Two classes of $C^*$-power-norms based on Hilbert $C^*$-modules

Abstract

Let $\mathfrak {A}$ be a $C^*$-algebra with the multiplier algebra $\mathcal {L}( \mathfrak {A})$. In this paper, we expand upon the concepts of “strongly type-$2$-multi-norm" introduced by Dales and “2-power-norm" introduced by Blasco, adapting them to the context of a left Hilbert $\mathfrak {A}$-module $\mathscr {E}$. We refer to these adapted notions as $\mathscr {P}_0(\mathscr {E})$ and $\mathscr {P}_2(\mathscr {E})$, respectively. Our objective is to establish key properties of these extended concepts. We establish that a sequence of norms $(\lVert \cdot \rVert _k :k\in \mathbb {N})$ belongs to $\mathscr {P}_0(\mathscr {E})$ if and only if, for every operator $T$ in the matrix space $\mathbb {M}_{n\times m}(\mathcal {L}( \mathfrak {A}))$, the norm of $T$ as a mapping from $\ell ^2_m(\mathfrak {A} )$ to $\ell ^2_n(\mathfrak {A} )$ equals the norm of the corresponding mapping from $(\mathscr {E}^m,\lVert \cdot \rVert _m )$ to $(\mathscr {E}^n,\lVert \cdot \rVert _n )$. This characterization is a novel contribution that enriches the broader theory of power-norms. In addition, we prove the inclusion $\mathscr {P}_0(\mathscr {E})\subseteq \mathscr {P}_2(\mathscr {E})$. Furthermore, we demonstrate that for the case of $\mathfrak {A}$ itself, we have $\mathscr {P}_0(\mathfrak {A})=\mathscr {P}_2(\mathfrak {A})=\lbrace ( \lVert \cdot \rVert _{\ell ^2_k(\mathfrak {A} ) } :k\in \mathbb {N} )\rbrace $. This extension of Ramsden's result shows that the only type-$2$-multi-norm based on ℂ is $(\lVert \cdot \rVert _{\ell ^2_k } :k\in \mathbb {N} )$. To provide concrete insights into our findings, we present several examples in the paper.