J. Aust. Math. Soc.  77 (2004), 91-110
Ideals of compact operators

Asvald Lima
  Department of Mathematics
  Agder College
  Gimlemoen 25J
  Serviceboks 422
  4604 Kristiansand
Eve Oja
  Faculty of Mathematics
  Tartu University
  Liivi 2-606
  EE-50409 Tartu

We give an example of a Banach space $X$ such that $\mathcal{K}(X,X)$ is not an ideal in $\mathcal{K}(X,X^{**})$. We prove that if $z^*$ is a weak $^*$ denting point in the unit ball of $Z^*$ and if $X$ is a closed subspace of a Banach space $Y$, then the set of norm-preserving extensions $HB(x^*\otimes z^*)\subseteq  \mathcal{L}(Z^*,Y)^*$ of a functional $x^*\otimes z^*\in (Z\otimes X)^*$ is equal to the set $HB(x^*)\otimes\{z^*\}$. Using this result, we show that if $X$ is an $M$-ideal in $Y$ and $Z$ is a reflexive Banach space, then $\mathcal{K}(Z,X)$ is an $M$-ideal in $\mathcal{K}(Z,Y)$ whenever $\mathcal{K}(Z,X)$ is an ideal in $\mathcal{K}(Z,Y)$. We also show that $\mathcal{K}(Z,X)$ is an ideal (respectively, an $M$-ideal) in $\mathcal{K}(Z,Y)$ for all Banach spaces $Z$ whenever $X$ is an ideal (respectively, an $M$-ideal) in $Y$ and $X^*$ has the compact approximation property with conjugate operators.
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