Local derivations of reflexive algebras

Wu, Jing

doi:10.1090/S0002-9939-97-03720-9

Local derivations of reflexive algebras
HTML articles powered by AMS MathViewer

by Jing Wu PDF

Proc. Amer. Math. Soc. 125 (1997), 869-873 Request permission

Abstract:

Let $\mathcal A$ be a reflexive algebra in Banach space $X$ such that both $O_+\not =O$ and $X_-\not =X$ in $\operatorname {Lat} \mathcal A$, the invariant subspace lattice of $\mathcal A$, then every derivation of $\mathcal A$ into itself is spatial. Furthermore, if $X$ is additionally reflexive, then the set of all inner derivations of $\mathcal A$ into itself is topologically algebraically reflexive.

References

Don Hadwin, Algebraically reflexive linear transformations, Linear and Multilinear Algebra 14 (1983), no. 3, 225–233. MR 718951, DOI 10.1080/03081088308817559
De Guang Han and Shu Yun Wei, Local derivations of nest algebras, Proc. Amer. Math. Soc. 123 (1995), no. 10, 3095–3100. MR 1246521, DOI 10.1090/S0002-9939-1995-1246521-1
Richard V. Kadison, Local derivations, J. Algebra 130 (1990), no. 2, 494–509. MR 1051316, DOI 10.1016/0021-8693(90)90095-6
David R. Larson, Reflexivity, algebraic reflexivity and linear interpolation, Amer. J. Math. 110 (1988), no. 2, 283–299. MR 935008, DOI 10.2307/2374503
David R. Larson and Ahmed R. Sourour, Local derivations and local automorphisms of ${\scr B}(X)$, Operator theory: operator algebras and applications, Part 2 (Durham, NH, 1988) Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence, RI, 1990, pp. 187–194. MR 1077437, DOI 10.1090/pspum/051.2/1077437
A. N. Loginov and V. S. Šul′man, Hereditary and intermediate reflexivity of $W^*$-algebras, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 6, 1260–1273, 1437 (Russian). MR 0405124
W. E. Longstaff, Strongly reflexive lattices, J. London Math. Soc. (2) 11 (1975), no. 4, 491–498. MR 394233, DOI 10.1112/jlms/s2-11.4.491