Ordered involutive operator spaces (Q2454889)

This paper is presented as a companion to [Stud.\ Math.\ 182, No.\,3, 227--262 (2007; Zbl 1130.46034)] and [Proc.\ Lond.\ Math.\ Soc.\ (3) 92, No.\,3, 682--712 (2006; Zbl 1118.46051)]. A \textit{selfadjoint} operator space is an operator space \(Y\) with a completely isometric involution \(*:Y\rightarrow Y\). An \textit{ordered} operator space is a selfadjoint operator space \(X\) together with a family of cones \((c_n)\) with \(c_n\subset M_n(X)\) and such that there are complete \(*\)-isometries \(T_n:X\rightarrow B(H)\) that are completely positive in the sense that \(T_n(c_n)\subset M_n(B(H))_+\). Let \(X\) be an ordered operator space, possibly nonunital. The authors construct a noncommutative Shilov boundary for a selfadjoint operator space, and this boundary is then used to construct a \(C^*\)-envelope of \(X\); this \(C^*\)-envelope (the ``ordered noncommutative Shilov boundary'') agrees with that of \textit{W.\,B.\thinspace Arveson} [Acta Math.\ 123, 141--224 (1969; Zbl 0194.15701)] and \textit{M.\,Hamana} [Publ.\ Res.\ Inst.\ Math.\ Sci.\ 15, 773--785 (1979; Zbl 0436.46046)] in the case where \(X\) is an operator system to start with. The ordered noncommutative Shilov boundary is then used to construct maximal and minimal unitizations of \(X\).