Hilbert空间中的Bessel列构成的 $C^*$-代数 $B_H(I)$

Let H be a separable Hilbert space, B H(I), B(H) and K(H) the sets of all Bessel sequences {f i}i∈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 αH : B H(I) → B(?2), β : B H(I) → 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), ?, ?) and(B H(I), ·, *) 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) := β-1(K(H)) is the unique proper closed self-adjoint ideal of the C*-algebra B H(I).

$C^*$-Algebra $B_H(I)$ Consisting of Bessel Sequences in a Hilbert Space

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 $B_H(I)$.