The principle of local reflexivity and an extension of the identity $\mathcal B(E,X^{**})\cong\mathcal B(E,X)^{**}$

TL;DR

This paper uses the Principle of Local Reflexivity to extend classical identities between operator spaces and tensor products, providing new insights into the reflexivity of operator spaces and generalizing the Goldstine theorem.

Contribution

It establishes a novel isomorphic identification involving ultrapowers and tensor products, extending classical operator space identities and improving understanding of reflexivity conditions.

Findings

Identifies duals of finite rank operators with ultrapower quotients.

Shows reflexivity of al B(E,X) implies equality with approximable operators.

Characterizes when al B(E) is reflexive as equivalent to E being finite-dimensional.

Abstract

By using the Principle of Local Reflexivity (PLR), we prove that for every two Banach spaces $E$ and $X$ there exists a suitable ultrafilter $\mathcal{U}$ such that $ \mathcal{F}(E,X)^*,$ the dual space of the finite rank operators, can be isomorphically identified with certain quotient of the ultrapower space $(E\widehat{\otimes} X^*)_\mathcal{U}$, of the projective tensor product space $E\widehat{\otimes} X^*.$ This generalizes the identity $\mathcal B(E,X^{**})\cong\mathcal B(E,X)^{**}$, where $E$ is finite-dimensional. We then serve our main result to improve some results on the reflexivity of $\mathcal B(E,X)$, the space of all bounded linear operators, by showing that: if $\mathcal B(E,X)$ is reflexive, then $\mathcal B(E,X)=\mathcal A(E,X)$, the space of all approximable operators. This particularly implies that, $\mathcal B(E)$ is reflexive if and only if $E$ is finite-dimensional. Finally, as more by-products of the PLR, some generalizations of the classical Goldstine weak$^*$-density theorem are also included.