Complements to various Stone-Weierstrass theorems for \(C^*\)-algebras and a theorem of Shultz
There are various versions (due to Kaplansky, Glimm, Longo, Popa and others) of the Stone-Weierstrass theorem for non-commutative \(C^*\)- algebras. The general setting of the non-commutative Stone-Weierstrass theorem is as follows: Let \(A\) be a \(C^*\)-algebra, \(S\) a set of states of \(A\) and \(B\) a \(C^*\)-subalgebra of \(A\) that separates \(S\cup\{0\}\). Then, under various assumptions on \(A\), \(B\) or \(S\) it can be concluded that \(B=A\). Typically, \(S\) is the set of pure states or the set of factorial states of \(A\). In this article complements of these theorems of the following form are proved: Let \(A\) be a \(C^*\)-algebra, \(S\) a set of states of \(A\) and \(B\) a \(C^*\)-subalgebra of \(A\). Let \(x\in A\) such that any two elements of \(S\cup\{0\}\) that coincide on \(B\), also agree on \(x\). Then, under various assumptions on \(A\), \(B\) or \(S\) it can be concluded that \(x\in B\).
Stone-Weierstrass theorem for non-commutative \(C^*\)-algebras
scientific article; zbMATH DE number 54719
scientific article