TWO CLASSES OF C*-POWER-NORMS BASED ON HILBERT C*-MODULES

Let U be aC*-algebra with themultiplier algebra L(U). 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 U-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 (U center dot U k : k. N) belongs to P0(E) if and only if, for every operator T in the matrix space Mnxm(L(U)), the norm of T as a mapping from U2 m(U) to U2 n(U) equals the norm of the corresponding mapping from (E m, U center dot U m) to (E n, U center dot Un). 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 U itself, we have P0(U) = P2(U) = {(U center dot U U 2k (U) : k. N)}. This extension of Ramsden's result shows that the only type-2-multi-norm based on C is (U center dot U U 2k : k. N). To provide concrete insights into our findings, we present several examples in the paper.