Spectrum-preserving linear mappings were studied for the first time by G. Frobenius [18]. He proved that a linear mapping Φ from Mn([Copf ]) onto Mn([Copf ]) 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).

The notion of stable isomorphism of bundle gerbes is considered. It has the consequence that the stable isomorphism classes of bundle gerbes over a manifold M are in bijective correspondence with H3(M, ℤ). Stable isomorphism sheds light on the local theory of bundle gerbes and enables a classifying theory for bundle gerbes to be developed using results of Gajer on B[Copf ]× bundles.

For a hyperbolic knot, the excellent component of the character curve is the one containing the complete hyperbolic structure on the complement of the knot. The paper explains a method of computing the excellent component of the character variety of tunnel number 1 knots.

The paper is concerned with finding sufficient smoothness conditions on the diffusion and drift coefficient of a one-dimensional stochastic diffusion to imply the existence and smoothness of a taboo density.