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

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