Gleason’s Problem Associated to a Real Ternary Algebra and Applications

This paper is an introduction of a new class of analytic functions defined on a ternary algebra, a three dimensional structure different from \({{\mathbb {C}}}\times {{\mathbb {R}}}\), i.e. a real commutative algebra given by the linear span of \(\{1,\mathbf{e},\mathbf{e}^2\}\), where \(\mathbf{e}\not \in {{\mathbb {C}}}\) is a generating unit. We define a single ternary conjugate and we build a new analytic function theory, different from previous approaches, on the basis of this single conjugation (akin to the quaternionic case). We give the solution to the Gleason problem which gives rise to Fueter-type variables and study the space of rational functions in this case.