Fujimoto, Ichiro; Takahasi, Sin-Ei Equivalent conditions for the general Stone-Weierstrass problem. (English) Zbl 0579.46040 Manuscr. Math. 53, 217-224 (1985). The setting of the Stone-Weierstraßproblem is generalized to the cone \(CP(A,H)\) of all completely positive linear maps of a unital \(C^*\)-algebras \(A\) into the \(C^*\)-algebra \(B(H)\) of all bounded linear operators on a Hilbert space \(H\). It is shown that some conditions are equivalent to that of Stone-Weierstraßconjecture, i.e., a \(C^*\)-subalgebra \(B\) of \(A\) which contains the identity of \(A\) separates the pure states of \(A\). One of such conditions is that \(B\) separates the pure completely positive linear maps \(P_ A(H)\) of \(A\) into \(B(H)\). As a consequence, a new generalized spectrum “\(\oplus_{\dim H\leq \alpha_ A}P_ A(H)\)” is introduced, where \(\alpha_ A=\sup_{\pi \in \hat A}\dim \pi\) and \(\hat A\) is the set of all unitary equivalence classes of the set of irreducible representations. This generalized spectrum contains pure states and irreducible representations as proper subsets. Reviewer: Ichiro Fujimoto Cited in 1 Document MSC: 46L05 General theory of \(C^*\)-algebras 46L30 States of selfadjoint operator algebras Keywords:Stone-Weierstrass problem; completely positive linear maps of a unital \(C^*\)-algebras; pure states; generalized spectrum; irreducible representations PDFBibTeX XMLCite \textit{I. Fujimoto} and \textit{S.-E. Takahasi}, Manuscr. Math. 53, 217--224 (1985; Zbl 0579.46040) Full Text: DOI EuDML References: [1] AKEMANN, C.A.: The general Stone-Weierstrass problem. J. Functional Anal.4, 277–294 (1969) · Zbl 0177.17603 · doi:10.1016/0022-1236(69)90015-9 [2] ANDERSON, J., BUNCE, J.W.: Stone-Weierstrass theorems for separable C*-algebras. J. Operator Theory6, 363–374 (1981) · Zbl 0501.46050 [3] ARVESON, W.B.: Subalgebras of C*-algebras. Acta Math.123, 141–224 (1969) · Zbl 0194.15701 · doi:10.1007/BF02392388 [4] DIXMIER, J.: C*-algebras. Amsterdam: North-Holland 1977 · Zbl 0372.46058 [5] GLIMM, J.: A Stone-Weierstrass theorem for C*-algebras. Ann. of Math.72, 216–244 (1960) · Zbl 0097.10705 · doi:10.2307/1970133 [6] KAPLANSKY, I.: The structure of certain operator algebras. Trans. Amer. Math. Soc.70, 219–255 (1951) · Zbl 0042.34901 · doi:10.1090/S0002-9947-1951-0042066-0 [7] SAKAI, S.: On the Stone-Weierstrass theorem of C*-algebras. Tohoku Math. J.22, 191–199 (1970) · Zbl 0211.16001 · doi:10.2748/tmj/1178242812 [8] SAKAI, S.: C*-algebras and W*-algebras. Berlin-Heidelberg-New York: Springer 1971 [9] SHULTZ, F.W.: Pure states as a dual object for C*-algebras. Commun. Math. Phys.82, 497–509 (1982) · Zbl 0488.46050 · doi:10.1007/BF01961237 [10] STINESPRING, W.F.: Positive functions on C*-algebras. Proc. Amer. Math. Soc.6, 211–216 (1955) · Zbl 0064.36703 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.