A study on state spaces in classical Banach spaces

Let X be a real or complex Banach space. Let S(X) denote the unit sphere of X. For x∈S(X), let Sx={x∗∈S(X∗):x∗(x)=1}. A lot of Banach space geometry can be determined by the ‘quantum’ of the state space Sx. In this paper, we mainly study the norm compactness and weak compactness of the state space in the space of Bochner integrable functions and c0-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∈L1(μ,X), we demonstrate that if Sf is norm compact, then f is a smooth point. When μ is the discrete measure, we show that if (xi)∈S(ℓ1(X)) and ∥xi∥≠0 for all i∈N, then S(xi) is weakly compact in ℓ∞(X∗) if and only if Sxi/∥xi∥ is weakly compact in X∗ for each i∈N and diam(Sxi/∥xi∥)→0. For discrete c0-sums, we show that for (xi)∈c0(X), S(xi) is weakly compact if and only if for each i0∈N such that ∥xi0∥=1, the state space Sxi0 is weakly compact.