2021/11/30 by Ramin Faal, Faal, Ramin, Hamid Reza Ebrahimi Vishki +1
Mathematics · #46B07 #46B10 #47L10 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.2112.00099
openalex publication_date 2021/11/30 · openalex created_date 2022/11/09 · openalex updated_date 2026/07/28
By using the Principle of Local Reflexivity (PLR), we prove that for every\ntwo Banach spaces E and X there exists a suitable ultrafilter \U\nsuch that \F(E,X)^*, the dual space of the finite rank operators,\ncan be isomorphically identified with certain quotient of the ultrapower space\n(E widehat\⊗ X^*)_\U, of the projective tensor product space\nE widehat\⊗ X^*. This generalizes the identity mathcal\nB(E,X**)\≅ mathcal B(E,X)**, where E is finite-dimensional. We then\nserve our main result to improve some results on the reflexivity of mathcal\nB(E,X), the space of all bounded linear operators, by showing that: if\n mathcal B(E,X) is reflexive, then mathcal B(E,X)= mathcal A(E,X), the\nspace of all approximable operators. This particularly implies that, mathcal\nB(E) is reflexive if and only if E is finite-dimensional. Finally, as more\nby-products of the PLR, some generalizations of the classical Goldstine\nweak^*-density theorem are also included.\n