[Paper Review] The flip is often discontinuous

Volker Runde|ArXiv.org|Feb 28, 2002

Advanced Operator Algebra Research7 references3 citations

TL;DR

This paper demonstrates that the flip automorphism on the algebra of elementary operators $\mathcal{E}\ell(\mathcal{K}(E))$, where $\mathcal{K}(E)$ is the algebra of compact operators on a reflexive Banach space $E$ with the approximation property, is discontinuous unless $\dim E < \infty$. The result follows from the non-coincidence of Arens products on the second dual of $\mathcal{K}(E)$, which fails to be Arens regular when $E$ is infinite-dimensional, implying that the flip cannot extend continuously to the compact operators. This resolves an open problem by showing the flip is often discontinuous in infinite dimensions.

ABSTRACT

Let $A$ be a Banach algebra. The flip on $A \otimes A^\op$ is defined through $A \otimes A^\op i a ensor b \mapsto b ensor a$. If $A$ is ultraprime, $\El(A)$, the algebra of all elementary operators on $A$, can be algebraically identified with $A \otimes A^\op$, so that the flip is well defined on $\El(\A)$. We show that the flip on $\El(A)$ is discontinuous if $A = K(E)$ for a reflexive Banach space $E$ with the approximation property.

Motivation & Objective

• To resolve the open problem of whether the flip on $\mathcal{E}\ell(\mathfrak{A})$ is continuous for ultraprime Banach algebras $\mathfrak{A}$.
• To investigate the continuity of the flip map on $\mathcal{E}\ell(\mathcal{K}(E))$ for reflexive Banach spaces $E$ with the approximation property.
• To determine the conditions under which the flip on $\mathcal{E}\ell(\mathfrak{A})$ extends continuously to the second dual.
• To clarify the role of Arens products and reflexivity in the continuity of the flip on operator algebras.

Proposed method

• Use the canonical algebraic isomorphism between $\mathcal{A} \otimes \mathcal{A}^{\mathrm{op}}$ and $\mathcal{E}\ell(\mathcal{A})$ for ultraprime Banach algebras $\mathcal{A}$, enabling the definition of the flip on $\mathcal{E}\ell(\mathcal{A})$.
• Define the flip as the anti-automorphism $T \mapsto T^\top$ induced by swapping tensor factors in $\mathcal{K}(E) \otimes \mathcal{K}(E)^{\mathrm{op}}$, which corresponds to transposition on elementary operators.
• Apply the theory of Arens products on the second dual: the first and second Arens products on $\mathcal{K}(E)^{**}$ are shown to differ when $E$ is infinite-dimensional and reflexive.
• Use the fact that $\mathcal{K}(E)^{**} \cong \mathcal{B}(E^{**})$ and that the second Arens product is given by $S \circ_2 T = (j^* \circ S^{**} \circ i^{**}) \circ T$, where $i, j$ are canonical embeddings.
• Show that the second Arens product has an identity only if $E$ is reflexive, and that this identity is $\mathrm{id}_{E^{**}}$, which fails to be a two-sided identity unless $E$ is finite-dimensional.
• Use the density of $\mathcal{E}\ell(\mathcal{K}(E))$ in $\mathcal{K}(\mathcal{K}(E))$ and the non-Arens-regularity of $\mathcal{K}(E)$ in infinite dimensions to conclude that the flip cannot extend continuously.

Experimental results

Research questions

• RQ1 Is the flip on $\mathcal{E}\ell(\mathcal{K}(E))$ continuous for a reflexive Banach space $E$ with the approximation property?
• RQ2 Under what conditions does the flip on $\mathcal{E}\ell(\mathfrak{A})$ extend continuously to the second dual of $\mathcal{K}(E)$?
• RQ3 How do the first and second Arens products on $\mathcal{K}(E)^{**}$ relate to the continuity of the flip?
• RQ4 Does the existence of a bounded approximate identity in $\mathcal{K}(E)$ imply continuity of the flip on $\mathcal{E}\ell(\mathcal{K}(E))$?
• RQ5 Can the flip on $\mathcal{E}\ell(\mathfrak{A})$ be continuous when $\mathfrak{A}$ contains $\mathcal{K}(E)$ and $\dim E = \infty$?

Key findings

• The flip on $\mathcal{E}\ell(\mathcal{K}(E))$ is discontinuous if $E$ is an infinite-dimensional reflexive Banach space with the approximation property.
• The flip is continuous on $\mathcal{E}\ell(\mathcal{K}(E))$ if and only if $\dim E < \infty$, establishing a sharp dichotomy.
• The second Arens product on $\mathcal{K}(E)^{**}$ has an identity if and only if $E$ is reflexive, which is a necessary condition for the flip to extend continuously.
• The canonical identification $\mathcal{K}(E)^{**} \cong \mathcal{B}(E^{**})$ implies that $\mathcal{K}(E)$ is Arens regular only when $\dim E < \infty$, which is equivalent to the flip being continuous.
• The algebra $\mathcal{E}\ell(\mathcal{K}(E))$ is norm dense in $\mathcal{K}(\mathcal{K}(E))$ for reflexive $E$ with the approximation property, allowing the extension of the flip to the compact operators.
• The failure of the second Arens product to coincide with the first on $\mathcal{K}(E)^{**}$ in infinite dimensions implies that the flip cannot be continuous on $\mathcal{E}\ell(\mathcal{K}(E))$.