C∗-algebra consisting of all g-Bessel sequences in a Hilbert space

Abstract

For Hilbert spaces [Formula: see text] and [Formula: see text], we use the notations [Formula: see text], [Formula: see text] and [Formula: see text] to denote the sets of all [Formula: see text]-Bessel sequences, [Formula: see text]-frames and Riesz bases in [Formula: see text] with respect to [Formula: see text], respectively. By defining a linear operation and a norm, we prove that [Formula: see text] becomes a Banach space and is isometrically isomorphic to the operator space [Formula: see text], where [Formula: see text]. In light of operator theory, it is proved that [Formula: see text] and [Formula: see text] are open sets in [Formula: see text]. This implies that both [Formula: see text]-frames and Riesz bases are stable under a small perturbation. By introducing a linear bijection [Formula: see text] from [Formula: see text] onto the [Formula: see text]-algebra [Formula: see text], a multiplication and an involution on the Banach space [Formula: see text] are defined so that [Formula: see text] becomes a unital [Formula: see text]-algebra that is isometrically isomorphic to the [Formula: see text]-algebra [Formula: see text], provided that [Formula: see text] and [Formula: see text] are isomorphic.