Tensor products of non-self-adjoint operator algebras
An arbitrary complex algebra \({\mathcal A}\) is said to be an operator algebra if there exists a representation \(\rho\) of \({\mathcal A}\) on some Hilbert space such that for every integer \(k\geq 1\) the algebra \({\mathcal M}_ k({\mathcal A})\) of \(k\times k\) matrices with entries from \({\mathcal A}\) is given the norm which is induced by \(\rho\). Such a family of norms on \({\mathcal M}_ k({\mathcal A})\) is called an operator norm. Given two unital operator algebras \({\mathcal A}_ 1\) and \({\mathcal A}_ 2\), a complete operator cross-norm is any operator norm on \({\mathcal A}_ 1\otimes {\mathcal A}_ 2\) which is a cross-norm and for which the natural inclusions of \({\mathcal A}_ i\) \((i=1,2)\) into \({\mathcal A}_ 1\otimes {\mathcal A}_ 1\) \((a_ 1\to a_ 1\otimes 1\), \(a_ 2\to 1\otimes a_ 2)\) induce isometries of \({\mathcal M}_ k({\mathcal A}_ i)\) into \({\mathcal M}_ k({\mathcal A}_ 1\otimes {\mathcal A}_ 2)\) for all k. In the present paper the authors introduce and study three natural complete operator cross-norms, called spatial, minimal and maximal. It is worth mentioning that, in general, the minimal and maximal norms may differ and to establish their equality is usually equivalent to the possibility of making use of Ando's dilation theorem for commuting contractions or the commutant lifting theorem of Sz.-Nagy and Foiaş.
Ando's dilation theorem for commuting contractions
commutant lifting theorem of Sz.-Nagy and Foiaş
scientific article; zbMATH DE number 4180124
scientific article