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Injective envelopes

Let $ X$ be a matricially normed space.

An operator space $ Z$ together with a completely isometric mapping $ \iota:X\to Z$ is called an injective envelope of $ X$ if $ Z$ is injective, and if $ \mathrm{id}_Z$ is the unique extension of $ \iota$ onto $ Z$.

This is the case if and only if $ Z$ is the only injective subspace of $ Z$ which contains the image of $ X$.

Each matricially normed space has an injective envelope. It is unique up to a canonical isomorphism.

A matricially normed space $ Z$ together with a completely isometric mapping $ \iota:X\to Z$ is called an essential extension of $ X$ if a complete contraction $ \varphi:Z\to Y$ is completely isometric if only $ \varphi\circ\iota$ is completely isometric.

$ \iota:X\to Z$ is an injective envelope if and only if $ Z$ is injective and $ \iota:X\to Z$ is an essential extension.

Every injective envelope $ \iota:X\to Z$ is a maximal essential extension, i. e. for each essential extension $ \tilde{\iota}:X\to \tilde{Z}$, there is a completely isometric mapping $ \varphi:\tilde{Z}\to Z$ such that $ \varphi\circ\tilde{\iota}=\iota$.



Prof. Gerd Wittstock 2001-01-07