Involution and the Haagerup tensor product (Q2747923)

For two \(C^*\)-algebras \({\mathcal A, B}\), consider the algebraic tensor product \({\mathcal A}\otimes{\mathcal B}\) with elements \(u=\sum_{j=1}^na_j\otimes b_j\), \(a_j\in{\mathcal A}\), \(b_j\in{\mathcal B}\), and the Haagerup norm on \({\mathcal A}\otimes{\mathcal B}\) defined by \(\|u\|_H=\inf \{\|\sum_{j=1}^na_ja_j^*\|^{1/2} \|\sum_{j=1}^nb_jb_j^*\|^{1/2}\}\), where the infimum is taken over all representations of \(u\in {\mathcal A}\otimes{\mathcal B}\). The Haagerup tensor product \({\mathcal A}\otimes_H {\mathcal B}\) is the Banach space which is the completed hull of \({\mathcal A}\otimes{\mathcal B}\) relative to the norm \(\|.\|_H\). Furthermore, \({\mathcal A}\otimes_H {\mathcal B}\) is a Banach algebra with the natural multiplication \((a\otimes b) (x\otimes y)=ax\otimes by\) for \(a, x\in{\mathcal A}\), \(b, y\in{\mathcal B}\). NEWLINENEWLINENEWLINEThe aim of the paper under review is to investigate involutions on the Haagerup tensor product \({\mathcal A}\otimes_H {\mathcal B}\). Along these lines, it is shown that the natural involution \(\Theta :{\mathcal A}\otimes{\mathcal B}\to{\mathcal A}\otimes{\mathcal B}\) given by \(\Theta (a\otimes b)=a^*\otimes b^*\) lifts to a continuous map \(\Theta_H\) on \({\mathcal A}\otimes_H {\mathcal B}\) if and only if either \({\mathcal A}\) or \({\mathcal B}\) are finite dimensional, or both \({\mathcal A}\) and \({\mathcal B}\) are infinite dimensional and subhomogeneous (i.e., there is some \(k\in{\mathbb N}\) such that every irreducible representation of \({\mathcal A}\) and \({\mathcal B}\) is on a Hilbert space of dimension not greater than \(k\)). Furthermore, \(\Theta_H\) is an isometry if and only if \({\mathcal A}\) and \({\mathcal B}\) are commutative. NEWLINENEWLINENEWLINEIf the isometric involution \(a\otimes b\mapsto b^*\otimes a^*\) is considered on \({\mathcal A}\otimes_H {\mathcal A}\), it follows that the Haagerup tensor product is an involutive Banach algebra. Among other results, in the paper under review it is shown for a unital \(C^*\)-algebra \({\mathcal A}\) that if \({\mathcal A}\otimes_H {\mathcal A}\) has a faithful \(*\)-representation on a Hilbert space, then \({\mathcal A}\) is commutative.