THE TRACE IN BANACH-JORDAN PAIRS

The algebraic trace form (as defined by O. Loos) of an element (x, y) of a (complex) Banach Jordan pair V, where x or y is in the socle, is equal to the sum of the products of all spectral values and their mult iplicity. The trace form is calculated for two examples, the Banach Jo rdan pair of bounded linear operators between two Banach spaces, and t he Banach Jordan pair of a quadratic form. Using analytic multifunctio ns, it is also shown that the complement of the socle of a Banach Jord an pair V is either dense or empty. In the last case, V has finite cap acity.