A study on state spaces in classical Banach spaces

Abstract

Let $X$ be a real or complex Banach space. Let $S(X)$ denote the unit sphere of $X$. For $x\in S(X)$, let $S_{x}=\{x^*\in S(X^*):x^*(x)=1\}$. A lot of Banach space geometry can be determined by the `quantum' of the state space $S_{x}$. In this paper, we mainly study the norm compactness and weak compactness of the state space in the space of Bochner integrable function and $c_{0}$-direct sums of Banach spaces. Suppose $X$ is such that $X^*$ is separable and let $μ$ be the Lebesgue measure on $[0,1]$. For $f\in L^1(μ,X)$, we demonstrate that if $S_{f}$ is norm compact, then $f$ is a smooth point. When $μ$ is the discrete measure, we show that if $ (x_i) \in S(\ell^{1}(X))$ and $ \|x_{i}\|\neq 0$ for all $i\in{\mathbb{N}}$, then $ S_{(x_i)}$ is weakly compact in $ \ell^\infty(X^*) $ if and only if $ S_{\frac{x_i}{\|x_i\|}} $ is weakly compact in $X^*$ for each $i\in{\mathbb{N}}$ and $\text{diam}\left(S_{\frac{x_i}{\|x_i\|}}\right) \to 0 $. For discrete $c_{0}$-sums, we show that for $(x_{i})\in c_{0}(X)$, $S_{(x_{i})}$ is weakly compact if and only if for each $i_{0}\in \mathbb{N}$ such that $\|x_{i_{0}}\|=1$, the state space $S_{x_{i_{0}}}$ is weakly compact.