On the purification of factor states

Abstract

Let\(\mathfrak{A}\) be aC*-algebra and\(\mathfrak{A}^ \circ \) be an opposite algebra. Notions of exact andj-positive states of\(\mathfrak{A}^ \circ \) ⊗\(\mathfrak{A}\) are introduced. It is shown, that any factor state ω of\(\mathfrak{A}\) can be extended to a pure exactj-positive state\(\tilde \omega \) of\(\mathfrak{A}^ \circ \) ⊗\(\mathfrak{A}\). The correspondence ω→\(\tilde \omega \) generalizes the notion of the purifications map introduced by Powers and Stormer. The factor states ω1 and ω2 are quasi-equivalent if and only if their purifications\(\tilde \omega _1 \) and\(\tilde \omega _2 \) are equivalent.