IJPAM: Volume 14, No. 2 (2004)

FRAMES AND BASES IN HILBERT MODULES
OVER LOCALLY $C^*$-ALGEBRAS

Amir Khosravi$^1$, M.S. Asgari$^2$
$^1$Faculty of Mathematical Sciences and Computer Engineering
University for Teacher Education
Taleghani Ave. 599, Tehran 15614, IRAN
e-mails: khosravi@saba.tmu.ac.ir, khosravi_amir@yahoo.com
$^2$Department of Mathematics
Science and Research Branch
Islamic Azad University
Tehran, IRAN
e-mail: msasgari@yahoo.com


Abstract.In this paper we develop the module frame theory in Hilbert modules over unital locally $C^*$-algebras that possess orthogonal bases, such that the Hilbert spaces and Hilbert $C^*$-modules situations appear as a special case. We show that like orthonormal bases if $\{x_i\}_{i\in I}$ is a frame in Hilbert module $M$ over a unital locally $C^*$-algebra $A$, then for any $x\in M$ the reconstruction formula $x=\sum_{i\in I} <x, S^{-1}x_i>x_i$ is valid, where $S$ is a positive $A$-linear bounded adjointable operator on $M$. Moreover we consider the canonical dual frame in Hilbert $A$-module.

Received: February 27, 2004

AMS Subject Classification: 46L99, 42C15, 47A05, 81R60, 46H25

Key Words and Phrases: frame, bases, $C^*$-algebra, locally $C^*$-algebra, Hilbert $C^*$-module, frame operator

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 14
Issue: 2