$C^*$-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
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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). |
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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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