Extensions of the Hilbert-multi-norm in Hilbert $C^{\ast }$-modules

Abstract

Dales and Polyakov introduced a multi-norm $\left({\Vert \cdot \Vert }_n^{(2,2)}:n\in \mathbb{N}\right)$ based on a Banach space $\mathcal{X}$ and showed that it is equal with the Hilbert-multi-norm $\left({\Vert \cdot \Vert }_n^{\mathcal{H}}:n\in \mathbb{N}\right)$ based on an infinite-dimensional Hilbert space $\mathcal{H}$. We enrich the theory and present three extensions of the Hilbert-multi-norm for a Hilbert $C^{\ast }$-module $\mathcal{X}$. We denote these multi-norms by $\left({\Vert \cdot \Vert }_n^{\mathcal{X}}:n\in \mathbb{N}\right)$, $\left({\Vert \cdot \Vert }_n^{\ast }:n\in \mathbb{N}\right)$, and $\left({\Vert \cdot \Vert }_n^{\mathcal{P}\left(\mathfrak{A}\right)}:n\in \mathbb{N}\right)$. We show that ${\Vert x\Vert }_n^{\mathcal{P}\left(\mathfrak{A}\right)}\ge {\Vert x\Vert }_n^{\mathcal{X}}\le {\Vert x\Vert }_n^{\ast }$ for each $x\in {\mathcal{X}}^n$. In the case when $\mathcal{X}$ is a Hilbert $\mathbb{K}\left(\mathcal{H}\right)$-module, for each $x\in {\mathcal{X}}^n$, we observe that ${\Vert \cdot \Vert }_n^{\mathcal{P}\left(\mathfrak{A}\right)}={\Vert \cdot \Vert }_n^{\mathcal{X}}$. Furthermore, if $\mathcal{H}$ is separable and $\mathcal{X}$ is infinite-dimensional, we prove that ${\Vert x\Vert }_n^{\mathcal{X}}={\Vert x\Vert }_n^{\ast }$. Among other things, we show that small and orthogonal decompositions with respect to these multi-norms are equivalent. Several examples are given to support the new findings.

16 pages, Accepted by Positivity