A simple, monotracial, stably projectionless
            <i>C</i>
            *‐algebra

Jacelon, Bhishan

doi:10.1112/jlms/jds049

Cited by 58 publications

(87 citation statements)

References 32 publications

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“…The Cu-semiring [0, ∞] is the Cuntz semigroup of the stably projectionless C * -algebra known as the Jacelon-Razak algebra. This algebra has been studied in [Jac13], where it is denoted by W. Following Robert, we denote the Jacelon-Razak algebra by R; see [Rob13a].…”

Section: The Realification Of a Semigroupmentioning

confidence: 99%

Tensor products and regularity properties of Cuntz semigroups

2018

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Abstract. The Cuntz semigroup of a C * -algebra is an important invariant in the structure and classification theory of C * -algebras. It captures more information than K-theory but is often more delicate to handle. We systematically study the lattice and category theoretic aspects of Cuntz semigroups.Given a C * -algebra A, its (concrete) Cuntz semigroup Cu(A) is an object in the category Cu of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu in [CEI08]. To clarify the distinction between concrete and abstract Cuntz semigroups, we will call the latter Cu-semigroups.We establish the existence of tensor products in the category Cu and study the basic properties of this construction. We show that Cu is a symmetric, monoidal category and relate Cu(A ⊗ B) with Cu(A) ⊗ Cu Cu(B) for certain classes of C * -algebras.As a main tool for our approach we introduce the category W of precompleted Cuntz semigroups. We show that Cu is a full, reflective subcategory of W. One can then easily deduce properties of Cu from respective properties of W, for example the existence of tensor products and inductive limits. The advantage is that constructions in W are much easier since the objects are purely algebraic.For every (local) C * -algebra A, the classical Cuntz semigroup W (A) together with a natural auxiliary relation is an object of W. This defines a functor from C * -algebras to W which preserves inductive limits. We deduce that the assignment A → Cu(A) defines a functor from C * -algebras to Cu which preserves inductive limits. This generalizes a result from [CEI08].We also develop a theory of Cu-semirings and their semimodules. The Cuntz semigroup of a strongly self-absorbing C * -algebra has a natural product giving it the structure of a Cu-semiring. For C * -algebras, it is an important regularity property to tensorially absorb a strongly self-absorbing C * -algebra. Accordingly, it is of particular interest to analyse the tensor products of Cu-semigroups with the Cu-semiring of a strongly self-absorbing C * -algebra. This leads us to define 'solid' Cu-semirings (adopting the terminology from solid rings), as those Cu-semirings S for which the product induces an isomorphism between S ⊗ Cu S and S. This can be considered as an analog of being strongly self-absorbing for Cu-semirings. As it turns out, if a strongly self-absorbing C * -algebra satisfies the UCT, then its Cu-semiring is solid. We prove a classification theorem for solid Cu-semirings. This raises the question of whether the Cuntz semiring of every strongly self-absorbing C * -algebra is solid.If R is a solid Cu-semiring, then a Cu-semigroup S is a semimodule over R if and only if R ⊗ Cu S is isomorphic to S. Thus, analogous to the case for C * -algebras, we can think of semimodules over R as Cu-semigroups that tensorially absorb R. We give explicit characterizations when a Cu-semigroup is such a semimodule for the cases that R is the Cu-semiring of a strongly self-absorbing C * -algebra satisfying the UCT. For instance, we show that ...

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Section: The Realification Of a Semigroupmentioning

confidence: 99%

Tensor products and regularity properties of Cuntz semigroups

2018

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“…In the unital case, this result is obtained as a combination of results by Toms [33] and Winter [42]. Many of the motivating examples of stably projectionless C * -algebras are known to be approximately subhomogeneous [12,17,31,37]. In fact, the class of approximately subhomogeneous, stably projectionless algebras is known to even include some crossed products of O 2 by R [10].…”

Section: Introductionmentioning

confidence: 84%

“…One might also add that, modulo the UCT, conditions (i)-(iii) should be equivalent to being classifiable (though in this form, such a statement is not well-formed because classifiability is a property of a class of C * -algebras, and not a property of a single C * -algebra). At the same time, certain examples of simple, stably projectionless C * -algebras have emerged -including certain crossed products of O 2 by R [21], a nuclear, separable, non-Z-stable example [31, Theorem 4.1], and others [12,17,37]. Certain tools, old and new, already allow one to understand some of the structure of these algebras under special hypotheses [6,27].…”

Section: Introductionmentioning

confidence: 99%

Nuclear dimension, $$\mathcal{Z }$$ Z -stability, and algebraic simplicity for stably projectionless $$C^*$$ C ∗ -algebras

Tikuisis

2013

Math. Ann.

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Abstract. The main result here is that a simple separable C * -algebra is Z-stable (where Z denotes the Jiang-Su algebra) if (i) it has finite nuclear dimension or (ii) it is approximately subhomogeneous with slow dimension growth. This generalizes the main results of [33,42] to the nonunital setting. As a consequence, finite nuclear dimension implies Z-stability even in the case of a separable C * -algebra with finitely many ideals. Algebraic simplicity is established as a fruitful weakening of being simple and unital, and the proof of the main result makes heavy use of this concept. IntroductionThe program to classify unital, simple, separable, nuclear C * -algebras has recently seen a small paradigm shift [13]. In light of the fact that there exist such C * -algebras that cannot be classified by ordered K-theory and traces [34], the current trend is to try to identify, using regularity properties, the C * -algebras which are (or should be) classifiable. This idea is crystallized in a conjecture (cf. [36, Section 3.5]), due to Andrew Toms and Wilhelm Winter, that among these C * -algebras, the following properties are equivalent:(i) finite nuclear dimension (defined in [44]); (ii) Z-stability (being isomorphic to one's tensor product with the Jiang-Su algebra Z, introduced in [18]); and (iii) almost unperforated Cuntz semigroup. One might also add that, modulo the UCT, conditions (i)-(iii) should be equivalent to being classifiable (though in this form, such a statement is not well-formed because classifiability is a property of a class of C * -algebras, and not a property of a single C * -algebra). At the same time, certain examples of simple, stably projectionless C * -algebras have emerged -including certain crossed products of O 2 by R [21], a nuclear, separable, non-Z-stable example [31, Theorem 4.1], and others [12,17,37]. Certain tools, old and new, already allow one to understand some of the structure of these algebras under special hypotheses [6,27]. However, most of the theory on regularity properties for C * -algebras was developed only in the unital case, and leads one to ask what obstructions (if any) exist with nonunital algebras. Certainly, the unital case carries with it a number of simplifications: for instance, the simplex of traces on a C * -algebra which take the value 1 at the unit (i.e. which are states) plays an indispensible role in the theory, and it is far from obvious what the correct replacement is in the nonunital case. Nonetheless, one hopes that the simplifications in the unital case are superficial, and that ultimately, one can find and prove analogues or generalizations of the known unital results.2010 Mathematics Subject Classification. 46L35, 46L80, 46L05, 47L40, 46L85.

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“…Let P = [0, ∞], with natural order and addition. Recall that P is isomorphic to the Cuntz semigroup of the Jacelon-Razak algebra (see [Jac13] and [Rob13]). The usual multiplication of real numbers extends to P. This gives P the structure of a commutative Cu-semiring.…”

Section: Cu-semirings and Cu-semimodulesmentioning

confidence: 99%

Abstract bivariant Cuntz semigroups II

2019

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AbstractWe previously showed that abstract Cuntz semigroups form a closed symmetric monoidal category. This automatically provides additional structure in the category, such as a composition and an external tensor product, for which we give concrete constructions in order to be used in applications. We further analyze the structure of not necessarily commutative {\mathrm{Cu}}-semirings, and we obtain, under mild conditions, a new characterization of solid {\mathrm{Cu}}-semirings R by the condition that {R\cong\llbracket R,R\rrbracket}.

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A simple, monotracial, stably projectionless C *‐algebra

Cited by 58 publications

References 32 publications

Tensor products and regularity properties of Cuntz semigroups

Tensor products and regularity properties of Cuntz semigroups

Nuclear dimension, $$\mathcal{Z }$$ Z -stability, and algebraic simplicity for stably projectionless $$C^*$$ C ∗ -algebras

Abstract bivariant Cuntz semigroups II

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