The Scalar Center for Quadratic Jordan Algebras
We propose that an element of a Jordan algebra J should be
considered ``central'' if it really is a scalar multiple of 1 in
some tight unital extension of J. For Jordan algebras with no
trivial ideals, this yields an acceptable center. Here it is
important that an algebra with no extreme elements has a
unique tight unital hull. Lopez has observed that a
necessary condition for centrality of an element c is that the
operators U(c), V(c) must be in the centroid. When the
algebra is nondegenerate, we show this condition is actually
sufficient, but we give a counterexample for general Jordan
algebras.
Bernard Fulgham < bif3c@virginia.edu >