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Tensor products of non-self-adjoint operator algebras. (English) Zbl 0716.46053
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.
Reviewer: F.H.Vasilescu

MSC:
46M05 Tensor products in functional analysis
46L05 General theory of \(C^*\)-algebras
47L30 Abstract operator algebras on Hilbert spaces
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