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 Mn× 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)= ( · ²ₖ(A ) :k∈ N ). This extension of Ramsden's result shows that the only type-$2$-multi-norm based on ℂ is ( · ²ₖ :k∈ N ). To provide concrete insights into our findings, we present several examples in the paper.