Consistency of a counterexample to Naimark's problem

Akemann, Charles A.; Weaver, Nik

doi:10.1073/pnas.0401489101

Cited by 40 publications

(57 citation statements)

References 17 publications

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“…By Lemma 4 of ref. 6 there is a separable C*-algebra A ␣ ϩ 1 ʕ B(H) that contains A ␣ , a ␣ , and q ␣ , and such that the restriction f ␣ ϩ 1 of g to A ␣ ϩ 1 is pure. To see this, write B(H) as the union of a continuous nested transfinite sequence of separable C*-algebras B ␥ such that B 0 is the C*-algebra generated by A ␣ , a ␣ , and q ␣ .…”

Section: (H)mentioning

confidence: 99%

ℬ( H ) has a pure state that is not multiplicative on any masa

Akemann¹,

Weaver²

2008

Proc. Natl. Acad. Sci. U.S.A.

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Section: (H)mentioning

confidence: 99%

ℬ( H ) has a pure state that is not multiplicative on any masa

Akemann¹,

Weaver²

2008

Proc. Natl. Acad. Sci. U.S.A.

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“…Perhaps it is worth pointing out that in general, heroic attempts to get rid of separability hypotheses for problems in operator algebras can force one to look carefully at the fundamentals of set theory. For example, Akemann and Weaver [AW04] have constructed a counterexample to Naimark's problem by making use of a set-theoretic principle that is known to be consistent with, but not provable from, the standard axioms of set theory. They also showed that the statement There is a counterexample to Naimark's problem that is generated by ℵ 1 elements is undecidable within standard set theory.…”

Section: Pure States Of Smentioning

confidence: 99%

The noncommutative Choquet boundary

Arveson

2007

J. Amer. Math. Soc.

102

123

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Let S S be an operator system–a self-adjoint linear subspace of a unital C ∗ C^* -algebra A A such that 1 ∈ S \mathbf 1\in S and A = C ∗ ( S ) A=C^*(S) is generated by S S . A boundary representation for S S is an irreducible representation π \pi of C ∗ ( S ) C^*(S) on a Hilbert space with the property that π ↾ S \pi \restriction _S has a unique completely positive extension to C ∗ ( S ) C^*(S) . The set ∂ S \partial _S of all (unitary equivalence classes of) boundary representations is the noncommutative counterpart of the Choquet boundary of a function system S ⊆ C ( X ) S\subseteq C(X) that separates points of X X . It is known that the closure of the Choquet boundary of a function system S S is the Šilov boundary of X X relative to S S . The corresponding noncommutative problem of whether every operator system has “sufficiently many" boundary representations was formulated in 1969, but has remained unsolved despite progress on related issues. In particular, it was unknown if ∂ S ≠ ∅ \partial _S\neq \emptyset for generic S S . In this paper we show that every separable operator system has sufficiently many boundary representations. Our methods use separability in an essential way.

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“…The additional axiom we add to Zermelo-Fraenkel set theory with the axiom of choice (ZFC) is Jensen's ♦ ℵ 1 , discussed below in section 3, and our construction is motivated by the work of Akemann and Weaver from ref. 6, where they use ♦ ℵ 1 to construct a counterexample to the Naimark problem. Our main theorem is: Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

Simple nuclear C *-algebras not isomorphic to their opposites

Farah

Hirshberg

2017

Proc. Natl. Acad. Sci. U.S.A.

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We show that it is consistent with Zermelo-Fraenkel set theory with the axiom of choice (ZFC) that there is a simple nuclear nonseparable C * -algebra, which is not isomorphic to its opposite algebra. We can furthermore guarantee that this example is an inductive limit of unital copies of the Cuntz algebra O 2 or of the canonical anticommutation relations (CAR) algebra.C*-algebras | Jensen's diamond | opposite algebra | Naimark's problem | Glimm dichotomy

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Consistency of a counterexample to Naimark's problem

Cited by 40 publications

References 17 publications

ℬ( H ) has a pure state that is not multiplicative on any masa

ℬ( H ) has a pure state that is not multiplicative on any masa

The noncommutative Choquet boundary

Simple nuclear C *-algebras not isomorphic to their opposites

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