An elementary proof of the characterization of isomorphisms of standard operator algebras

Asadi, Mohammad; Khosravi, A.

doi:10.1090/S0002-9939-06-08375-4

An elementary proof of the characterization of isomorphisms of standard operator algebras
HTML articles powered by AMS MathViewer

by Mohammad B. Asadi and A. Khosravi PDF

Proc. Amer. Math. Soc. 134 (2006), 3255-3256 Request permission

Abstract:

This study provides an elementary proof of the well-known fact that any isomorphism $\pi : \mathcal {A}\to \mathcal {B}$ of standard operator algebras on normed spaces $X , Y$, respectively, is spatial; i.e., there exists a topological isomorphism $T : X \to Y$ such that $\pi (A) = TAT^{-1}$ for any $A \in \mathcal {A}$. In particular, $\pi$ is continuous.

References

Paul R. Chernoff, Representations, automorphisms, and derivations of some operator algebras, J. Functional Analysis 12 (1973), 275–289. MR 0350442, DOI 10.1016/0022-1236(73)90080-3
Ali A. Jafarian and A. R. Sourour, Spectrum-preserving linear maps, J. Funct. Anal. 66 (1986), no. 2, 255–261. MR 832991, DOI 10.1016/0022-1236(86)90073-X
Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
Peter emrl, Applying projective geometry to transformations on rank one idempotents, J. Funct. Anal. 210 (2004), no. 1, 248–257. MR 2052121, DOI 10.1016/j.jfa.2003.07.009
Peter emrl, Invertibility preserving linear maps and algebraic reflexivity of elementary operators of length one, Proc. Amer. Math. Soc. 130 (2002), no. 3, 769–772. MR 1866032, DOI 10.1090/S0002-9939-01-06177-9