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

Dijana Ilišević

Studia Mathematica (2005)

  • Volume: 171, Issue: 2, page 103-123
  • ISSN: 0039-3223

Abstract

top
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.

How to cite

top

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 -

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.