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

Let A be a C^*-algebra with the multiplier algebra L (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 A-module E. We refer to these adapted notions as P₀ (E) and P₂ (E), respectively. Our objective is to establish key properties of these extended concepts. We establish that a sequence of norms (ₖ: k N) belongs to P₀ (E) if and only if, for every operator T in the matrix space M₍ ₌ (L (A) ), the norm of T as a mapping from ²ₘ (A) to ²ₙ (A) equals the norm of the corresponding mapping from (Eᵐ, ₘ) to (Eⁿ, ₙ). This characterization is a novel contribution that enriches the broader theory of power-norms. In addition, we prove the inclusion P₀ (E) P₂ (E). Furthermore, we demonstrate that for the case of A itself, we have P₀ (A) = P₂ (A) = (ℂ䂵 (₀): k N). This extension of Ramsden's result shows that the only type-2-multi-norm based on ℂ is (ℂ䂵: k N). To provide concrete insights into our findings, we present several examples in the paper.