The recognition problem for table algebras and reality-based algebras

Given a finite-dimensional noncommutative semisimple algebra A over C with involution, we show that A always has a basis B for which (A,B) is a reality-based algebra. For algebras that have a one-dimensional representation δ, we show that there always exists an RBA-basis for which δ is a positive degree map. We characterize all RBA-bases of the 5-dimensional noncommutative semisimple algebra for which the algebra has a positive degree map, and give examples of RBA-bases of C⊕Mn(C) for which the RBA has a positive degree map, for all n≥2.