Fraïssé Limits of C*-Algebras

Eagle, Christopher J.; Farah, Ilijas; Hart, Bradd; Kadets, Borys; Kalashnyk, Vladyslav; Lupini, Martino

doi:10.1017/jsl.2016.14

Cited by 22 publications

(42 citation statements)

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“…Among those instances are operator algebras. In [Eag+15], a more generalized version of Fraïssé theory for metric structures was presented, where the axioms of Fraïssé class were relaxed, and so the bijective correspondence established in the original theory no longer holds and the limit structures would have less homogeneity, though it is still powerful as a construction method. Using this version, the authors of the paper succeeded in realizing a family of AF algebras including the UHF algebras, the hyperfinite II 1 factor and the Jiang-Su algebra as (generalized) Fraïssé limits of a class of finite dimensional C*-algebras with distinguished traces, the class of finite dimensional factors and the class of dimension drop algebras with distinguished traces respectively.…”

Section: Introductionmentioning

confidence: 99%

The Jiang–su Algebra as a Fraïssé Limit

Masumoto¹

2017

J. symb. log.

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Section: Introductionmentioning

confidence: 99%

The Jiang–su Algebra as a Fraïssé Limit

Masumoto¹

2017

J. symb. log.

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“…(3) the sort C is interpreted as the complex numbers, and its domains are interpreted correctly; (4) the interpretation of the domains D λ,S n is the ball of radius n of the range of E λ centered at the origin (see also [12, Example 2.2.1]); (5) the interpretation of the domain D λ,S n is the ball of radius n of the range of P λ ⊗ E λ centered at the origin; (6) the interpretation of the sort S (0) is isomorphic to C(G) as a C*-algebra; (7) the interpretation of the sort S (1) is isomorphic to C(G) ⊗ A as a C(G)-bimodule; (8) the norm on S (1) is the minimal (injective) tensor product norm (see Lemma 3.4); (9) the function symbols for T :…”

Section: 42mentioning

confidence: 99%

“…This includes UHF-algebras (or matroid algebras), AF-algebras, AI-algebras, AT-algebras, countable inductive limits of one-dimensional NCCW-complexes, and several other classes. (5) having nuclear dimension at most n; (6) having decomposition rank at most n; (7) having real rank at most n; (8) having stable rank at most n.…”

Section: The Rokhlin Propertymentioning

confidence: 99%

“…Items (1),(3),(4),(5),(8), (9) can be obtained by observing that the corresponding properties are definable by a uniform family of L C* G -formulas as shown in [12, Theorem 2.5.1, Theorem 2.5.2] and [22,Proposition 4.3]; see also [22,Remark 3.2]. The items (2), (6), (7) can alternatively be obtained by applying the fact that the corresponding properties are definable by a uniform family of L C*,nuc G -formulas as shown in [12,Section 5]; see also [22,Remark 3.5]. The language L C*,nuc G is the nuclear language for C*-algebras introduced in [22, Subsection 3.3].…”

Section: The Rokhlin Propertymentioning

confidence: 99%

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Model theory and Rokhlin dimension for compact quantum group actions

Gardella

Kalantar

Lupini

2019

J. Noncommut. Geom.

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We show that, for a given compact or discrete quantum group G, the class of actions of G on C*-algebras is first-order axiomatizable in the logic for metric structures. As an application, we extend the notion of Rokhlin property for G-C*-algebra, introduced by Barlak, Szabó, and Voigt in the case when G is second countable and coexact, to an arbitrary compact quantum group G. All the the preservations and rigidity results for Rokhlin actions of second countable coexact compact quantum groups obtained by Barlak, Szabó, and Voigt are shown to hold in this general context. As a further application, we extend the notion of equivariant order zero dimension for equivariant *-homomorphisms, introduced in the classical setting by the first and third authors, to actions of compact quantum groups. This allows us to define the Rokhlin dimension of an action of a compact quantum group on a C*-algebra, recovering the Rokhlin property as Rokhlin dimension zero. We conclude by establishing a preservation result for finite nuclear dimension and finite decomposition rank when passing to fixed point algebras and crossed products by compact quantum group actions with finite Rokhlin dimension.

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“…The universal minimal flow has been explicitly computed for many topological groups, typically given as automorphism groups of naturally arising mathematical structures. Examples of extremely amenable Polish groups include the group of order automorphisms of Q [61], the group of unitary operators on the separable infinite-dimensional Hilbert space [30], the automorphism group of the hyperfinite II 1 factor and of infinite type UHF C * -algebras [14,21], or the isometry group of the Urysohn space [62]. Examples of nontrivial metrizable universal minimal flows include the universal minimal flow of the group of orientation preserving homeomorphisms of the circle, which is equivariantly homeomorphic to the circle itself [61], the universal minimal flow of the group S ∞ of permutations of N, which can be identified with the space of linear orders on N [22], and the universal minimal flow of the homeomorphism group Homeo(2 N ) of the Cantor set 2 N , which can be seen as the canonical action of Homeo(2 N ) on the space of maximal chains of closed subsets of 2 N [23, 37,70].…”

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confidence: 99%

The Ramsey property for Banach spaces and Choquet simplices, and applications

Bartošová

Lopez-Abad

Lupini

et al. 2017

Comptes Rendus. Mathématique

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Abstract. We show that the Gurarij space G and its noncommutative analog NG both have extremely amenable automorphism group. We also compute the universal minimal flows of the automorphism groups of the Poulsen simplex P and its noncommutative analogue NP. The former is P itself, and the latter is the state space of the operator system associated with NP. This answers a question of Conley and Törnquist. We also show that the pointwise stabilizer of any closed proper face of P is extremely amenable. Similarly, the pointwise stabilizer of any closed proper biface of the unit ball of the dual of the Gurarij space (the Lusky simplex) is extremely amenable.These results are obtained via the Kechris-Pestov-Todorcevic correspondence, by establishing the approximate Ramsey property for several classes of finite-dimensional operator spaces and operator systems (with distinguished linear functionals), including: Banach spaces, exact operator spaces, function systems with a distinguished state, and exact operator systems with a distinguished state. This is the first direct application of the Kechris-Pestov-Todorcevic correspondence in the setting of metric structures. The fundamental combinatorial principle that underpins the proofs is the Dual Ramsey Theorem of Graham and Rothschild.In the second part of the paper, we obtain factorization theorems for colorings of matrices and Grassmannians over R and C, which can be considered as continuous versions of the Dual Ramsey Theorem for Boolean matrices and of the Graham-Leeb-Rothschild Theorem for Grassmannians over a finite field.Recall that given a topological group G, a compact G-space or G-flow is a compact Hausdorff space X endowed with a continuous action of G. Such a G-flow X is called minimal when every orbit is dense. There is a natural notion of morphism between G-flows, given by a (necessarily surjective) G-equivariant continuous map (factor). A minimal G-flow is universal if it factors onto any other minimal G-flow. It is a classical fact that any topological group G admits a unique (up to isomorphism of G-flows) universal minimal flow, usually denoted by M (G) [18,33]. For any locally compact non compact Polish group G, the universal minimal G-flow is nonmetrizable. At the opposite end, non locally compact topological groups often have metrizable universal minimal flows, or even reduced to a single point. A topological group for which M (G) is a singleton is called extremely amenable. (Amenability of G is equivalent to the assertion that every compact G-space has an invariant Borel measure. Thus any extremely amenable group is, in particular, amenable.)The universal minimal flow has been explicitly computed for many topological groups, typically given as automorphism groups of naturally arising mathematical structures. Examples of extremely amenable Polish groups include the group of order automorphisms of Q [61], the group of unitary operators on the separable infinite-dimensional Hilbert space [30], the automorphism group of the hyperfinite II 1 factor and of in...

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Fraïssé Limits of C*-Algebras

Cited by 22 publications

References 38 publications

The Jiang–su Algebra as a Fraïssé Limit

The Jiang–su Algebra as a Fraïssé Limit

Model theory and Rokhlin dimension for compact quantum group actions

The Ramsey property for Banach spaces and Choquet simplices, and applications

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