A (bounded, linear) operator H on a Banach space is said to be hermitian if ∥exp(itH)∥ = 1 for all real t. An operator N on is said to be normal if N = H + iK, where H and K are commuting hermitian operators. These definitions generalize those familiar concepts of operators on Hilbert spaces. Also, the normal derivations defined in [1] are normal operators. For more details about hermitian operators and normal operators on general Banach spaces, see [4]. The main result concerning normal operators in the present paper is the following theorem.

A distributive p-algebra is an algebra 〈L; ∨, ∧, *, 0, 1〉 for which 〈L, ∨, ∧, 0, 1〉 is a bounded distributive lattice and * is a unary operation on L such that a ∧ x = 0 if and only if x ≤ a* (i.e. a pseudocomplementation). A distributive double p-algebra is an algebra 〈L; ∨, ∧, *, +, 0, 1〉 in which the deletion of + gives a distributive p-algebra and the deletion of * gives a dual distributive p-algebra, that is a ∨ (x = 1 if and only if x ≥ a+.

For various discrete commutative rings a concept of uniform distribution has already been introduced and studied, for example, for the ring of rational integers by Niven [9] (see also Kuipers and Niederreiter [2, Ch. 5]), for the rings of Gaussian and Eisenstein integers by Kuipers, Niederreiter, and Shiue [3], for rings of algebraic integers by Lo and Niederreiter [4], [7], and for finite fields by Gotusso [1] and Niederreiter and Shiue [8]. In the present paper, we shall show that a satisfactory theory of uniform distribution can also be developed in a noncommutative setting, namely for matrix rings over the rational integers.