Two classes of \(C^*\)-power-norms based on Hilbert \(C^*\)-modules (Q6568018)

Let \(\left(E,\left\|\cdot\right\|\right)\) be a normed space over \(\mathbb{C}\). A power-norm based on \(E\) is a sequence \((\left\|\cdot\right\|_n : n\in \mathbb{N})\) such that \(\left\|\cdot\right\|_n\) is a norm on \({E}^n\) for each \(n \in \mathbb{N}\), \(\left\|x_0\right\|_1=\left\|x_0\right\|\), where \(x_0 \in E\), and the following conditions hold for each \(n \in \mathbb{N}\) and \(x=(x_1,\ldots,x_n) \in {E}^n\):\N\begin{itemize}\N\item for each \(\sigma \in \mathfrak{S}_n\), where \(\mathfrak{S}_n\) is the set of all permutation of \(\left\lbrace 1,\ldots,n\right\rbrace \), we have \N\[\N\left\| (x_{\sigma(1)},\ldots,x_{\sigma(n)})\right\|_n = \left\| x \right\|_n ; \N\]\N\item for each \(\alpha_1,\ldots,\alpha_n \in \mathbb{C}\), \N\[\N\left\|(\alpha_1x_1,\ldots,\alpha_nx_n) \right\|_n\leq (\max_{1\leq i\leq n } \left| \alpha_i \right| )\left\| x \right\|_n; \N\]\N\item for each \(x_1,\ldots,x_{n-1} \in E\), \N\[\N\left\|(x_1,\ldots,x_{n-1},0) \right\|_n=\left\|(x_1,\ldots,x_{n-1}) \right\|_{n-1}.\N\]\N\end{itemize}\NThen, \(\left( \left\|\cdot\right\|_n: n\in\mathbb{N} \right)\) is a power-norm based on \(E\), and \(\left( \left( {E}^n,\left\|\cdot\right\|_n\right):n\in\mathbb{N} \right) \) is called a power-normed space. A power-norm \(\left( \left\|\cdot\right\|_n: n\in\mathbb{N} \right)\) based on a Hilbert module \(E\) over a C*-algebra of coefficients \(\mathfrak{A}\) is called a C*-power-norm based on \(E\) if, in addition to the above conditions, we have\N\begin{itemize}\N\item for each \(a_1,\ldots,a_n \in \mathfrak{A}\),\N\[\N\left\|(x_1a_1,\ldots,x_na_n) \right\|_n\leq (\max_{1\leq i\leq n}\left\|a_i\right\| )\left\| x \right\|_n.\N\]\N\end{itemize}\NA power-norm is a multi-norm based on \(E\), and \(\left( \left( {E}^n,\left\|\cdot\right\|_n\right):n\in\mathbb{N} \right) \) is a multi-normed space if, in addition to the first three above conditions, we have\N\begin{itemize}\N\item for each \(x_1,\ldots,x_{n-1} \in E\), \N\[\N \left\|\left( x_1,\ldots,x_{n-1},x_{n-1}\right)\right\|_n\leq\left\|\left( x_1,\ldots,x_{n-2},x_{n-1}\right) \right\|_{n-1}. \N\]\N\end{itemize}\NFor a power-normed space \(\left(\left({E}^n,\left\|\cdot\right\|_n\right):n\in\mathbb{N} \right)\), it holds that \N\[\N\max_{1\leq i\leq n}\left\| x_i \right\| \leq \left\|(x_1,\ldots,x_{n}) \right\|_{n}\leq \sum_{i=1}^{n}\left\| x_i \right\|. \N\]\N\textit{H. G. Dales} and \textit{M. E. Polyakov} [Diss. Math. 488, 165 p. (2012; Zbl 1381.46020)] presented the concept of Hilbert multi-norm based on a Hilbert space \(H\) by\N\[\N\left\|\left(x_1,\ldots,x_n\right) \right\|^{H}_n= \sup\left\|p_1x_1 +\dots+p_nx_n\right\|= \sup\left(\left\|p_1x_1 \right\|^2+\dots+\left\|p_nx_n\right\|^2 \right)^{1/2}\N\]\Nfor each \(x_1,\ldots,x_n\in E\), where the supremum is taken over all families \((p_i, 1 \leq i \leq n)\) of mutually orthogonal projections summing to \(1_{\mathcal{L}(H)}\).\N\N\textit{O. Blasco} [Positivity 21, No. 2, 593--632 (2017; Zbl 1392.46007)] introduced the two classes of power-norms based on a normed space \(E\). These concept generalize the notions of multi-norms introduced by Dales and Polyakov and of \(p\)-multi-norm introduced under a different name by \textit{P. Ramsden} [Semigroup Forum 79, No. 3, 515--530 (2009; Zbl 1213.43002)] and extensively studied in [\textit{H. G. Dales} et al., Diss. Math. 524, 115 p. (2017; Zbl 1380.46011)].\N\NIn this paper, the authors study 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 Hilbert \(\mathfrak{A}\)-module \(E\) (which is also a Banach space). They reformulate these results for Hilbert modules and obtain several relations between C*-power-norms.