Two classes of C^*-power-norms based on Hilbert C^*-modules - Abedi, Sajjad; Moslehian, Mohammad Sal (Mathematica Scandinavica, 2024)
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 P_0(E) and P_2(E), respectively. Our objective is to establish key properties of these extended concepts. We establish that a sequence of norms (· Vert _k :kN) belongs to P_0(E) if and only if, for every operator T in the matrix space M_n× m(L( A)), the norm of T as a mapping from ^2_m(A ) to ^2_n(A ) equals the norm of the corresponding mapping from (E^m,· Vert _m ) to (E^n,· Vert _n ). This characterization is a novel contribution that enriches the broader theory of power-norms. In addition, we prove the inclusion P_0(E)P_2(E). Furthermore, we demonstrate that for the case of A itself, we have P_0(A)=P_2(A)=( · Vert _^2_k(A ) :kN ) brace . This extension of Ramsden’s result shows that the only type-2-multi-norm based on ℂ is (· Vert _^2_k :kN ). To provide concrete insights into our findings, we present several examples in the paper.