Haagerup norm on C${}^*$-algebras

Question

Let $E$ and $F$ be C${}^*$-algebras. Let $\|\cdot \|_h $ denote the Haagerup norm defined on the algebraic tensor product $E \odot F$.

Is the completion of $E \odot F$ with respect to $\|\cdot\|_h$ a C${}^*$-algebra?

Answer 1

$\newcommand{\Hi}{\mathcal{H}} \newcommand{\Li}{\mathcal{L}}$

Call $E \otimes_h F$ the completion of $E \odot F$ under $\| \cdot \|_h$. The answer is that $E \otimes_h F$ is usually not a C${}^*$-algebra.

In general, $E \otimes_h F$ is not even an operator algebra. In fact, David Blecher proved on his Ph.D disertation the following result

Theorem 1. $E \otimes_h F$ is an operator algebra if and only if $E$ or $F$ is finite dimensional.

The best general result we can get is:

Theorem 2. $E \otimes_h F$ is a Banach algebra that is isometrically isomorphic to a subalgebra of $\mathcal{L}(\mathcal{L}(\mathcal{H}))$ for some Hilbert space $\mathcal{H}$.

Edit: Here's more detail on how to prove theorem 2. $E \otimes_h F$ is seen as a Banach algebra by extending the usual multiplication $$ (a \otimes b)(c \otimes d)=(ac)\otimes (db) $$ from $E \odot F$ to all of $E \otimes_h F$.

Let $(\varphi_E, \Hi_E)$ and $(\varphi_F, \Hi_F)$ the universal representations for the C${}^*$-algebras $E$ and $F$ respectively. Define $\varphi: E \odot F \to \Li(\Li(\Hi_E \oplus \Hi_F))$ on elementary tensors by $$ \varphi(a \otimes b)(T):= (\varphi_E(a) \oplus 0)T(0 \oplus \varphi_F(b)) $$ for any $T \in \Li(\Hi_E \oplus \Hi_F)$. One can check that this extends to $ E \otimes_h F$ and gives an isometric algebra homomorphism from $E \otimes_h F$ to $\mathcal{L}(\mathcal{L}(\mathcal{H}_E \oplus \Hi_F))$.