Affine mappings of invertible operators

Harris, Lawrence; Kadison, Richard

doi:10.1090/S0002-9939-96-03445-4

Affine mappings of invertible operators
HTML articles powered by AMS MathViewer

by Lawrence A. Harris and Richard V. Kadison PDF

Proc. Amer. Math. Soc. 124 (1996), 2415-2422 Request permission

Abstract:

The infinite-dimensional analogues of the classical general linear group appear as groups of invertible elements of Banach algebras. Mappings of these groups onto themselves that extend to affine mappings of the ambient Banach algebra are shown to be linear exactly when the Banach algebra is semi-simple. The form of such linear mappings is studied when the Banach algebra is a C*-algebra.

References

M. D. Choi, D. Hadwin, E. Nordgren, H. Radjavi, and P. Rosenthal, On positive linear maps preserving invertibility, J. Funct. Anal. 59 (1984), no. 3, 462–469. MR 769376, DOI 10.1016/0022-1236(84)90060-0
G. Frobenius, Über die Darstellung der endlichen Gruppen durch lineare Substitutionen, I. Sitzungberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 1897, pp. 994–1015.
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
L. Harris and R. Kadison, Schurian algebras and spectral additivity, J. of Algebra 180 (1996), 175–186.
Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. XXXI, American Mathematical Society, Providence, R.I., 1974. Third printing of the revised edition of 1957. MR 0423094
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
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. II, Pure and Applied Mathematics, vol. 100, Academic Press, Inc., Orlando, FL, 1986. Advanced theory. MR 859186, DOI 10.1016/S0079-8169(08)60611-X
Marvin Marcus, Linear transformations on matrices, J. Res. Nat. Bur. Standards Sect. B 75B (1971), 107–113. MR 309953, DOI 10.6028/jres.075B.008
Marvin Marcus and Roger Purves, Linear transformations on algebras of matrices: the invariance of the elementary symmetric functions, Canadian J. Math. 11 (1959), 383–396. MR 105425, DOI 10.4153/CJM-1959-039-4
B. Russo, Linear mappings of operator algebras, Proc. Amer. Math. Soc. 17 (1966), 1019–1022. MR 198269, DOI 10.1090/S0002-9939-1966-0198269-4
Jaroslav Zemánek, Concerning spectral characterizations of the radical in Banach algebras, Comment. Math. Univ. Carolinae 17 (1976), no. 4, 689–691. MR 428040