Communications of the
Korean Mathematical Society CKMS

Let $\mathcal{X}$ be a countably generated Hilbert module over a locally $C^*$-algebra $\mathcal{A}$ in multiplier module $M(\mathcal{X})$ of $\mathcal{X}$. We propose the necessary and sufficient condition such that a sequence $\{h_n:n\in \mathbb N\}$ in $M(\mathcal{X})$ is a standard frame of multipliers in $\mathcal{X}$. We also show that if $T$ in $b(L_{\mathcal{A}}(\mathcal{X}))$, the space of bounded maps in set of all adjointable maps on $\mathcal{X}$, is surjective and $\{h_n:n\in \mathbb N\}$ is a standard frame of multipliers in $\mathcal{X}$, then $\{T\circ h_n:n\in \mathbb N\}$ is a standard frame of multipliers in $\mathcal{X}$, too.

Keywords: locally $C^*$-algebras, Hilbert modules over locally $C^*$-algebras, bounded module maps, countably generated Hilbert modules, standard frames of multipliers

MSC numbers: Primary 46L08; Secondary 46L05, 42C15