Advances in Operator Theory and Inequalities

Inspired by state space theory in $C^*$-algebras, we study state spaces in Banach spaces--the family of norm-one support functionals for the unit ball at a given unit vector. This generalizes the state space of a unital $C^*$-algebra because it is the family of norm-one support functionals for the unit ball at the multiplicative unit. The state space of $X$ associated with a unit vector $x\in X$ is defined by $S_{x}=\{x^*\in X^*:\|x^*\|=1 \,\& \,x^*(x)=1\}$, where $X^*$ is the dual space of $X$. It is clear that $S_x$ is non-empty and weak$^*$-compact in $X^*$. We study the compactness of $S_x$ with respect to the norm and weak topologies on $X^*$. In this talk, we mainly discuss the norm compactness and weak compactness of the state space in the space of Bochner integrable functions and $c_{0}$, $\ell_1$-direct sums of Banach spaces.