Quadratic functionals on modules over complex Banach *-algebras with an approximate identity

Dijana Ilišević. "Quadratic functionals on modules over complex Banach *-algebras with an approximate identity." Studia Mathematica 171.2 (2005): 103-123. <http://eudml.org/doc/284825>.

@article{DijanaIlišević2005,
abstract = {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.},
author = {Dijana Ilišević},
journal = {Studia Mathematica},
keywords = {quadratic functional equation; Banach -algebra; approximate identity; -algebra; -algebra; trace class; standard operator algebra; Hilbert module; sesquilinear form; Jordan -derivation; double centralizer},
language = {eng},
number = {2},
pages = {103-123},
title = {Quadratic functionals on modules over complex Banach *-algebras with an approximate identity},
url = {http://eudml.org/doc/284825},
volume = {171},
year = {2005},
}

TY - JOUR
AU - Dijana Ilišević
TI - Quadratic functionals on modules over complex Banach *-algebras with an approximate identity
JO - Studia Mathematica
PY - 2005
VL - 171
IS - 2
SP - 103
EP - 123
AB - 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.
LA - eng
KW - quadratic functional equation; Banach -algebra; approximate identity; -algebra; -algebra; trace class; standard operator algebra; Hilbert module; sesquilinear form; Jordan -derivation; double centralizer
UR - http://eudml.org/doc/284825
ER -