The Choquet boundary of an operator system. (English) Zbl 1344.46041
The authors establish that every pure unital completely positive map from an operator system into \(\mathcal{B}(H)\) has a pure maximal dilation which is a boundary representation. In fact, they show that every operator system has sufficiently many boundary representations to generate the \(C^*\)-envelope, solving a long-standing open problem of W. B. Arveson [Acta Math. 123, 141–224 (1969; Zbl 0194.15701)].
Reviewer: Ghadir Sadeghi (Sabzevār)
MSC:
| 46L07 | Operator spaces and completely bounded maps |
| 46L52 | Noncommutative function spaces |
| 47L55 | Representations of (nonselfadjoint) operator algebras |
| 47A20 | Dilations, extensions, compressions of linear operators |
Citations:
Zbl 0194.15701References:
| [1] | J. Agler, “An abstract approach to model theory” in Surveys of Some Recent Results in Operator Theory, Vol. II , Pitman Res. Notes Math. Ser. 192 , Longman Sci. Tech., Harlow, U.K., 1988, 1-23. · Zbl 0788.47005 |
| [2] | W. Arveson, Subalgebras of C\ast -algebras , Acta Math. 123 (1969), 141-224. · Zbl 0194.15701 · doi:10.1007/BF02392388 |
| [3] | W. Arveson, Subalgebras of C\ast -algebras, II , Acta Math. 128 (1972), 271-308. · Zbl 0245.46098 · doi:10.1007/BF02392166 |
| [4] | W. Arveson, The noncommutative Choquet boundary , J. Amer. Math. Soc. 21 (2008), 1065-1084. · Zbl 1207.46052 · doi:10.1090/S0894-0347-07-00570-X |
| [5] | D. Blecher, Z.-J. Ruan, and A. Sinclair, A characterization of operator algebras , J. Funct. Anal. 89 (1990), 188-201. · Zbl 0714.46043 · doi:10.1016/0022-1236(90)90010-I |
| [6] | M. D. Choi and E. G. Effros, Injectivity and operator spaces , J. Funct. Anal. 24 (1977), 156-209. · Zbl 0341.46049 · doi:10.1016/0022-1236(77)90052-0 |
| [7] | M. A. Dritschel and S. A. McCullough, Boundary representations for families of representations of operator algebras and spaces , J. Operator Theory 53 (2005), 159-167. · Zbl 1119.47311 |
| [8] | D. R. Farenick, Extremal matrix states on operator systems , J. Lond. Math. Soc. (2) 61 (2000), 885-892. · Zbl 0960.46038 · doi:10.1112/S0024610799008613 |
| [9] | D. R. Farenick, Pure matrix states on operator systems , Linear Algebra Appl. 393 (2004), 149-173. · Zbl 1073.46044 · doi:10.1016/j.laa.2003.12.006 |
| [10] | M. Hamana, Injective envelopes of operator systems , Publ. Res. Inst. Math. Sci. 15 (1979), 773-785. · Zbl 0436.46046 · doi:10.2977/prims/1195187876 |
| [11] | A. Hopenwasser, Boundary representations on C\ast -algebras with matrix units , Trans. Amer. Math. Soc. 177 (1973), 483-490. · Zbl 0264.46058 · doi:10.2307/1996610 |
| [12] | C. Kleski, Boundary representations and pure completely positive maps , J. Operator Theory 71 (2014), 45-62. · Zbl 1349.46061 · doi:10.7900/jot.2011oct22.1927 |
| [13] | C. Kleski, personal communication, May 2013. |
| [14] | P. S. Muhly and B. Solel, “An algebraic characterization of boundary representations” in Nonselfadjoint Operator Algebras, Operator Theory, and Related Topics , Oper. Theory Adv. Appl. 104 , Birkhäuser, Basel, 1998, 189-196. · Zbl 0899.46038 · doi:10.1007/978-3-0348-8779-3_13 |
| [15] | V. Paulsen, Completely Bounded Maps and Operator Algebras , Cambridge Stud. Adv. Math. 78 , Cambridge Univ. Press, Cambridge, 2002. · Zbl 1029.47003 |
| [16] | B. Sz.-Nagy, C. Foiaş, H. Bercovici, and L. Kérchy, Harmonic Analysis of Operators on Hilbert Space , 2nd ed., Springer, New York, 2010. |
| [17] | C. Webster and S. Winkler, The Krein-Milman theorem in operator convexity , Trans. Amer. Math. Soc. 351 , no. 1 (1999), 307-322. · Zbl 0908.47042 · doi:10.1090/S0002-9947-99-02364-8 |
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