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Spectrum‐Preserving Linear Mappings between Banach Algebras or Jordan–Banach Algebras
Author(s) -
Aupetit Bernard
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610700001514
Subject(s) - invertible matrix , mathematics , spectrum (functional analysis) , pure mathematics , morphism , matrix (chemical analysis) , linear operators , banach space , mathematical analysis , physics , chemistry , chromatography , quantum mechanics , bounded function
Spectrum‐preserving linear mappings were studied for the first time by G. Frobenius [ 18 ]. He proved that a linear mapping Φ from M n (C) onto M n (C) which preserves the spectrum has one of the forms Φ( x ) = axa −1 or Φ( x ) = a t xa −1 , for some invertible matrix a . (Incidentally the hypothesis that Φ is onto is superfluous; see Proposition 2.1(i).) This result was extended by J. Dieudonné [ 17 ] supposing Φ onto and satisfying SpΦ( x ) ⊂ Sp x , for every n × n matrix x . Several results of M. Nagasawa, S. Banach and M. Stone, R. V. Kadison, A. Gleason and J. P. Kahane and W. Żelazko led I. Kaplansky in [ 22 ] to the following problem: given two Banach algebras with unit and Φ a linear mapping from A into B such that Φ(1) = 1 and SpΦ( x ) ⊂ Sp x , for every x ∈ A , is it true that Φ is a Jordan morphism? With this general formulation, this question cannot be true (see [ 2 ], p. 28).