There exist injective Tauberian operators on ${{L}_{1}}\left( 0,\,1 \right)$ that have dense, nonclosed range. This gives injective nonsurjective operators on ${{\ell }_{\infty }}$ that have dense range. Consequently, there are two quasi-complementary noncomplementary subspaces of ${{\ell }_{\infty }}$ that are isometric to ${{\ell }_{\infty }}$ .

The Arens products are the standard way of extending the product from a Banach algebra to its bidual ″. Ultrapowers provide another method which is more symmetric, but one that in general will only give a bilinear map, which may not be associative. We show that if is Arens regular, then there is at least one way to use an ultrapower to recover the Arens product, a result previously known for C*-algebras. Our main tool is a principle of local reflexivity result for modules and algebras.

Let ${{({{\mathcal{M}}_{i}})}_{i}}{{_{\in }}_{I}},{{({{\mathcal{N}}_{j}})}_{j\in J}}$ be families of von Neumann algebras and $\mathcal{U},\,{\mathcal{U}}'$ be ultrafilters in $I$ , $J$ , respectively. Let $1\,\le \,p\,<\,\infty $ and $n\,\in \,\mathbb{N}$ . Let ${{x}_{1}},...,{{x}_{n}}\,\text{in}\,\prod {{L}_{p}}({{\mathcal{M}}_{i}})$ and ${{y}_{1}},...,{{y}_{n\,}}\text{in }\Pi \,{{L}_{p}}({{\mathcal{N}}_{j}})$ be bounded families. We show the following equality

$$\underset{i,\mathcal{U}}{\mathop{\lim }}\,\,\underset{j,{\mathcal{U}}'}{\mathop{\lim }}\,\,{{\left\| \sum\limits_{k=1}^{n}{{{x}_{k}}(i)\,\otimes \,{{y}_{k}}(j)} \right\|}_{{{L}_{p}}({{\mathcal{M}}_{i}}\otimes {{\mathcal{N}}_{j}})}}=\,\underset{j,{\mathcal{U}}'}{\mathop{\lim }}\,\,\underset{i,\mathcal{U}}{\mathop{\lim }}\,\,{{\left\| \sum\limits_{k=1}^{n}{{{x}_{k}}(i)\,\,\otimes \,{{y}_{k}}(j)} \right\|}_{{{L}_{p}}({{\mathcal{M}}_{i}}\otimes {{\mathcal{N}}_{j}})}}$$

For $p\,=\,1$ this Fubini type result is related to the local reflexivity of duals of ${{C}^{*}}$ -algebras. This fails for $P=\infty $ .