\(C^*\)-algebras generated by operator systems
First recall the definition of an operator system. Consider a complex vector space \(X\) with an involution \(x\to x^*\) and a distinguished self-adjoint element \(1\) (the unit of \(X\)). Assume that there is given a family of complete norms \(\|\cdot\|_n\) on the spaces \(X\otimes M_n(\mathbb{C})\) where \(M_n(\mathbb{C})\) is the complex vector space of \(n\times n\) matrices over \(\mathbb{C}\) and \(n= 1,2,3,\dots\)\ . Then \(X\) is called an operator system if there exists a \(C^*\)-algebra \(A\) and a self-adjoint unital embedding \(\iota: X\to A\) such that for all \(n\), the maps \(\iota\otimes id_n: X\otimes M_n(\mathbb{C})\to A\otimes M_n(\mathbb{C})\) are isometric when \(X\otimes M_n(\mathbb{C})\) is considered with the given norm \(\|\cdot\|_n\) and \(A\otimes M_n(\mathbb{C})\) is considered with the unique \(C^*\)-tensor product norm. So, roughly speaking, an operator system is a closed self-adjoint unital subspace of a \(C^*\)-algebra with unit. If \(X\) and \(Y\) are two operator systems and \(\phi\) a linear map from \(X\) to \(Y\), consider the maps \(\phi_n\) defined as \(\phi\otimes id_n\) from \(X\otimes M_n(\mathbb{C})\) to \(Y\otimes M_n(\mathbb{C})\). If \(\phi_n\) is a positive map for all \(n\), then \(\phi\) is called completely positive. If \(\phi_n\) is isometric for all \(n\), then \(\phi\) is called completely isometric. A \(C^*\)-system is an operator system \(X\) such that the double dual \(X^{**}\) is a \(C^*\)-algebra and such that the canonical embedding of \(X\) in \(X^{**}\) is a unital complete isometry. For any \(C^*\)-algebra \(A\) and any closed left ideal \(L\) of \(A\), the quotient vector space \(A/(L+ L^*)\) (where \(L^*= \{x^*\mid x\in L\}\)) carries a natural operator system structure. It is a \(C^*\)-system and one of the main results of the paper says that any separable \(C^*\)-system is of this form. It is also shown that any (inseparable) \(C^*\)-system is an inductive limit of separable \(C^*\)-systems. To any operator system \(X\) one can associate its universal \(C^*\)-algebra. It is characterized by the fact that there is a unital complete isometry \(u: X\to A\) such that the image generates \(A\) as a \(C^*\)-algebra and such that for any other completely isometric map \(\theta\) from \(X\) to some \(C^*\)-algebra \(B\), there is a *-homomorphism \(\pi: A\to B\) such that \(\pi\circ u=\theta\). It is shown in the paper that any operator system \(X\) has a unique universal \(C^*\)-algebra associated with it. It is denoted by \(C^*_u(X)\). On the other hand, when \(X\) is a \(C^*\)-system, the regular \(C^*\)-algebra of \(X\) is the \(C^*\)-algebra generated by \(X\) in \(X^{**}\) and it is denoted by \(C^*_r(X)\). Recall that for \(C^*\)-systems, \(X^{**}\) is a \(C^*\)-algebra and \(X\) sits in \(X^{**}\) as an operator system. By the definition of the universal \(C^*\)-algebra, for any \(C^*\)-system \(X\) there exists a *-homomorphism \(\sigma: C^*_u(X)\to C^*_r(X)\) (the extension of the identity map). This map is surjective, but in general not injective. If e.g. \(X\) is non-trivial and \(\sigma\) is injective, then \(X\) must contain \(M_n(\mathbb{C})\) for any \(n\). On the other hand, there exists a separable nuclear \(C^*\)-system for which \(\sigma\) is an isomorphism. Recall that an operator system is called nuclear if (roughly speaking) the identity map on \(X\) can be approximated by maps that factor through some \(M_n(\mathbb{C})\). Strange enough however, this operator system cannot be embedded in a nuclear \(C^*\)-algebra. The results and techniques used in the paper are important for the study of the structure of exact \(C^*\)-algebras.
closed self-adjoint unital subspace of a \(C^*\)-algebra with unit
scientific article; zbMATH DE number 1199813