A group invariant Bishop-Phelps theorem

Abstract:

We show that for any Banach space and any compact topological group $G\subset L(X)$ such that the norm of $X$ is $G$-invariant, the set of norm attaining $G$-invariant functionals on $X$ is dense in the set of all $G$-invariant functionals on $X$, where a mapping $f$ is called $G$-invariant if for every $x\in X$ and every $g\in G$, $f\big (g(x)\big )=f(x)$. In contrast, we show also that the analog of Bollobás result does not hold in general. A version of Bollobás and James’ theorems is also presented.