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

Dijana Ilišević¹

¹Department of Mathematics University of Zagreb Bijenič ka 30 P.O. Box 335 10002 Zagreb, Croatia

Studia Mathematica, Tome 171 (2005) no. 2, pp. 103-123

Résumé

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.

Détail

DOI : 10.4064/sm171-2-1

Keywords: problem representability quadratic functionals sesquilinear forms studied article setting module algebra belongs certain class complex banach ast algebras approximate identity class includes * algebras * algebras their trace classes each quadratic functional acting module represented unique sesquilinear form form generally takes values larger algebra given quadratic functional does special cases module complex vector space compatible vector space underlying algebra quadratic functional positive definite values * algebra trace class * algebra resulting sesquilinear form takes values algebra particular every normed module * algebra * algebra without nonzero commutative closed two sided ideals pre hilbert module furthermore representation theorem quadratic functionals acting modules standard operator algebras obtained