Unbounded <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>C</mml:mi><mml:mo>*</mml:mo></mml:msup></mml:mrow></mml:math>-seminorms, biweights, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi></mml:mi><mml:mo>*</mml:mo></mml:msup></mml:mrow></mml:math>-representations of partial <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi></mml:mi><mml:mo>*</mml:mo></mml:msup></mml:mrow></mml:math>-algebras: A review

Trapani, Camillo

doi:https://doi.org/10.1155/IJMMS/2006/79268

International Journal of Mathematics and Mathematical Sciences

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Volume 2006 | Article ID 079268 | https://doi.org/10.1155/IJMMS/2006/79268

Unbounded C-seminorms, biweights, and -representations of partial *-algebras: A review

Camillo Trapani¹

Received01 Mar 2006

Revised30 Jun 2006

Accepted18 Jul 2006

Published03 Oct 2006

Abstract

The notion of (unbounded) C*-seminorms plays a relevant role in the representation theory of *-algebras and partial *-algebras. A rather complete analysis of the case of *-algebras has given rise to a series of interesting concepts like that of semifinite C*-seminorm and spectral C*-seminorm that give information on the properties of *-representations of the given *-algebra A and also on the structure of the *-algebra itself, in particular when A is endowed with a locally convex topology. Some of these results extend to partial *-algebras too. The state of the art on this topic is reviewed in this paper, where the possibility of constructing unbounded C*-seminorms from certain families of positive sesquilinear forms, called biweights, on a (partial) *-algebra A is also discussed.

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Copyright © 2006 Camillo Trapani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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International Journal of Mathematics and Mathematical Sciences

Unbounded C*-seminorms, biweights, and *-representations of partial *-algebras: A review

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Unbounded C-seminorms, biweights, and -representations of partial *-algebras: A review