Vector-valued invariant means on spaces of bounded linear maps
Volume 132 / 2013
Abstract
Let ${\mathcal A}$ be a Banach algebra and let ${\mathcal M}$ be a $W^*$-algebra. For a homomorphism $\varPhi $ from ${\mathcal A}$ into ${\mathcal M}$, we introduce and study ${\mathcal M}$-valued invariant $\varPhi $-means on the space of bounded linear maps from ${\mathcal A}$ into ${\mathcal M}$. We establish several characterizations of existence of an ${\mathcal M}$-valued invariant $\varPhi $-mean on $B({\mathcal A},{\mathcal M})$. We also study the relation between existence of an ${\mathcal M}$-valued invariant $\varPhi $-mean on $B({\mathcal A},{\mathcal M})$ and amenability of ${\mathcal A}$. Finally, for a character $\phi $ of ${\mathcal A}$, we give some descriptions for $\phi $-amenability of $\mathcal A$ in terms of ${\mathcal M}$-valued invariant $\varPhi $-means.
