Quadratic functionals on modules over complex Banach *-algebras with an approximate identity
The problem of representability of quadratic functionals by sesquilinear forms is studied in this article in the setting of a module over an algebra that belongs to a certain class of complex Banach ∗-algebras with an approximate identity. That class includes C∗-algebras as well as H∗-algebras and their trace classes. Each quadratic functional acting on such a module can be represented by a unique sesquilinear form. That form generally takes values in a larger algebra than the given quadratic functional does. In some special cases, such as when the module is also a complex vector space compatible with the vector space of the underlying algebra, and when the quadratic functional is positive definite with values in a C∗-algebra or in the trace class for an H∗-algebra, the resulting sesquilinear form takes values in the same algebra. In particular, every normed module over a C∗-algebra, or an H∗-algebra, without nonzero commutative closed two-sided ideals is a pre-Hilbert module. Furthermore, the representation theorem for quadratic functionals acting on modules over standard operator algebras is also obtained. Introduction. Let M be a right module over a ∗-ring R. Then a mapping Q : M → R is called a quadratic functional if it satisfies the parallelogram law Q(f + g) +Q(f − g) = 2Q(f) + 2Q(g) for all f, g ∈ M and the homogeneity equation Q(fa) = a∗Q(f)a for all f ∈ M and a ∈ R. A biadditive form S : M ×M → R is called sesquilinear if S(fa, g) = a∗S(f, g), S(f, ga) = S(f, g)a for all f, g ∈ M and a ∈ R. For every sesquilinear form S : M × M → R, the mapping Q : M → R defined by Q(f) = S(f, f) is a quadratic functional. The question is if each 2000 Mathematics Subject Classification: Primary 46L, 46K15, 46K50, 47A07, 46H25, 46L08; Secondary 16W, 39B52, 47B47.