Spectra of elements in operator space tensor products of \(\mathrm{C}^{\ast}\)-algebras
The study of various tensor products in the category of operator spaces and \(C^*\)-algebras has been an essential part of non-commutative functional analysis. Projective tensor product, the Haagerup tensor product and the Schur tensor product are few of the most important ones which have attracted the attention of many working in the area. Recently, Defant and Wiesner introduced a generalized tensor product that includes all the three tensor products mentioned above, and called this construction the \(\lambda\)-tensor product [\textit{A.~Defant} and \textit{D.~Wiesner}, J. Funct. Anal. 266, No.~9, 5493--5525 (2014; Zbl 1305.46040)]. In the present paper, the authors discuss necessary conditions for the injectivity of the canonical mapping \(i_{\lambda}\) from the \(\lambda\)-tensor product \(E \otimes_{\lambda} F\) into the operator space injective tensor product \(E \otimes_{\min} F\). Let \(A\) and \(B\) be two \(C^*\)-algebras, then \(A \otimes_{\lambda} B\) and \(A \otimes_{\min} B\) respectively denote the (completed) \(\lambda\)-tensor product and the minimal \(C^*\)-tensor product of the two. The authors examine the spectra of elementary tensors and a problem related to characterizing elements using the spectra of some members of the algebra. The results are proved in terms of Property-\(P\), which the authors introduce as: A Banach algebra \(A\) is said to have Property-\(P\) if for \(u, v \in A\), \(\sigma (ux) \cup \{0\} = \sigma(vx) \cup \{0\}\) for every \(x \in A\) if and only if \(u = v\), where \(\sigma(x)\) denotes the spectrum of an element \(x \in A\).
scientific article; zbMATH DE number 7437483
scientific article