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 $S_x=\{x^{\ast }\in S(X^{\ast }):x^{\ast }(x)=1\}$. 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 $X^{\ast }$ is separable and let $\hat{I}¼$ be the Lebesgue measure on $[0,1]$. For $f\in L^1(\hat{I}¼,X)$, we demonstrate that if $S_f$ is norm compact, then $f$ is a smooth point. When $\hat{I}¼$ is the discrete measure, we show that if $(x_i)\in S({\ell }^1(X))$ and $\Vert x_i\Vert \ne 0$ for all $i\in \mathbb{N}$, then $S_{(x_i)}$ is weakly compact in ${\ell }^{\infty }(X^{\ast })$ if and only if $S_{\frac{x_i}{\Vert x_i\Vert }}$ is weakly compact in $X^{\ast }$ for each $i\in \mathbb{N}$ and $\text{diam}\left(S_{\frac{x_i}{\Vert x_i\Vert }}\right)\to 0$. 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 $\Vert x_{i_0}\Vert =1$, the state space $S_{x_{i_0}}$ is weakly compact.

To appear in Colloquium Mathematicum