hilbert \cez-modules. An (essential) extension of a Hilbert \cez-module $V$ over a \cez-algebra \as is defined as a quadruple $(W, \bss, \Phi, \varphi)$ consisting of a Hilbert \bss-module $W$, an injective morphism of \cez-algebras $\varphi : \ass \rightarrow \bss$ such that $\mbox{; ; ; Im}; ; ; \, \varphi$ is an (essential) ideal in \bss, and a morphism of Hilbert \cez-modules $\Phi : V \rightarrow W$ such that $\mbox{; ; ; Im}; ; ; \, \Phi$ is an ideal submodule of $W$. This leads to the exact sequence $0 \rightarrow V \rightarrow W \rightarrow W/\mbox{; ; ; Im}; ; ; \, \Phi \rightarrow 0$ of Hilbert \cez-modules. It is proved that for each Hilbert \ass-module $V$ there exists the largest essential extension $(V_d, \ogrr(\ass), \Gamma, \gamma)$ such that for any other essential extension $(W, \bss, \Phi, \varphi, )$ of $V$ one can embed $W$ into $V_d$. It is also shown that the \cez-algebras of all adjointable operators acting on $V$ and $V_d$, respectively, are isomorphic.

C*-algebra ; Hilbert C*-module ; adjointable operator

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