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Pure state extensions of the trace on the Choi algebra. (English) Zbl 0672.46037

Let G be the free group on two generators u and v which satisfy \(u^ 2=v^ 3=e\). The authors construct a subset F in G satisfying
(1) there is a sequence \(\{z_ 1\}\) of words in u and v such that \(\cap (z_ iF)=\{e\}.\)
Actually the authors construct sets \(F=F_{\theta}\) depending upon the parameter \(\theta\) equal to sequences of positive integers. Now let L be the left regular representation of G on \(\ell^ 2(G)\) and let A be the \(C^*\)-algebra generated by \(L_ u\), \(L_ v\) and the projection \(E=\chi_ F\). The algebra A is isomorphic to the Cuntz algebra \({\mathcal O}_ 2\). The authors show that A acts irreducibly on \(\ell^ 2(G)\) by analyzing the commutant of A. Since the Choi algebra B is embedded in \({\mathcal O}_ 2\) and since \(x\to <x\chi_ e,\chi_ e>\) induces a trace on B, the authors obtain an example of a tracial state on B that has an extension to a pure state on \({\mathcal O}_ 2\). The authors will show elsewhere that the states on \({\mathcal O}_ 2\) constructed by using \(\theta\) ’s with different tails give mutually inequivalent representations.
The authors consider the present work as part of a more general project of analyzing the kinds of extensions that are possible for states.
Reviewer: H.Halpern

MSC:

46L30 States of selfadjoint operator algebras
46L35 Classifications of \(C^*\)-algebras
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