Reflexivity of operator spaces

Baker, J. M.

doi:10.1090/S0002-9939-1982-0656104-0

Reflexivity of operator spaces
HTML articles powered by AMS MathViewer

by J. M. Baker PDF

Proc. Amer. Math. Soc. 85 (1982), 366-368 Request permission

Abstract:

For reflexive Banach spaces $E$ and $F$ (with $E$ or $F$ having the approximation property), the space of opeartors from $E$ into $F$ (the inductive tensor product of ${E^ * }$ with $F$) is reflexive if and only if the operator space coincides with the inductive tensor product of ${E^ * }$ with $F$. Consequently, $E$ must be finite-dimensional if either the projective tensor product of $E$ with ${E^ * }$ is reflexive, or the inductive tensor product of $E$ with ${E^ * }$ is reflexive and $E$ has the approximation property.

References

Linear operators

Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
J. R. Holub, Hilbertian operators and reflexive tensor products, Pacific J. Math. 36 (1971), 185–194. MR 301487
J. R. Holub, Reflexivity of $L(E,\,F)$, Proc. Amer. Math. Soc. 39 (1973), 175–177. MR 315407, DOI 10.1090/S0002-9939-1973-0315407-4
William H. Ruckle, Reflexivity of $L(E,\,F)$, Proc. Amer. Math. Soc. 34 (1972), 171–174. MR 291777, DOI 10.1090/S0002-9939-1972-0291777-X
Robert Schatten, A Theory of Cross-Spaces, Annals of Mathematics Studies, No. 26, Princeton University Press, Princeton, N. J., 1950. MR 0036935