Nuclear \(C^ *\)-algebras and related questions
If \({\mathfrak A}_ 1\) and \({\mathfrak A}_ 2\) are \(C^*\)-algebras due to their exact representations \(\rho_ j: {\mathfrak A}_ j\to B(H_ J)\), where B(H) denotes the algebra of all bounded operators in the Hilbert space H, the minimal norm \(\| \|_{\min}\) can be defined on their tensor product \({\mathfrak A}_ 1\otimes {\mathfrak A}_ 2\). On the other hand there exists the maximal norm \(\| a\|_{\max}=\sup_{\gamma}\| a\|_{\gamma}\) where the supremum is taken over all possible \(C^*\)- norms. The \(C^*\)-algebra \({\mathfrak A}\) is called nuclear, if for any \(C^*\)-algebra \({\mathfrak B}\) the norms \(\| a\|_{\min}=\| a\|_{\max}\) coincide on the tensor product \({\mathfrak A}\otimes {\mathfrak B}\) (i.e. the \(C^*\)-norm is defined there uniquely). The example of such nuclear algebra is \(M_ n({\mathbb{C}})\)-the \(n\times n\) matrices over the complex field. The results obtained by the different authors in the last 10 years on nuclear \(C^*\)-algebras are exposed (73 references).
scientific article; zbMATH DE number 4009103
scientific article