In connection with the study of perturbation methods for differential eigenvalue problems, Aronszajn put forth a theory of systems (X, Y; A, B) consisting of a pair of linear transformations A, B:X → Y (see [1]; cf. also [2]). Here X and Y are complex vector spaces, possibly of infinite dimension. The algebraic aspects of this theory, where no restrictions of topological nature are imposed, where developed in [3] and [5]. We hasten to point out that the category of C2-systems (definition in § 1) in which this algebraic investigation takes place is equivalent to the category of all right modules over the ring of matrices of the form

The purpose of this note is to show that the values of an irreducible (Brauer) character are the characteristic values of a matrix with non-negative rational integers. The construction of these integral matrices is done by a description of a representation of the Grothendieck ring of the category of modules over the group algebra. In particular a result of Solomon on characters and a result of Burnside on vanishing of a non-linear character on some conjugate class are generalized.

All functions mentioned in this paper will be real-valued. If f1, f2, g are nonnegative functions on a set S that satisfy g ≦ f1 + f2, the Riesz decomposition problem associated with these data is to find functions gi on S such that

The formula

always furnishes a solution. The problem becomes more interesting if one asks under what conditions one can find solutions that are, roughly speaking, as smooth as the data.

A class of (universal) algebras [4; 5] of the same type or species τ is called equationally complete if the variety [4] Var() generated by has exactly two subvarieties —namely Var() itself and the class of all one element algebras. I t follows that equationally complete varieties are the atoms in the complete lattice of all varieties of a given type of algebras. J. Kalicki, D. Scott [7; 8; 9; 10] and others [2; 3; 6; 12] have considered several questions about equational completeness. A good many of these results have appeared in books. (See, for instance, [4, 5], to which we also refer the reader for more extensive bibliography on subjects related to equational completeness.)