Mathematics

Displays of Polish Isometry Groups in Banach Lattices

Mary Angelica Gramcko-Tursi

Abstract: A Polish group $G$ is displayable in a Banach lattice $(X, \|\cdot \|)$ if there exists a group homomorphism $\rho$ from $G$ into the lattice isometries of $X$ such that 1) $G$ is homeomorphic to $\rho(G)$, and 2) $X$ can be renormed with an equivalent lattice norm $\|| \cdot |\|$ so that $\rho(G)$ is the group of lattice isometries on $(X, \| | \cdot | \| )$. When is a group $G$ displayable in a Banach lattice $X$? This question has been explored in the context of Banach spaces and surjective linear isometries. In this talk based on ongoing work, we first survey some the known results and techniques for displays in Banach spaces to provide context. We then prove displayability results for certain classes of Banach lattices. In particular, if $X$ is either order continuous or an $AM$ space, $X$ can be renormed using various techniques, so that the identity is the only lattice isometry on $X$. Finally, we expand on these techniques to give general conditions sufficient for $G$ to be a display on $X$. This talk will be accessible to grad students of all levels.