The $p$-class in a dual $B^{\ast }$-algebra

Abstract:

In this paper, we introduce and study the class ${A_p}(0 < p \leqslant \infty )$ in a dual ${B^ \ast }$-algebra $A$. We show that, for $1 \leqslant p \leqslant \infty ,{A_p}$ is a dual ${A^ \ast }$-algebra which is a dense two-sided ideal of $A$. If $1 < p < \infty$, we obtain that ${A_p}$ is uniformly convex and hence reflexive. We also identify the conjugate space of ${A_p}(1 \leqslant p < \infty )$. This is a generalization of the class ${C_p}$ of compact operators on a Hilbert space.