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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm171-2-1,
author = {Dijana Ili\v sevi\'c},
title = {Quadratic functionals on modules over complex Banach *-algebras with an approximate identity},
journal = {Studia Mathematica},
volume = {166},
year = {2005},
pages = {103-123},
zbl = {1088.46028},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm171-2-1}
}
Dijana Ilišević. Quadratic functionals on modules over complex Banach *-algebras with an approximate identity. Studia Mathematica, Tome 166 (2005) pp. 103-123. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm171-2-1/