Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Affine mappings of invertible operators

Authors: Lawrence A. Harris and Richard V. Kadison
Journal: Proc. Amer. Math. Soc. 124 (1996), 2415-2422
MSC (1991): Primary 46L05
MathSciNet review: 1340389
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] M-D. Choi, D. Hadwin, E. Nordgren, H. Radjavi, and P. Rosenthal, On positive linear maps preserving invertibility, J. Functional Anal. 59 (1984), 462--469. MR 86a:46071
  • [2] 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.
  • [3] B. Fuglede and R. Kadison, Determinant theory in finite factors, Ann. of Math. 55 (1952), 520--530. MR 14:660a
  • [4] L. Harris and R. Kadison, Schurian algebras and spectral additivity, J. of Algebra 180 (1996), 175--186.
  • [5] E. Hille and R. Phillips, Functional Analysis and Semi-Groups, AMS, Providence, 1957. MR 54:11077
  • [6] A. Jafarian and A. Sourour, Spectrum-preserving linear maps, J. Functional Anal. 66 (1986), 255--261. MR 87m:47011
  • [7] R. Kadison, A generalized Schwarz inequality and algebraic invariants for operator algebras, Ann. of Math. 56 (1952), 493--502. MR 14:481c
  • [8] R. Kadison and J. Ringrose, Fundamentals of the Theory of Operator Algebras, Academic Press, Orlando, Vol. I, 1983, Vol. II, 1986. MR 88d:46106
  • [9] M. Marcus, Linear transformations on matrices, J. Res. Natl. Bur. Stand. 75B (1971), 107--113. MR 46:9056
  • [10] M. Marcus and R. Purves, Linear transformations on algebras of matrices: The invariance of the elementary symmetric functions, Canadian J. Math. 11 (1959), 383--396. MR 21:4167
  • [11] B. Russo, Linear mappings of operator algebras, Proc. Amer. Math. Soc. 17 (1966), 1019--1022. MR 33:6428
  • [12] J. Zemánek, Concerning spectral characterizations of the radical in Banach algebras, Comment. Math. Univ. Carolinæ17 (4) (1976), 689--691. MR 55:1070

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 46L05

Retrieve articles in all journals with MSC (1991): 46L05

Additional Information

Lawrence A. Harris
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027

Richard V. Kadison
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395

Keywords: Banach algebra, C*-algebra, invertible elements
Received by editor(s): February 13, 1995
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society