Necessary and sufficient conditions are given for a (complete) commutative algebra that is regular in the sense of von Neumann to have a non-zero derivation. In particular, it is shown that there exist non-zero derivations on the algebra L(M) of all measurable operators affiliated with a commutative von Neumann algebra M, whose Boolean algebra of projections is not atomic. Such derivations are not continuous with respect to measure convergence. In the classical setting of the algebra S[0,1] of all...

In recent years much work has been done analyzing maps, not assumed to be linear, between uniform algebras that preserve the norm, spectrum, or subsets of the spectra of algebra elements, and it is shown that such maps must be linear and/or multiplicative. Letting A and B be uniform algebras on compact Hausdorff spaces X and Y, respectively, it is shown here that if λ ∈ ℂ / 0 and T: A → B is a surjective map, not assumed to be linear, satisfying $$∥T\left(f\right)T\left(g\right)+\lambda ∥=∥fg+\lambda ∥\forall f,g\in A,$$ then T is an ℝ-linear isometry and there exist an...