CENTRAL TORSION UNITS OF INTEGRAL REALITY-BASED ALGEBRAS WITH A POSITIVE DEGREE MAP

A reality-based algebra (RBA) is a finite-dimensional associative algebra that has a distinguished basis B containing 1A, where 1A is the identity element of A, that is closed under a pseudo-inverse condition. If the RBA has a one-dimensional representation taking positive values on B, then we say that the RBA has a positive degree map. When the structure constants relative to a standardized basis of an RBA with positive degree map are all integers, we say that the RBA is integral. Group algebras of finite groups are examples of integral RBAs with a positive degree map, and so it is natural to ask if properties known to hold for group algebras also hold for integral RBAs with positive degree map. In this article we show that every central torsion unit of an integral RBA with algebraic integer coefficients is a trivial unit of the form ζb, for some ζ is a root of unit in C and b is an element of degree 1 in B.