Categoria: Operator Algebras

Joan Bosa (Universitat Autònoma de Barcelona)

Stable elements and property (S)

Abstract: We study the relation (and differences) between stability and Property (S) in the simple and stably finite framework. This leads us to characterize stable elements in terms of its support, and study these concepts from different sides : hereditary subalgebras, projections in the multiplier algebra and order properties in the Cuntz semigroup. We use these  approaches to show both that cancellation at infinity on the Cuntz semigroup just holds when its Cuntz equivalence is given by isomorphism at the level of Hilbert right-modules, and that different notions as Regularity, -comparison, Corona Factorization Property, property R, etc.. are equivalent under mild assumptions.

Eduard Vilalta (Universitat Autònoma de Barcelona)

The range problem for the Cuntz semigroup of AI-algebras

Abstract:

A C*-algebra A is said to be a (separable) AI-algebra if A is isomorphic to an inductive limit of the form $\lim\limits_n (C[0,1]\otimes M_n)$ with $F_n$ a finite dimensional C*-algebra for every n. Whenever A is unital and commutative, A is isomorphic to C(X) with X an inverse limit of finite disjoint copies of unit intervals.

In this 2-session talk, we will study the range problem for the Cuntz semigroup of AI-algebras. That is, we will study whether or not one can determine a natural set of properties that an abstract Cuntz semigroup must satisfy in order to be isomorphic to the Cuntz semigroup of an AI-algebra.

During the first part of the talk, we will focus on unital commutative AI-algebras. In this case, one is able to solve the range problem for this class, thus giving a list of properties that an abstract Cuntz semigroup S satisfies if and only if S is isomorphic to the Cuntz semigroup of such an algebra. In order to prove this result, we first introduce the notion of almost chainable spaces and prove that a compact metric space X is almost chainable if and only if C(X) is an AI-algebra. We also characterize when S is isomorphic to the Cuntz semigroup of lower-semicontinuous functions $X\to 0,1,\dots,\infty$ for some T1-space X. The results in this first session will appear in [4].

In the second session, we will present a local characterization for the Cuntz semigroup of any AI-algebra resembling Shen’s local characterization of dimension groups[3], later used in the celebrated Effros-Handelman-Shen theorem[2]. One of the key features in the proof of our result will be the notion of Cauchy sequences for Cu-morphisms (with respect to the distance introduced in [1]) and the fact that, under the right assumptions, they have a unique limit; see [5].

[1] Ciuperca, A. and Elliott, G. “A remark on invariants for C*-algebras of stable rank one”, Int. Math. Res. Not. IMRN(2008)
[2] Effros, E. G. and Handelman, D. E. and Shen, C. L. “Dimension groups and their affine representations”, Amer. J. Math.102(1980), 385–407.
[3] Shen, C. L. “On the classification of the ordered groups associated with the approximately finite dimensional C*-algebras” ,Duke Math. J.46(1979), 613–633.
[4] Vilalta, E. “The Cuntz semigroup of unital commutative AI-algebras”, in preparation.
[5] Vilalta, E. “A local characterization for the Cuntz semigroup of AI-algebras”, (preprint) arXiv:2102.13557 [math.OA]

Joachim Zacharias (University of Glasgow)

AF-embeddings and quotients of the Cantor set

Abstract: The classical Aleksandrov-Uryson Theorem says that every compact metric space X is a quotient of the Cantor set S, hence the C*-algebra C(X) of continuous functions on X embeds into C(S), an AF algebra, i.e. an inductive limit of finite dimensional C*-algebras. Thus every separable commutative C*-algebra is AF-embeddable. Whilst this cannot be true for arbitrary separable non-commutative C*-algebras such embeddings into AF-algebras have been established in many cases. We explore how the proof of the classical A-U-Theorem can be mimicked to obtain AF-embeddings and related results for classes of non-commutative C*-algebras.

Maria Stella Adamo (University of Rome “Tor Vergata”)

Cuntz-Pimsner algebras associated to C*-correspondences over commutative C*-algebras

Abstract: In this talk, structural properties of Cuntz-Pimsner algebras arising by full, minimal, non-periodic, and finitely generated C*-correspondences over commutative C*-algebras will be discussed. A broad class of examples is provided considering the continuous sections $Gamma(V,varphi)$ of a complex locally trivial vector bundle $V$ on a compact metric space $X$ twisted by a minimal homeomorphism $varphi: Xto X$. In this case, we identify a “large enough” C*-subalgebra that captures the fundamental properties of the containing Cuntz-Pimsner algebra. Lastly, we will examine conditions when these C*-algebras can be classified using the Elliott invariant. This is joint work in progress with Archey, Forough, Georgescu, Jeong, Strung, Viola.

Álvaro Sánchez (Universitat Autònoma de Barcelona)

Natural embedding of H0 (G) into K0 (Cr ∗ (G)) for rank 3 Deaconu-Renault groupoids, and HK conjecture II

Abstract: Take an étale groupoid G such that G (0) is compact, metrizable and totally disconnected.
By definition of Cr ∗ (G), we can always consider the canonical inclusion

ι : C(G(0)) →Cr ∗(G), which induces an homomorphism in K-theory

K0 (ι) : K0(C(G (0))) → K0 (Cr ∗ (G)).
Now, since there are no non-unit elements in G (0), K0 (C(G (0) )) = C(G (0) , Z). For any U compact open bisection of G, u = χU is a partial isometry of Cr ∗ (G) such that uu∗ = χs(U), and u∗ u = χr(U ) , which means χs(U ) and χr(U ) belong to the same equivalence class in K0(Cr ∗ (G)).

Then the differential map δ1 : Cc (G, Z) → C(G (0) , Z) defining the homology groups verifies (K0 (ι) ◦ δ1 )(u) = 0 (since δ1 (u) = χs(U ) − χr(U ) ). From here we deduce that imδ1 ⊆ ker(K0 (ι)). Since every étale groupoid has a countable basis consisting in open bisections, it follows that there exists a canonical homomorphism Φ : H0 (G) → K0 (Cr ∗(G)) such that Φ([f ]) := K0 (ι)(f ). The question about when this map is injective is open, even for some simple groupoids. We prove this result for the Deaconu-Renault groupoids of
rank 3, i. e. the groupoids arising from N3 acting over a Cantor set by surjective local homeomorphisms. We use this to prove the HK-conjecture for this family of groupoids.

Álvaro Sánchez (Universitat Autònoma de Barcelona) delivered the talk:

Natural embedding of H0 (G) into K0 (Cr ∗ (G)) for rank 3 Deaconu-Renault groupoids, and HK conjecture I

Abstract: Take an étale groupoid G such that G (0) is compact, metrizable and totally disconnected. By definition of Cr ∗ (G), we can always consider the canonical inclusion ι : C(G (0)) → Cr ∗(G), which induces an homomorphism in K-theory K0 (ι) : K0(C(G (0))) → K0 (Cr ∗ (G)).
Now, since there are no non-unit elements in G (0), K0 (C(G (0) )) = C(G (0) , Z). For any U compact open bisection of G, u = χU is a partial isometry of Cr ∗ (G) such that uu∗ = χs(U ) ,and u∗ u = χr(U ) , which means χs(U ) and χr(U ) belong to the same equivalence class inK0(Cr∗ G)).

Then the differential map δ1 : Cc (G, Z) → C(G (0) , Z) defining the homology groups verifies (K0 (ι) ◦ δ1 )(u) = 0 (since δ1 (u) = χs(U ) − χr(U ) ). From here we deduce that imδ1 ⊆ ker(K0 (ι)). Since every étale groupoid has a countable basis consisting in open bisections, it follows that there exists a canonical homomorphism Φ : H0 (G) → K0 (Cr ∗(G)) such that Φ([f ]) := K0 (ι)(f ). The question about when this map is injective is open, even for some simple groupoids. We prove this result for the Deaconu-Renault groupoids of
rank 3, i. e. the groupoids arising from N3 acting over a Cantor set by surjective local homeomorphisms. We use this to prove the HK-conjecture for this family of groupoids.