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For p ≥ 1, a subset K of a Banach space X is said to be relatively p-compact if $K\subset {\sum }_{n=1}^{\infty }\alpha ₙxₙ:\alpha ₙ\in Ball\left(l_{p^{\text{'}}}\right)$, where p’ = p/(p-1) and $xₙ\in l_p^s\left(X\right)$. An operator T ∈ B(X,Y) is said to be p-compact if T(Ball(X)) is relatively p-compact in Y. Similarly, weak p-compactness may be defined by considering $xₙ\in l_p^w\left(X\right)$. It is proved that T is (weakly) p-compact if and only if T* factors through a subspace of $l_p$ in a particular manner. The normed operator ideals $(K_p,{\kappa }_p)$ of p-compact operators and $(W_p,{\omega }_p)$ of weakly p-compact operators, arising from these factorizations,...

The notion of ℱ-approximate order unit norm for ordered ℱ-bimodules is introduced and characterized in terms of order-theoretic and geometric concepts. Using this notion, we characterize the inductive limit of matrix order unit spaces.