Quadratic functionals on modules over complex Banach $\ast $-algebras with an approximate identity

Volume 171 / 2005

Dijana Ilišević Studia Mathematica 171 (2005), 103-123 MSC: Primary 46L, 46K15, 46K50, 47A07, 46H25, 46L08; Secondary 16W, 39B52, 47B47. DOI: 10.4064/sm171-2-1


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 $\ast$-algebras with an approximate identity. That class includes ${\rm C}^*$-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 ${\rm C}^*$-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 ${\rm C}^*$-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.


  • Dijana IliševićDepartment of Mathematics
    University of Zagreb
    Bijenič ka 30
    P.O. Box 335
    10002 Zagreb, Croatia

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