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Conference on

Structure and Classification of C∗-Algebras

Centre de Recerca MatematicaBellaterra

June 6 to 10, 2011

Organisers

Ramon Antoine, Universitat Autonoma de Barcelona

Pere Ara, Universitat Autonoma de Barcelona

Joan Bosa, Universitat Autonoma de Barcelona

Francesc Perera, Universitat Autonoma de Barcelona

Scientific Committee

George A. Elliott, University of Toronto

Andrew S. Toms, Purdue University

Nathanial P. Brown, The Pennsylvania State University

Joachim Cuntz, Universitat Munster

Marius Dadarlat, Purdue University

Mikael Rørdam, University of Copenhagen

Speakers

Nathanial P. Brown, The Pennsylvania State University

Marius Dadarlat, Purdue University

George A. Elliott, University of Toronto

Ilan Hirshberg, Ben Gurion University of The Negev

Takeshi Katsura, University of Copenhagen

Eberhard Kirchberg, Humboldt Universitat zu Berlin

Huaxin Lin, University of Oregon

Christopher N. Phillips, University of Oregon

Leonel Robert, University of Copenhagen

Mikael Rørdam, University of Copenhagen

Luis Santiago, Universitat Autonoma de Barcelona

Mark Tomforde, University of Houston

Andrew S. Toms, Purdue University

Wilhelm Winter, University of Nottingham

Acknowledgements: Conference on “Structure and Classification of C∗-Algebras” is

supported by the AGAUR (ref. 2010ARCS 1-00098), by the Ingenio Mathematica Pro-

gramme (ref. PMII-C5-0332), and the Ministerio de Ciencia e Innovacion (ref. MTM2010-

10838-E).

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Contents

1. Practical Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2. Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3. Abstracts of Main Talks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Nathanial P. Brown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Analogies in the structure of C∗- and W ∗-algebras.

Marius Dadarlat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Group quasi-representations and index theory.

George A. Elliott. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15An invariant for non-simple C∗-algebras of stable rank one.

Ilan Hirshberg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15The higher dimensional Rokhlin property.

Takeshi Katsura . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Towards the classification of Cuntz-Krieger algebras and graphalgebras.

Eberhard Kirchberg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Local morphisms and C∗-extensions.

Huaxin Lin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Locally AH-algebras and the tracial rank.

Christopher N. Phillips. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Towards the classification of outer actions of finite groups onpurely infinite algebras.

Leonel Robert. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18The Cuntz semigroup: axioms and variations.

Mikael Rørdam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Divisibility properties for C∗-algebras.

Luis Santiago . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19A stably projectionless C∗-algebra.

Mark Tomforde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Classifying nonsimple graph C∗-algebras up to stableisomorphism.

Andrew S. Toms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Classification and C∗-algebras.

Wilhelm Winter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Dynamics and dimension.

4. Abstracts of Contributed Talks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Dawn Archey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Living without projections.

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Etienne Blanchard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Amalgamated products of C∗-bundles.

Joan Bosa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21The Cuntz semigroup of some C(X)-algebras.

Jose R. Carrion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21On invariants of almost-flat bundles associated withalmost-homomorphisms of groups.

Christina Cerny . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22On Fowler’s Toeplitz- and Cuntz-Pimsner algebras over Nk

product systems.

Caleb Eckhardt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Free products of nuclear C∗-algebras.

Taylor Hines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22The radius of comparison for crossed products and themean topological dimension.

David Kerr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Sofic entropy and noncommutative dynamics.

Alla Kuznetsova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Characterization of nuclearity of a C∗-algebra in terms ofa compact subgroup of the group of its automorphisms.

Hyun Ho Lee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23A lifting problem through K-theory and deformation.

Xin Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Semigroup C∗-algebras and amenability of semigroups.

Hiroki Matui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Group actions on simple stably finite C∗-algebra.

Eduard Ortega . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Graph C∗-algebras and crossed product by endomorphisms.

Cornel Pasnicu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24The Cuntz semigroup, a Riesz type interpolation property,comparison and the ideal property.

Efren Ruiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Classification of singular graph algebras.

Yasuhiko Sato. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Gauge actions on the Jiang-Su algebra with the Rohlin property.

David Sherman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Unified representation theorems for Hilbert space operators.

Tatiana Shulman. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Projective C∗-algebras and noncommutative shape theory.

Mitsuharu Takeori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Application of E-theory to the classification of C∗-algebras.

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Hannes Thiel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Inductive limits of projective C∗-algebras.

Stuart White . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Perturbations of some crossed product algebras.

6. List of participants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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1. Practical Information

Programme and list of participants:You can check the latest version of the programme as well as the list of partic-

ipants at the conference’s web page.

Lodging arrangements:A list with the participants and their lodging arranged through the CRM is

posted on the conference’s web page. Please, check your particular lodging ar-rangement there (http://www.crm.cat/calgebras) and contact us as soon aspossible if you find any inaccuracy.

Lecture room:The Conference will take place in the CRM “Auditori” located in the Sciences

Building (Edifici de Ciencies), Universitat Autonoma de Barcelona in Bellaterra.(http://www.crm.es/General/LocationEng.htm)

Administration:The CRM Administration will be available to the participants from Monday

to Friday from 9:00 am to 1:00 pm.

Computer facilities:The computer space of the CRM will be available for the participants of the

Activity.

Printing and photocopying policy:Printing and photocopying at the CRM is only permitted to research visitors

and staff.Participants to CRM activities can use the OCE∗ for printing and photocopy-

ing.The OCE is located at floor -1 (one level below the CRM) and next to the

front desk (looking at it, on the left hand side). They offer black and white andcolor printing and photocopying and from paper or pen-drive sources.

Prices are 0,045 euro/page for black and white, and 0,33 euro/page for colorcopies.

Their opening hours are Monday to Friday from 8:30 to 14:00 and from 15:00to 20:00.∗OCE is the Faculty printing and photocopying service

Breaks:Coffee and cookies will be served during the morning breaks to all participants.

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Picture: A group picture will be taken on Wednesday, June 8 before the coffeebreak. We will inform you of the place to meet. The picture will be posted onthe Activity’s web page.

Questionnaire:Following the directions of the CRM Governing Board, we give a questionnaire

to all the people participating in activities at the CRM in order to assess theirlevel of satisfaction. The questionnaire is anonymous and not mandatory, but wewould greatly appreciate it if you could answer the questions and return it to us.Thank you for your cooperation.

Local emergency numbers:

Medical emergency campus number(inside the University premises)

1800 / 1900 during officehours 2525 at other times

UAB’s Science Faculty reception office(inside the University premises)

1055

General emergency (police, fire- fighters,ambulances)

112

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2. Schedule

Monday, June 6

09:00 – 09:30 Registration

09:30 – 10:25 M. Rørdam

Divisibility properties for C∗-algebras

10:30 – 11:00 Coffee Break

11:00 – 11:55 E. Kirchberg

Local morphisms and C∗-extensions.

12:00 – 12:30 S. White

Perturbations of some crossed product algebras.

12:35 – 13:05 D. Archey

Living without projections.

Lunch

15:00 – 15:55 L. Santiago

A stably projectionless C∗-algebra.

16:00 – 16:25 Break

16:25 – 16:55 T. Shulman

Projective C∗-algebras and noncommutative shape theory.

17:00 – 17:30 H. Thiel

Inductive limits of projective C∗-algebras.

17:35 – 18:05 J. Bosa

The Cuntz semigroup of some C(X)-algebras.

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Tuesday, June 7

09:30 – 10:25 M. Dadarlat

Group quasi-representations and index theory.


11:00 – 11:55 M. Tomforde

Classifying nonsimple graph C∗-algebras up to stableisomorphism.

12:00 – 12:30 E. Blanchard

Amalgamated products of C∗-bundles.

12:35 – 13:05 E. Ruiz

Classification of singular graph algebras.

Lunch

15:00 – 15:55 T. Katsura

Towards the classification of Cuntz-Krieger algebras andgraph algebras.

16:00 – 16:25 Break

16:25 – 16:55 J. R. Carrion

On invariants of almost-flat bundles associated with almost-homomorphisms of groups.

17:00 – 17:30 X. Li

Semigroup C∗-algebras and amenability of semigroups.

17:35 – 18:05 A. Kuznetsova

Characterization of nuclearity of a C∗-algebra in terms ofa compact subgroup of the group of its automorphisms.

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Wednesday, June 8

09:30 – 10:25 N. P. Brown

Analogies in the structure of C∗- and W ∗-algebras.


11:00 – 11:55 H. Lin

Locally AH-algebras and the tracial rank.

12:00 – 12:30 H. Matui

Group actions on simple stably finite C∗-algebra.

12:35 – 13:05 C. Eckhardt

Free products of nuclear C∗-algebras.

Lunch

15:00 SOCIAL ACTIVITY AND DINNER

Thursday, June 9

09:30 – 10:25 N. C. Phillips

Towards the classification of outer actions of finite groups onpurely infinite algebras.


11:00 – 11:55 W. Winter

Dynamics and dimension.

12:00 – 12:30 C. Pasnicu

The Cuntz semigroup, a Riesz type interpolation property, com-parison and the ideal property.

12:35 – 13:05 Y. Sato

Gauge actions on the Jiang-Su algebra with the Rohlinproperty.

Lunch

15:00 – 15:55 A. S. Toms

Classification and C∗-algebras.

16:00 – 16:25 Break

16:25 – 16:55 C. Cerny

On Fowler’s Toeplitz- and Cuntz-Pimsner algebras over Nk

product systems.

17:00 – 17:30 M. Takeori

Application of E-theory to the classification of C∗-algebras.

17:35 – 18:05 T. Hines

The radius of comparison for crossed products and the meantopological dimension.

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Friday, June 10

09:30 – 10:25 G. A. Elliott

An invariant for non-simple C∗-algebras of stable rank one.


11:00 – 11:55 L. Robert

The Cuntz semigroup: axioms and variations.

12:00 – 12:30 D. Kerr

Sofic entropy and noncommutative dynamics.

12:35 – 13:05 H. Lee

A lifting problem through K-theory and deformation.

Lunch

15:00 – 15:55 I. Hirshberg

The higher dimensional Rokhlin propert.

16:00 – 16:25 Break

16:25 – 16:55 E. Ortega

Graph C∗-algebras and crossed product by endomorphisms.

17:00 – 17:30 D. Sherman

Unified representation theorems for Hilbert space operators.

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3. Abstracts of Main Talks

Nathanial P. BrownAnalogies in the structure of C∗- and W ∗-algebras.

Abstract: Recent work of Winter and collaborators has revealed tantalizinganalogies between the structure of factors and simple C∗-algebras. I’ll try toexplain these analogies and how they relate to some important open C∗-questions.

Contact address: [email protected]

Marius DadarlatGroup quasi-representations and index theory .

Abstract: For a countable discrete group G we discuss the existence of discreteasymptotic homomorphisms (πk : C∗(G) → Mn(k))k∈N which act nontrivially onthe rational K-theory group of C∗(G). We will present certain generalizations ofthe Exel-Loring formula for two almost commuting unitaries (which correspondsin our framework to the group G = Z2), 1

2πiTr(log[v, u]) = Tr(Bott(u, v)), u, v ∈

U(n), ‖uv − vu‖ < ε and explain how these formulas relate to index theory.


George A. ElliottAn invariant for non-simple C∗-algebras of stable rank one.

Abstract: An invariant based on the Cuntz semigroup and including the al-gebraic K1-group information needed only globally in the simple case for theclassification of inductive limits of circle algebras can be formulated in the gen-eral (non-simple) stable rank one case. (This is joint work with Alin Ciuperca,Leonel Robert, and Luis Santiago.) Certain difficulties still remain in the way ofclassifying arbitrary inductive limits of circle algebras using this invariant.


Ilan HirshbergThe higher dimensional Rokhlin propert .

Abstract: The Rokhlin property for an automorphism of a C∗-algebra has beenused extensively for studying C∗-dynamical systems and their crossed products.However, the Rokhlin property requires the existence of many projections, andthus does not occur in many natural examples.

I will survey joint work in progress with Winter and Zacharias concerning ahigher dimensional version of the Rokhlin property. This generalized Rokhlin

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property is very common, and crossed products by actions satisfying this prop-erty have good permanence properties with respect to nuclear dimension andZ-absorption.


Takeshi KatsuraTowards the classification of Cuntz-Krieger algebras and graphalgebras.

Abstract: I would like to survey the recent progress on the classification ofCuntz-Krieger algebras and graph algebras using K-theory by Soren Eilers andhis many collaborators including myself, and then talk on a different approachby myself using semigroups.

Simple graph algebras were classified by their ordered K0-groups and K1-groups. For non-simple C∗-algebras, we need more complicated invariants toclassify. One candidate for the invariants is the so-called filtrated K-theory (withorder). Almost all known results so far on the classification of graph algebrasuse this invariant. After looking what are known and what are not known alongthese lines, I would like to introduce another invariant involving semigroups, anddiscuss its relation to filtrated K-theory.


Eberhard KirchbergLocal morphisms and C∗-extensions.

Abstract: We describe a strategy for the construction of extensions and of ideal-equivariant semi-splittings of extensions that satisfy given topological conditionson the primitive ideal spaces of the C*-algebras in question. An example isthe next theorem, where a “Dini space” is a second countable, locally compactand sober (or: tidy) T0 spaces X. (To simplify notations, all algebras B hereare supposed to be separable, stable and tensorially O2– absorbing, with theexception C =M(B)/B or C =M(B).)

Theorem. Let X a Dini space, U an open subset of X. Suppose that thereare stable, amenable and separable C*-algebras A and B and homeomorphismsµB from Prim(B) onto U and µA from Prim(A) onto X \ U . (We suppose – inaddition – that A ∼= A⊗O2.)

Then there exists a unique (up to unitary equivalence) Busby invariant β : A→Q(B) :=M(B)/B, such that Prim(E) is homeomorphic to X (in a natural way)for the corresponding extension 0→ B → E → A→ 0 , with E := π−1β(A), withnatural epimorphism π : M(B)→ Q(B).

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The homeomorphism µE from Prim(E) onto X is “natural” in the sense thatµE(Prim(B)) = U and µE|Prim(B) = µB, the isomorphism β : A → E/B andµE|Prim(E/B) induce the homeomorphism µA from Prim(A) onto X \ U .

Corollary (joint work with O.B.Ioffe):All coherent Dini spaces X are homeomorphic to primitive ideal spaces Prim(A)of amenable and separable C∗-algebras A.

(X is coherent if the intersection C1 ∩ C2 of any two compact Gδ-subsetsC1, C2 ⊂ X is again compact.)

We denote by A0 the unique separable amenable C*-algebra A0 (considered byMortenson and Rørdam) with A0⊗O2

∼= A0 and Prim(A0) = (0, 1]lsc. The latterT0 space is (0, 1] with the new lattice of open sets O((0, 1]lsc) := {(t, 1] ; t ∈ [0, 1]}.

Let A strongly purely infinite and stable, and let SE(A0, A) denote the(Rørdam-type) semigroup of unitary homotopy classes [h] of *-monomorphismsh ∈ Mon(A0, A) with Cuntz sum [h1] + [h2] := [h1 ⊕ h2] as addition. Theorder-preserving homeomorphisms f ∈ Homeo+[0, 1] operate on A0 as groupof automorphisms (e.g. by functional calculus on generic element of A0). Thus,Homeo+[0, 1] operates on SE(A0, A) from the right.

We introduce a natural isomorphism between the semigroup SE(A0, C) andthe the semigroup D1(PrimC) of the Dini functions on Prim(C) of norm = 1with f ⊕ g := max(f, g) as addition, (and with strictly monotonous functionalcalculus by h ∈ Homeo[0, 1]+ as extra structure).

The (additive and Homeo[0, 1]+-action preserving) morphisms T fromSR(A0, B) into SR(A0, C) will be called local morphisms from A into C.

The corresponding structure-preserving morphisms DT from D1(A) into D1(C)are in one-to-one correspondence to lower semi-continuous and monotone uppersemi-continuous actions Ψ: I(C) → I(A) (of Prim(C) on A), defined by theproperty that Ψ(J) is the closed ideal of A generated by all ϕ ∈ Mon(A0, A),with T [ϕ](A0) ⊂ J .

The map Ψ: I(C) → I(A) (between the lattices of closed ideals) is lowers.c. and monotone upper s.c., iff, Ψ respects limits of increasing or decreasingsequences and satisfies Ψ(I ∩ J) = Ψ(I) ∩Ψ(J) (but not necessarily Ψ(I + J) =Ψ(I) + Ψ(J)).

This 1-1-relations imply that, for exact A and strongly purely infinite C, thereexists C*-algebra morphisms hT : A⊗O2 → C compatible to a given local mor-phism T : SR(A0, A) → SR(A,C) in the sense that T [ϕ] = [hT ◦ ϕ]. It is uniqueup to unitary homotopy.

This observations allows to reduce all questions (about extensions or lifts) tothe case A = A0, and then to pure topological considerations.


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Huaxin LinLocally AH-algebras and the tracial rank .

Abstract: We give a uniqueness theorem for almost multiplicative maps fromhomogeneous algebras to a unital separable simple C∗-algebra with tracial rankat most one. From this, we provide a classification of unital separable simplelocally AH-algebras with slow dimension growth.

The proof is independent of Gong’s decomposition theorem. We also showthat every unital separable simple C∗-algebra with finite tracial rank which alsosatisfies the UCT has tracial rank at most one.


Christopher N. PhillipsTowards the classification of outer actions of finite groups on purelyinfinite algebras.

Abstract: UCT Kirchberg algebras (purely infinite simple separable nuclearC∗-algebras satisfying the Universal Coefficient Theorem) are known to be deter-mined up to isomorphism by K-theoretic invariants. More recently, a K-theoreticclassification has been given for actions of finite groups on such algebras satisfyingthe Rokhlin property. We describe progress toward such a classification underthe much less restrictive condition that the action be pointwise outer, with thebest results being possible when the group is cyclic of prime order.


Leonel RobertThe Cuntz semigroup: axioms and variations.

Abstract: I will discuss some of the basic properties of the Cuntz semigroupof a C∗-algebra. I will first recall the axioms of the category Cu introduced byCoward, Elliott, and Ivanescu. I will then talk about some other properties ofthe Cuntz semigroup that are not listed among these axioms. Finally, I will goover a number of variations on the Cuntz semigroup construction that lead tonew and interesting invariants.


Mikael RørdamDivisibility properties for C∗-algebras.

Abstract:


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Luis SantiagoA stably projectionless C∗-algebra .

Abstract: In this talk I will introduce a simple C∗-algebra that is stably pro-jectionless and selfabsorbing. Then I will give some indication on why this C∗-algebra may be relevant to the classification of a certain class of stably projec-tionless C∗-algebras.


Mark TomfordeClassifying nonsimple graph C∗-algebras up to stable isomorphism .

Abstract: We discuss how K-theory can be used to provide complete stableisomorphism invariants for certain classes of nonsimple graph C∗-algebras. More-over, we will show how these invariants can be calculated from data determinedby the graph, and describe the range of the invariants.


Andrew S. TomsClassification and C∗-algebras.

Abstract: What does it mean to classify a category of objects? This talk willexplore the interplay between two approaches through the lens of C∗-algebratheory, namely, complete invariants and Borel complexity. We will discuss theidea of Borel reducibility, machinery from the world of descriptive set theorymeant to quantify how difficult it is to assign invariants to isomorphism classes ofa category in a computable way. We’ll also see examples of interaction betweenthese two approaches in which a classification by invariants leads to new resultson the Borel complexity of nuclear separable C∗-algebras. If time permits, we’lldiscuss the Borel computability of some C∗-algebra invariants such as the Cuntzsemigroup and the Elliott invariant.


Wilhelm WinterDynamics and dimension .

Abstract: Noncommutative topological dimension has played a key role in recentadvances in the structure and classification theory of nuclear C∗-algebras. Weintroduce dynamical versions of topological dimension which can be used to carryover some of these ideas from crossed product C∗-algebras to the underlyingdynamical systems.


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4. Abstracts of Contributed Talks

Dawn ArcheyLiving without projections.

Abstract: Many C∗-algebras have few projections, but many tricks for workingwith C∗-algebras use projections. For example, decomposing the identity intoorthogonal projections. In this talk I will discuss some techniques for workingwith C∗-algebras that have few projections. If there is time I will show an argu-ment for proving a certain type of C∗-algebra has stable rank one by performinga decomposition of the identity into (non-orthogonal) positive elements. I expectto also touch on the Cuntz semigroup and strict comparison of positive elements.


Etienne BlanchardAmalgamated products of C∗-bundles.

Abstract: We describe which classical amalgamated products of continuousC∗-bundles are continuous C∗-bundles and we analyse the involved extensionproblems for continuous C∗-bundles.


Joan BosaThe Cuntz semigroup of some C(X)-algebras.

Abstract: In this talk we show that, if A is a C(X)-algebra with stable rankone, no K1-obstructions and dimX ≤ 1, then the Cuntz semigroup of A can berecovered as the sheaf of continuous sections of an etale bundle.


Jose R. CarrionOn invariants of almost-flat bundles associated withalmost-homomorphisms of groups.

Abstract: To an almost-multiplicative map from a group to the unitary groupof a unital C∗-algebra A (which we assume has a trace) we associate a HilbertA-module bundle that is “almost flat” and study a related invariant.

This is joint work with Marius Dadarlat.


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Christina CernyOn Fowler’s Toeplitz- and Cuntz-Pimsner algebras over Nk productsystems.

Abstract: We will start by recalling Fowler’s definition of a discrete product sys-tem of Hilbert C∗-correspondences, which is a family of Hilbert correspondencesover a discrete monoid satisfying certain compatibility conditions, and how to as-sociate C∗-algebras to it that generalise Pimsner’s Toeplitz- and Cuntz-Pimsneralgebras of a single correspondence. When restricting to product systems overNk, important theorems from Pimsner’s paper, namely that the Toeplitz alge-bra of a single correspondence can be represented by creation operators on theFock module faithfully and the existence of six-term cyclic exact sequences inKK-theory for a slightly modified Toeplitz extension of the Cuntz-Pimsner al-gebra generalise to the Toeplitz- and Cuntz-Pimsner algebras over these productsystems. We conclude by exploring which new algebras occur in this case.


Caleb EckhardtFree products of nuclear C∗-algebras.

Abstract: In ’93 Bozejko and Picardello showed that the free product ofamenable groups is weakly amenable. We’d like to have a version of this re-sult for nuclear C∗-algebras, namely: Does the reduced free product of nuclearC∗-algebras have the completely contractive approximation property? Ricard andXu showed that the answer is yes if the states involved are “CP -approximable”.We’ll show that not all states on nuclear C∗-algebras are CP -approximable, butwith an additional faithfullness condition one can guarantee CP -approximabilityof states on homogeneous C∗-algebras.


Taylor HinesThe radius of comparison for crossed products and the meantopological dimension .

Abstract: The radius of comparison of a C∗-algebra A is an invariant extend-ing the topological (covering) dimension for noncommutative spaces. In the casethat A is the crossed product of a topological dynamical system on a finite-dimensional space, several results exist which bound the radius of comparison interms of the dimension of the underlying space. Our work is an attempt to extendresults of this type to dynamical systems on infinite-dimensional spaces using themean topological dimension. This talk summarizes recent progress by Q. Lin,N.C. Phillips, A. Toms and others towards giving the radius of comparison of a

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minimal system in terms of its mean dimension. We also discuss current conjec-tures and recent results which give evidence for the conjecture that the radius ofcomparison of the crossed product algebra of a minimal system is approximatelyhalf the mean dimension.


David KerrSofic entropy and noncommutative dynamics.

Abstract: Recently Lewis Bowen showed that the classical theory of entropy formeasure-preserving dynamics can be extended far beyond the realm of amenabil-ity to the setting of sofic groups. Hanfeng Li and I subsequently developed anoperator algebra approach to sofic entropy that yields both measure and topo-logical dynamical invariants. However, while there are several notions of entropyfor actions of amenable groups on noncommutative C∗-algebras, there appear tobe serious obstructions to formulating a noncommutative version of sofic entropy,and I will discuss some of the issues involved.


Alla KuznetsovaCharacterization of nuclearity of a C∗-algebra in terms of acompact subgroup of the group of its automorphisms.

Abstract:


Hyun Ho LeeA lifting problem through K-theory and deformation .

Abstract: We consider the lifting problem from the corona algebra of C(X)⊗ B.In fact, we investigate when a projection in the corona algebra is liftable to aprojection in the multiplier algebra. We begin with the classical example corre-sponding to the case C(X) ⊗ K where K is the ideal of compact operators inB(H). Then we introduce some tools to generalize this classical case.


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Xin LiSemigroup C∗-algebras and amenability of semigroups.

Abstract: With each left cancellative semigroup, we associate reduced and fullsemigroup C∗-algebras. With the help of these new constructions, we can thencharacterize (left) amenability of (certain cancellative) semigroups in analogy tothe group case.


Hiroki MatuiGroup actions on simple stably finite C∗-algebra .

Abstract: I will discuss recent progress of the classification of (strongly) outeractions of certain discrete amenable groups on unital simple C∗-algebras.


Eduard OrtegaGraph C∗-algebras and crossed product by endomorphisms.

Abstract: We use the characterization of certain graph C∗-algebras as crossedproduct by endomorphisms to initiate the study of the gauge invariant ideals,simplicity and purely infiniteness of more general crossed products.

We also propose a condition on the endomorphism that is equivalent to condi-tion (L) for graphs.


Cornel PasnicuThe Cuntz semigroup, a Riesz type interpolation property,comparison and the ideal property .

Abstract: We define a Riesz type interpolation property for the Cuntz semi-group of a C∗-algebra and prove it is satisfied by the Cuntz semigroup of everyC∗-algebra with the ideal property. Related to this, we obtain two characteriza-tions of the ideal property in terms of the Cuntz semigroup of the C∗-algebra.Some additional characterizations are proved in the special case of the stable,purely infinite C∗-algebras, and two of them are expressed in language of theCuntz semigroup. We introduce a notion of comparison of positive elements forevery unital C∗-algebra that has (normalized) quasitraces. We prove that largeclasses of C∗-algebras (including large classes of AH algebras) with the idealproperty have this comparison property.This is joint work with Francesc Perera.


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Efren RuizClassification of singular graph algebras.

Abstract: We discuss the classification problem of non-simple graph C∗-algebrasand its algebraic analog Leavitt path algebras. We will show that if G1 and G2

are graphs with finitely many vertices with no breaking vertices and every vertexof Gi is singular, then C∗(G1) is strongly Morita equivalent to C∗(G2) if and onlyif LC(G1) is Morita equivalent to LC(G2).This is joint work with Søren Eilers, Adam Sørensen, and Mark Tomforde.


Yasuhiko SatoGauge actions on the Jiang-Su algebra with the Rohlin property .

Abstract: Let A be a unital separable simple nuclear C∗-algebra with a uniquetracial state and α is an automorphism of A. Suppose that A has the property(SI) and α has the weak Rohlin property which are defined in J. Funct. Anal.259 (2010), 453-476. We construct a gauge action σ on the Jiang-Su algebra Zwhich has the weak Rohlin property and prove that the automorphism α⊗ σ ofA⊗ Z and α of A are outer conjugate. The proof of this theorem is a variationof Rørdam’s technique which implies the absorption of strongly self absorbingC∗-algebras. Using this result we further prove that two automorphisms of A areouter conjugate if A is classifiable by the K-groups, they have the weak Rohlinproperty, and they are asymptotically unitarily equivalent.


David ShermanUnified representation theorems for Hilbert space operators.

Abstract: I’ll present a unified approach to some representation theorems forHilbert space operators, obtained by grafting a few new ideas onto fundamentalresults of Agler, Arveson, Hadwin, and Voiculescu. Most of the theorems to bediscussed were already proved for one or more of four well-studied categories –C∗-algebras, hereditary manifolds, operator algebras, and operator systems – butI will give formulations that apply simultaneously to any of these, and any othercategories built in a similar way. One interesting feature is that each “operatorcategory” is associated with an operator topology. I will explain what makes atopology compatible with a category, leading to some new and natural topologies.


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Tatiana ShulmanProjective C∗-algebras and noncommutative shape theory .

Abstract: The goal of commutative shape theory is to separate out the globalproperties of a topological space X from the possibly pathological local struc-ture of X. The idea it to write X as projective limit X = limXn of “nice”spaces, namely of absolute neighborhood retracts, and then consider only thosetopological properties of X which can be determined from the homotopy typeof the Xn’s and the connecting maps. A noncommutative analogue of this the-ory was developed by Blackadar, and as noncommutative analogues of absoluteretracts and absolute neighborhood retracts he introduced notions of projectiveand semiprojective C∗- algebras. Some statements, which are true in commuta-tive shape theory, might not hold in noncommutative setting or it might be veryhard to prove their noncommutative analogues. For example, it is relatively easyto identify spaces that are absolute retracts. Nevertheless projective C∗-algebrasare traditionally regarded as extremely rare phenomena, because even C∗- alge-bras C(X), where X is an absolute retract, need not be projective due to purelynoncommutative effects. Also it remains an open question (by B. Blackadar) ifeach separable C∗-algebra is an inductive limit of semiprojective ones. This isa noncommutative analogue of very important statement in commutative shapetheory. The purpose of my talk is to give a partial answer to this question andalso to prove that projective C∗-algebras are not rare but form a large class ofC∗-algebras. It turns out that these questions are related with some questionsin operator theory, namely with Olsen’s questions about best approximation bycompact operators. We introduce a formula (“a generalized spectral radius for-mula”) which allows us for almost all operators to answer one of Olsen’s questionsand also allows to construct new examples of semiprojective C∗-algebras.Coauthor: Terry Loring


Mitsuharu TakeoriApplication of E-theory to the classification of C∗-algebras.

Abstract: In many classification results for nuclear C∗-algebras, one of the mostremarkable results is Kirchberg’s classification of non-simple nuclear C∗-algebrasby the generalization of Kasparov KK-theory. His result shows that stable,separable, nuclear, strongly purely infinite C∗-algebras with the primitive idealspace homeomorphic to a topological space X are classified up to isomorphismby KK(X)-theory. As in the Kasparov KK-theory, however, KK(X)-theoryhas some technical restrictions. For instance, the bivariant functor KK(X) ingeneral fails to be half-exact. To resolve these technical problems, M. Dadarlatand R. Meyer recently introduced the E(X)-theory which is the generalization ofConnes and Higson’s E-theory. This theory was formulated based on the theoryof C∗-algebras over topological spaces which extends the theory of C∗-algebra

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bundles. Here a C∗-algebra over a (possibly non-Hausdorff) topological space Xis defined as a C∗-algebra that possesses a continuous map from its primitive idealspace to X, and if X is a locally compact Hausdorff space, a C∗-algebra over X isin fact equivalent to a C0(X)-algebra. Using a new technique with E(X)-theory,the author of this paper has reached to a similar classification result to Kirch-berg’s that for tight separable nuclear C∗-algebras A and B over X, A and B areE(X)-equivalent if and only if A⊗O∞⊗K ' B⊗O∞⊗K. Here a tight C∗-algebraover X is a C∗-algebra with the primitive ideal space homeomorphic to X. Thisimplies that stable Kirchberg algebras over a topological space X are classifiedup to X-equivariant isomorphism by their E(X)-theory; a Kirchberg algebra overX is a tight, separable, nuclear, strongly purely infinite C∗-algebra over X. IfX is a one-point space, a Kirchberg algebra over X is nothing but a Kirchbergalgebra, that is, a simple, separable, nuclear, purely infinite C∗-algebra.

In this short communication, I shall briefly explain technical advantages ofE(X)-theory and its application to the classification of non-simple C∗-algebras.For instance, it can prove the following result; If A and B are stable Kirchberg al-gebras over X in the generalized bootstrap class NX , they are X-equivariantly iso-morphic if and only if there is an element that induces invertible mapsK∗(A(U)) → K∗(B(U)) for all open subsets U in X. Here A(U) is the closedideal of A corresponding to each open subset U by the map Prim(A)→ X.

The other result derived from E(X)-theory is that every continuous separablenuclear C∗-algebras over X is E(X)-equivalent to a Kirchberg algebra over X.This implies that classifying Kirchberg algebras over X is equivalent to classifyingcontinuous separable nuclear C∗-algebras over X up to E(X)-equivalence. Thelatter class can be shown to form a triangulated subcategory in the categoryE(X), closed under direct sums.

In this paper, I shall also introduce a new relation between KK(X)-theoryand E(X)-theory. Specifically, for separable C∗-algebras A and B over X, thereexists an isomorphism

E(X;A,B) −→ KK(X;SA,Q(B ⊗K)),

where SA is a suspension of A.


Hannes ThielInductive limits of projective C∗-algebras.

Abstract: We show that the class of inductive limits of projective C∗-algebrasconsists precisely of the C∗-algebras with trivial shape. It follows that all con-tractible C∗-algebras are inductive limits of projective C∗-algebras. This is thenoncommutative analogue of a classical result about contractible spaces.

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We deduce from our findings that a C*-algebra is (weakly) projective if andonly if it is (weakly) semiprojective and has trivial shape. Moreover, we confirma conjecture of Loring by showing that a C∗-algebra is projective if and onlyif it is semiprojective and contractible. These results are also noncommutativeanalogues of classical results in shape theory.

If time permits, we indicate how to extend the above results to the study ofinductive limits of semiprojective C∗-algebras. We show that this class has niceclosure properties and is thus quite large. This is a step forward in confirminga conjecture of Blackadar, which predicts that every C∗-algebra is an inductivelimit of semiprojective C∗-algebras.


Stuart WhitePerturbations of some crossed product algebras.

Abstract: Kadison and Kastler equipped the collection of all operator algebrason a Hilbert space with a metric and asked whether sufficiently close algebrasare necessarily (spatially) isomorphic. This was established for injective vonNeumann algebras in the late 70’s. In this talk I’ll discuss the situation forcertain non-injective II1 factors arising from crossed products.This is joint work with: Cameron, Christensen, Sinclair, Smith and Wiggins.


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5. List of Participants

Name Institution

Ramon Antoine Universitat Autonoma de Barcelona

Pere Ara Universitat Autonoma de Barcelona

Dawn Archey Marymount Manhattan College

Selcuk Barlak University of Munster

Etienne Blanchard Institut Math. de Jussieu

Joan Bosa Universitat Autonoma de Barcelona

Nathanial P. Brown The Pennsylvania State University

Jose Carrion Purdue University

Christina Cerny University of Nottingham

Chi Weng Cheong Purdue University

Marius Dadarlat Purdue University

Liam Dickson University of Glasgow

Caleb Eckhardt Purdue University

George A. Elliott University of Toronto

Dominic Enders University of Munster

Martin Engbers Westfalische Wilhelms Universitat Munster

Amaury Freslon Universite de Paris VII

Thierry Giordano University of Ottawa

Kenneth Goodearl University of California at Santa Barbara

Taylor Hines Purdue University

Ilan Hirshberg Ben Gurion University of The Negev

Mitchell Hitchcock Texas A&M University

Siri-Malen Høynes Norwegian University of Science and Technology

Bhishan Jacelon The University of Glasgow

Mirka Johanesova University of Nottingham

Takeshi Katsura University of Copenhagen

David Kerr Texas A&M University

Sunho Kim Seoul National University

Eberhard Kirchberg Humboldt Universitat zu Berlin

Alla Kuznetsova Kazan Federal University

Hyun Ho Lee University of Ulsan

Jae Hyup Lee Seoul National University

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Name Institution

Xin Li University of Munster

Huaxin Lin University of Oregon

Jose Luis Lugo Purdue University

Hiroki Matui Chiba University

Yasushi Nagai Keio University

Ping Wong Ng University of Louisiana at Lafayette

Rui Okayasu Osaka Kyoiku University

Eduard Ortega Norwegian University of Science and Technology

Rui Palma University of Oslo

Cornel Pasnicu University of Puerto Rico

Francesc Perera Universitat Autonoma de Barcelona

Christopher N. Phillips University of Oregon

Leonel Robert University of Copenhagen

Mikael Rørdam University of Copenhagen

Efren Ruiz University of Hawaii at Hilo

Luis Santiago Universitat Autonoma de Barcelona

Yasuhiko Sato Kyoto University

David Sherman University of Virginia

Tatiana Shulman University of Copenhagen

Nicolai Stammeier Universitat Munster

Karen Strung University of Nottingham

Mitsuharu Takeori University of Goettingen

Hannes Thiel University of Copenhagen

Mark Tomforde University of Houston

Andrew Toms Purdue University

Maria GraziaViola Lakehead University

Stuart White University of Glasgow

Wilhelm Winter University of Nottingham

Joachim Zacharias University of Nottingham

Aleksey Zelenberg The Pennsylvania State University

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