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Spectrum‐preserving linear mappings were studied for the first time by G. Frobenius [18]. He proved that a linear mapping Φ from Mn(C) onto Mn(C) which preserves the spectrum has one of the forms Φ(x) = axa−1 or Φ(x) = atxa−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).

Using an appropriate definition of the multiplicity of a spectral value, we introduce a new definition of the trace and determinant of elements with finite spectrum in Jordan-Banach algebras. We first extend a result obtained by J. Zemánek in the associative case, on the connectedness of projections which are close to each other spectrally (Theorem 2.3). Secondly we show that the rank of the Riesz projection associated to a finite-rank element a and a finite subset of its spectrum is equal to the sum of the multiplicities of these spectral values (Theorem 2.6). Then we turn to the study of properties such as linearity and continuity of the trace and multiplicativity of the determinant.

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