$C^*$-Algebra $B_H(I)$ Consisting of Bessel Sequences in a Hilbert Space
Received:December 12, 2013  Revised:November 22, 2014
Key Words: $C^*$-algebra   Bessel sequence   Hilbert space   frame   Riesz basis  
Fund Project:Supported by the National Natural Science Foundation of China (Grant Nos.11401359; 11371012; 11301318), China Postdoctoral Science Foundation (Grant No.\,2014M552405) and the Natural Science Research Program of Shaanxi Province (Grant No.\,2014JQ1010).
Author NameAffiliation
Zhihua GUO College of Mathematics and Information Science, Shaanxi Normal University, Shaanxi 710062, P. R. China 
Maoren YIN Department of Mathematics, Junior College of Xinzhou Teachers University, Shanxi 034000, P. R. China 
Huaixin CAO College of Mathematics and Information Science, Shaanxi Normal University, Shaanxi 710062, P. R. China 
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Abstract:
      Let $H$ be a separable Hilbert space, $B_H(I)$, $B(H)$ and $K(H)$ the sets of all Bessel sequences $\{f_i\}_{i\in I}$ in $H$, bounded linear operators on $H$ and compact operators on $H$, respectively. Two kinds of multiplications and involutions are introduced in light of two isometric linear isomorphisms $\alpha_H:B_H(I)\rightarrow B(\ell^2), \beta:B_H(I)\rightarrow B(H)$, respectively, so that $B_H(I)$ becomes a unital $C^*$-algebra under each kind of multiplication and involution. It is proved that the two $C^*$-algebras $(B_H(I), \circ, \sharp)$ and $(B_H(I), \cdot, *)$ are $*$-isomorphic. It is also proved that the set $F_H(I)$ of all frames for $H$ is a unital multiplicative semi-group and the set $R_H(I)$ of all Riesz bases for $H$ is a self-adjoint multiplicative group, as well as the set $K_H(I):=\beta^{-1}(K(H))$ is the unique proper closed self-adjoint ideal of the $C^*$-algebra $B_H(I)$.
Citation:
DOI:10.3770/j.issn:2095-2651.2015.02.009
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