Conference on Structure and Classification of C -Algebras

Transcription

1 Conference on Structure and Classification of C -Algebras Centre de Recerca Matemàtica Bellaterra June 6 to 10, 2011

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3 Organisers Ramon Antoine, Universitat Autònoma de Barcelona Pere Ara, Universitat Autònoma de Barcelona Joan Bosa, Universitat Autònoma de Barcelona Francesc Perera, Universitat Autònoma de Barcelona Scientific Committee George A. Elliott, University of Toronto Andrew S. Toms, Purdue University Nathanial P. Brown, The Pennsylvania State University Joachim Cuntz, Universität Münster 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 Universität 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 Autònoma de Barcelona Mark Tomforde, University of Houston Andrew S. Toms, Purdue University Wilhelm Winter, University of Nottingham

4 Acknowledgements: Conference on Structure and Classification of C -Algebras is supported by the AGAUR (ref. 2010ARCS ), by the Ingenio Mathematica Programme (ref. PMII-C5-0332), and the Ministerio de Ciencia e Innovación (ref. MTM E).

5 5 Contents 1. Practical Information Schedule Abstracts of Main Talks Nathanial P. Brown Analogies in the structure of C - and W -algebras. Marius Dadarlat Group quasi-representations and index theory. George A. Elliott An invariant for non-simple C -algebras of stable rank one. Ilan Hirshberg The higher dimensional Rokhlin property. Takeshi Katsura Towards the classification of Cuntz-Krieger algebras and graph algebras. Eberhard Kirchberg Local morphisms and C -extensions. Huaxin Lin Locally AH-algebras and the tracial rank. Christopher N. Phillips Towards the classification of outer actions of finite groups on purely infinite algebras. Leonel Robert The Cuntz semigroup: axioms and variations. Mikael Rørdam Divisibility properties for C -algebras. Luis Santiago A stably projectionless C -algebra. Mark Tomforde Classifying nonsimple graph C -algebras up to stable isomorphism. Andrew S. Toms Classification and C -algebras. Wilhelm Winter Dynamics and dimension. 4. Abstracts of Contributed Talks Dawn Archey Living without projections.

6 6 Etienne Blanchard Amalgamated products of C -bundles. Joan Bosa The Cuntz semigroup of some C(X)-algebras. José R. Carrion On invariants of almost-flat bundles associated with almost-homomorphisms of groups. Christina Cerny On Fowler s Toeplitz- and Cuntz-Pimsner algebras over N k product systems. Caleb Eckhardt Free products of nuclear C -algebras. Taylor Hines The radius of comparison for crossed products and the mean topological dimension. David Kerr Sofic entropy and noncommutative dynamics. Alla Kuznetsova Characterization of nuclearity of a C -algebra in terms of a compact subgroup of the group of its automorphisms. Hyun Ho Lee A lifting problem through K-theory and deformation. Xin Li Semigroup C -algebras and amenability of semigroups. Hiroki Matui Group actions on simple stably finite C -algebra. Eduard Ortega Graph C -algebras and crossed product by endomorphisms. Cornel Pasnicu The Cuntz semigroup, a Riesz type interpolation property, comparison and the ideal property. Efren Ruiz Classification of singular graph algebras. Yasuhiko Sato Gauge actions on the Jiang-Su algebra with the Rohlin property. David Sherman Unified representation theorems for Hilbert space operators. Tatiana Shulman Projective C -algebras and noncommutative shape theory. Mitsuharu Takeori Application of E-theory to the classification of C -algebras.

7 Hannes Thiel Inductive limits of projective C -algebras. Stuart White Perturbations of some crossed product algebras. 6. List of participants

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9 9 1. Practical Information Programme and list of participants: You can check the latest version of the programme as well as the list of participants 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 arrangement there ( and contact us as soon as possible if you find any inaccuracy. Lecture room: The Conference will take place in the CRM Auditori located in the Sciences Building (Edifici de Ciències), Universitat Autònoma de Barcelona in Bellaterra. ( 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 photocopying. 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 and color 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 color copies. Their opening hours are Monday to Friday from 8:30 to 14:00 and from 15:00 to 20:00. OCE is the Faculty printing and photocopying service Breaks: Coffee and cookies will be served during the morning breaks to all participants.

10 10 Picture: A group picture will be taken on Wednesday, June 8 before the coffee break. We will inform you of the place to meet. The picture will be posted on the 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 their level of satisfaction. The questionnaire is anonymous and not mandatory, but we would 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) UAB s Science Faculty reception office (inside the University premises) General emergency (police, fire- fighters, ambulances) 1800 / 1900 during office hours 2525 at other times

11 11 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.

12 12 Tuesday, June 7 09:30 10:25 M. Dadarlat 10:30 11:00 Coffee Break 11:00 11:55 M. Tomforde 12:00 12:30 E. Blanchard 12:35 13:05 E. Ruiz 15:00 15:55 T. Katsura 16:00 16:25 Break 16:25 16:55 J. R. Carrion 17:00 17:30 X. Li Group quasi-representations and index theory. Classifying nonsimple graph C -algebras up to stable isomorphism. Amalgamated products of C -bundles. Classification of singular graph algebras. Lunch Towards the classification of Cuntz-Krieger algebras and graph algebras. On invariants of almost-flat bundles associated with almosthomomorphisms of groups. 17:35 18:05 A. Kuznetsova Semigroup C -algebras and amenability of semigroups. Characterization of nuclearity of a C -algebra in terms of a compact subgroup of the group of its automorphisms.

13 Wednesday, June 8 09:30 10:25 N. P. Brown Analogies in the structure of C - and W -algebras. 10:30 11:00 Coffee Break 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 on purely infinite algebras. 10:30 11:00 Coffee Break 11:00 11:55 W. Winter Dynamics and dimension. 12:00 12:30 C. Pasnicu The Cuntz semigroup, a Riesz type interpolation property, comparison and the ideal property. 12:35 13:05 Y. Sato Gauge actions on the Jiang-Su algebra with the Rohlin property. 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 N k 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 mean topological dimension. 13

14 14 Friday, June 10 09:30 10:25 G. A. Elliott An invariant for non-simple C -algebras of stable rank one. 10:30 11:00 Coffee Break 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.

15 15 3. Abstracts of Main Talks Nathanial P. Brown Analogies in the structure of C - and W -algebras. Abstract: Recent work of Winter and collaborators has revealed tantalizing analogies between the structure of factors and simple C -algebras. I ll try to explain these analogies and how they relate to some important open C -questions. Contact address: nbrown@math.psu.edu Marius Dadarlat Group quasi-representations and index theory. Abstract: For a countable discrete group G we discuss the existence of discrete asymptotic homomorphisms (π k : C (G) M n(k) ) k N which act nontrivially on the rational K-theory group of C (G). We will present certain generalizations of the Exel-Loring formula for two almost commuting unitaries (which corresponds in our framework to the group G = Z 2 1 ), Tr(log[v, u]) = Tr(Bott(u, v)), u, v 2πi U(n), uv vu < ε and explain how these formulas relate to index theory. Contact address: mdd@math.purdue.edu George A. Elliott An invariant for non-simple C -algebras of stable rank one. Abstract: An invariant based on the Cuntz semigroup and including the algebraic K 1 -group information needed only globally in the simple case for the classification of inductive limits of circle algebras can be formulated in the general (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 of classifying arbitrary inductive limits of circle algebras using this invariant. Contact address: elliott@math.toronto.edu Ilan Hirshberg The higher dimensional Rokhlin propert. Abstract: The Rokhlin property for an automorphism of a C -algebra has been used extensively for studying C -dynamical systems and their crossed products. However, the Rokhlin property requires the existence of many projections, and thus does not occur in many natural examples. I will survey joint work in progress with Winter and Zacharias concerning a higher dimensional version of the Rokhlin property. This generalized Rokhlin

16 16 property is very common, and crossed products by actions satisfying this property have good permanence properties with respect to nuclear dimension and Z-absorption. Contact address: Takeshi Katsura Towards the classification of Cuntz-Krieger algebras and graph algebras. Abstract: I would like to survey the recent progress on the classification of Cuntz-Krieger algebras and graph algebras using K-theory by Soren Eilers and his many collaborators including myself, and then talk on a different approach by myself using semigroups. Simple graph algebras were classified by their ordered K 0 -groups and K 1 - groups. For non-simple C -algebras, we need more complicated invariants to classify. One candidate for the invariants is the so-called filtrated K-theory (with order). Almost all known results so far on the classification of graph algebras use this invariant. After looking what are known and what are not known along these lines, I would like to introduce another invariant involving semigroups, and discuss its relation to filtrated K-theory. Contact address: katsura@math.keio.ac.jp Eberhard Kirchberg Local morphisms and C -extensions. Abstract: We describe a strategy for the construction of extensions and of idealequivariant semi-splittings of extensions that satisfy given topological conditions on the primitive ideal spaces of the C*-algebras in question. An example is the next theorem, where a Dini space is a second countable, locally compact and sober (or: tidy) T 0 spaces X. (To simplify notations, all algebras B here are supposed to be separable, stable and tensorially O 2 absorbing, with the exception C = M(B)/B or C = M(B).) Theorem. Let X a Dini space, U an open subset of X. Suppose that there are stable, amenable and separable C*-algebras A and B and homeomorphisms µ B from Prim(B) onto U and µ A from Prim(A) onto X \ U. (We suppose in addition that A = A O 2.) 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 1 β(a), with natural epimorphism π : M(B) Q(B).

17 The homeomorphism µ E from Prim(E) onto X is natural in the sense that µ E (Prim(B)) = U and µ E Prim(B) = µ B, the isomorphism β : A E/B and µ E Prim(E/B) induce the homeomorphism µ A 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 C 1 C 2 of any two compact G δ -subsets C 1, C 2 X is again compact.) We denote by A 0 the unique separable amenable C*-algebra A 0 (considered by Mortenson and Rørdam) with A 0 O 2 = A0 and Prim(A 0 ) = (0, 1] lsc. The latter T 0 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(A 0, A) denote the (Rørdam-type) semigroup of unitary homotopy classes [h] of *-monomorphisms h Mon(A 0, A) with Cuntz sum [h 1 ] + [h 2 ] := [h 1 h 2 ] as addition. The order-preserving homeomorphisms f Homeo + [0, 1] operate on A 0 as group of automorphisms (e.g. by functional calculus on generic element of A 0 ). Thus, Homeo + [0, 1] operates on SE(A 0, A) from the right. We introduce a natural isomorphism between the semigroup SE(A 0, C) and the the semigroup D 1 (PrimC) of the Dini functions on Prim(C) of norm = 1 with f g := max(f, g) as addition, (and with strictly monotonous functional calculus by h Homeo[0, 1] + as extra structure). The (additive and Homeo[0, 1] + -action preserving) morphisms T from SR(A 0, B) into SR(A 0, C) will be called local morphisms from A into C. The corresponding structure-preserving morphisms D T from D 1 (A) into D 1 (C) are in one-to-one correspondence to lower semi-continuous and monotone upper semi-continuous actions Ψ: I(C) I(A) (of Prim(C) on A), defined by the property that Ψ(J) is the closed ideal of A generated by all ϕ Mon(A 0, A), with T [ϕ](a 0 ) J. The map Ψ: I(C) I(A) (between the lattices of closed ideals) is lower s.c. and monotone upper s.c., iff, Ψ respects limits of increasing or decreasing sequences 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, there exists C*-algebra morphisms h T : A O 2 C compatible to a given local morphism T : SR(A 0, A) SR(A, C) in the sense that T [ϕ] = [h T ϕ]. It is unique up to unitary homotopy. This observations allows to reduce all questions (about extensions or lifts) to the case A = A 0, and then to pure topological considerations. Contact address: kirchbrg@mathematik.hu-berlin.de 17

18 18 Huaxin Lin Locally AH-algebras and the tracial rank. Abstract: We give a uniqueness theorem for almost multiplicative maps from homogeneous algebras to a unital separable simple C -algebra with tracial rank at most one. From this, we provide a classification of unital separable simple locally AH-algebras with slow dimension growth. The proof is independent of Gong s decomposition theorem. We also show that every unital separable simple C -algebra with finite tracial rank which also satisfies the UCT has tracial rank at most one. Contact address: hlin@uoregon.edu Christopher N. Phillips Towards the classification of outer actions of finite groups on purely infinite algebras. Abstract: UCT Kirchberg algebras (purely infinite simple separable nuclear C -algebras satisfying the Universal Coefficient Theorem) are known to be determined up to isomorphism by K-theoretic invariants. More recently, a K-theoretic classification has been given for actions of finite groups on such algebras satisfying the Rokhlin property. We describe progress toward such a classification under the much less restrictive condition that the action be pointwise outer, with the best results being possible when the group is cyclic of prime order. Contact address: ncp@darkwing.uoregon.edu Leonel Robert The Cuntz semigroup: axioms and variations. Abstract: I will discuss some of the basic properties of the Cuntz semigroup of a C -algebra. I will first recall the axioms of the category Cu introduced by Coward, Elliott, and Ivanescu. I will then talk about some other properties of the Cuntz semigroup that are not listed among these axioms. Finally, I will go over a number of variations on the Cuntz semigroup construction that lead to new and interesting invariants. Contact address: leonel.robert@gmail.com Mikael Rørdam Divisibility properties for C -algebras. Abstract: Contact address: rordam@math.ku.dk

19 Luis Santiago A stably projectionless C -algebra. Abstract: In this talk I will introduce a simple C -algebra that is stably projectionless 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 projectionless C -algebras. Contact address: luis.santiago@uab.cat Mark Tomforde Classifying nonsimple graph C -algebras up to stable isomorphism. Abstract: We discuss how K-theory can be used to provide complete stable isomorphism invariants for certain classes of nonsimple graph C -algebras. Moreover, we will show how these invariants can be calculated from data determined by the graph, and describe the range of the invariants. Contact address: tomforde@math.uh.edu Andrew S. Toms Classification and C -algebras. Abstract: What does it mean to classify a category of objects? This talk will explore the interplay between two approaches through the lens of C -algebra theory, namely, complete invariants and Borel complexity. We will discuss the idea of Borel reducibility, machinery from the world of descriptive set theory meant to quantify how difficult it is to assign invariants to isomorphism classes of a category in a computable way. We ll also see examples of interaction between these two approaches in which a classification by invariants leads to new results on the Borel complexity of nuclear separable C -algebras. If time permits, we ll discuss the Borel computability of some C -algebra invariants such as the Cuntz semigroup and the Elliott invariant. Contact address: atoms@purdue.edu Wilhelm Winter Dynamics and dimension. Abstract: Noncommutative topological dimension has played a key role in recent advances in the structure and classification theory of nuclear C -algebras. We introduce dynamical versions of topological dimension which can be used to carry over some of these ideas from crossed product C -algebras to the underlying dynamical systems. Contact address: wilhelm.winter@nottingham.ac.uk 19

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21 21 4. Abstracts of Contributed Talks Dawn Archey Living without projections. Abstract: Many C -algebras have few projections, but many tricks for working with C -algebras use projections. For example, decomposing the identity into orthogonal projections. In this talk I will discuss some techniques for working with C -algebras that have few projections. If there is time I will show an argument for proving a certain type of C -algebra has stable rank one by performing a decomposition of the identity into (non-orthogonal) positive elements. I expect to also touch on the Cuntz semigroup and strict comparison of positive elements. Contact address: darchey@mmm.edu Etienne Blanchard Amalgamated products of C -bundles. Abstract: We describe which classical amalgamated products of continuous C -bundles are continuous C -bundles and we analyse the involved extension problems for continuous C -bundles. Contact address: blanchar@math.jussieu.fr Joan Bosa The Cuntz semigroup of some C(X)-algebras. Abstract: In this talk we show that, if A is a C(X)-algebra with stable rank one, no K 1 -obstructions and dim X 1, then the Cuntz semigroup of A can be recovered as the sheaf of continuous sections of an etale bundle. Contact address: jbosa@mat.uab.cat José R. Carrion On invariants of almost-flat bundles associated with almost-homomorphisms of groups. Abstract: To an almost-multiplicative map from a group to the unitary group of a unital C -algebra A (which we assume has a trace) we associate a Hilbert A-module bundle that is almost flat and study a related invariant. This is joint work with Marius Dadarlat. Contact address: jcarrion@math.purdue.edu

22 22 Christina Cerny On Fowler s Toeplitz- and Cuntz-Pimsner algebras over N k product systems. Abstract: We will start by recalling Fowler s definition of a discrete product system of Hilbert C -correspondences, which is a family of Hilbert correspondences over a discrete monoid satisfying certain compatibility conditions, and how to associate C -algebras to it that generalise Pimsner s Toeplitz- and Cuntz-Pimsner algebras of a single correspondence. When restricting to product systems over N k, important theorems from Pimsner s paper, namely that the Toeplitz algebra of a single correspondence can be represented by creation operators on the Fock module faithfully and the existence of six-term cyclic exact sequences in KK-theory for a slightly modified Toeplitz extension of the Cuntz-Pimsner algebra generalise to the Toeplitz- and Cuntz-Pimsner algebras over these product systems. We conclude by exploring which new algebras occur in this case. Contact address: christina.cerny@gmx.de Caleb Eckhardt Free products of nuclear C -algebras. Abstract: In 93 Bozejko and Picardello showed that the free product of amenable groups is weakly amenable. We d like to have a version of this result for nuclear C -algebras, namely: Does the reduced free product of nuclear C -algebras have the completely contractive approximation property? Ricard and Xu 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, but with an additional faithfullness condition one can guarantee CP -approximability of states on homogeneous C -algebras. Contact address: ceckhard@gmail.com Taylor Hines The radius of comparison for crossed products and the mean topological dimension. Abstract: The radius of comparison of a C -algebra A is an invariant extending the topological (covering) dimension for noncommutative spaces. In the case that A is the crossed product of a topological dynamical system on a finitedimensional space, several results exist which bound the radius of comparison in terms of the dimension of the underlying space. Our work is an attempt to extend results of this type to dynamical systems on infinite-dimensional spaces using the mean 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

23 minimal system in terms of its mean dimension. We also discuss current conjectures and recent results which give evidence for the conjecture that the radius of comparison of the crossed product algebra of a minimal system is approximately half the mean dimension. Contact address: thines@math.purdue.edu 23 David Kerr Sofic entropy and noncommutative dynamics. Abstract: Recently Lewis Bowen showed that the classical theory of entropy for measure-preserving dynamics can be extended far beyond the realm of amenability to the setting of sofic groups. Hanfeng Li and I subsequently developed an operator algebra approach to sofic entropy that yields both measure and topological dynamical invariants. However, while there are several notions of entropy for actions of amenable groups on noncommutative C -algebras, there appear to be serious obstructions to formulating a noncommutative version of sofic entropy, and I will discuss some of the issues involved. Contact address: kerr@math.tamu.edu Alla Kuznetsova Characterization of nuclearity of a C -algebra in terms of a compact subgroup of the group of its automorphisms. Abstract: Contact address: alla-kuznetsova@rambler.ru Hyun Ho Lee A 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 a projection in the multiplier algebra. We begin with the classical example corresponding to the case C(X) K where K is the ideal of compact operators in B(H). Then we introduce some tools to generalize this classical case. Contact address: hyun.lee7425@gmail.com

24 24 Xin Li Semigroup C -algebras and amenability of semigroups. Abstract: With each left cancellative semigroup, we associate reduced and full semigroup C -algebras. With the help of these new constructions, we can then characterize (left) amenability of (certain cancellative) semigroups in analogy to the group case. Contact address: xinli.math@uni-muenster.de Hiroki Matui Group actions on simple stably finite C -algebra. Abstract: I will discuss recent progress of the classification of (strongly) outer actions of certain discrete amenable groups on unital simple C -algebras. Contact address: matui@math.s.chiba-u.ac.jp Eduard Ortega Graph C -algebras and crossed product by endomorphisms. Abstract: We use the characterization of certain graph C -algebras as crossed product 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 condition (L) for graphs. Contact address: eduardor@math.ntnu.no Cornel Pasnicu The Cuntz semigroup, a Riesz type interpolation property, comparison and the ideal property. Abstract: We define a Riesz type interpolation property for the Cuntz semigroup of a C -algebra and prove it is satisfied by the Cuntz semigroup of every C -algebra with the ideal property. Related to this, we obtain two characterizations 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 the Cuntz semigroup. We introduce a notion of comparison of positive elements for every unital C -algebra that has (normalized) quasitraces. We prove that large classes of C -algebras (including large classes of AH algebras) with the ideal property have this comparison property. This is joint work with Francesc Perera. Contact address: cpasnicu@gmail.com

25 25 Efren Ruiz Classification of singular graph algebras. Abstract: We discuss the classification problem of non-simple graph C -algebras and its algebraic analog Leavitt path algebras. We will show that if G 1 and G 2 are graphs with finitely many vertices with no breaking vertices and every vertex of G i is singular, then C (G 1 ) is strongly Morita equivalent to C (G 2 ) if and only if L C (G 1 ) is Morita equivalent to L C (G 2 ). This is joint work with Søren Eilers, Adam Sørensen, and Mark Tomforde. Contact address: ruize@hawaii.edu Yasuhiko Sato Gauge actions on the Jiang-Su algebra with the Rohlin property. Abstract: Let A be a unital separable simple nuclear C -algebra with a unique tracial 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), We construct a gauge action σ on the Jiang-Su algebra Z which has the weak Rohlin property and prove that the automorphism α σ of A Z and α of A are outer conjugate. The proof of this theorem is a variation of Rørdam s technique which implies the absorption of strongly self absorbing C -algebras. Using this result we further prove that two automorphisms of A are outer conjugate if A is classifiable by the K-groups, they have the weak Rohlin property, and they are asymptotically unitarily equivalent. Contact address: ysato@math.kyoto-u.ac.jp David Sherman Unified representation theorems for Hilbert space operators. Abstract: I ll present a unified approach to some representation theorems for Hilbert space operators, obtained by grafting a few new ideas onto fundamental results of Agler, Arveson, Hadwin, and Voiculescu. Most of the theorems to be discussed were already proved for one or more of four well-studied categories C -algebras, hereditary manifolds, operator algebras, and operator systems but I will give formulations that apply simultaneously to any of these, and any other categories built in a similar way. One interesting feature is that each operator category is associated with an operator topology. I will explain what makes a topology compatible with a category, leading to some new and natural topologies. Contact address: dsherman@virginia.edu

26 26 Tatiana Shulman Projective C -algebras and noncommutative shape theory. Abstract: The goal of commutative shape theory is to separate out the global properties of a topological space X from the possibly pathological local structure of X. The idea it to write X as projective limit X = lim X n of nice spaces, namely of absolute neighborhood retracts, and then consider only those topological properties of X which can be determined from the homotopy type of the X n s and the connecting maps. A noncommutative analogue of this theory was developed by Blackadar, and as noncommutative analogues of absolute retracts and absolute neighborhood retracts he introduced notions of projective and semiprojective C - algebras. Some statements, which are true in commutative shape theory, might not hold in noncommutative setting or it might be very hard to prove their noncommutative analogues. For example, it is relatively easy to identify spaces that are absolute retracts. Nevertheless projective C -algebras are traditionally regarded as extremely rare phenomena, because even C - algebras C(X), where X is an absolute retract, need not be projective due to purely noncommutative effects. Also it remains an open question (by B. Blackadar) if each separable C -algebra is an inductive limit of semiprojective ones. This is a noncommutative analogue of very important statement in commutative shape theory. The purpose of my talk is to give a partial answer to this question and also to prove that projective C -algebras are not rare but form a large class of C -algebras. It turns out that these questions are related with some questions in operator theory, namely with Olsen s questions about best approximation by compact operators. We introduce a formula ( a generalized spectral radius formula ) which allows us for almost all operators to answer one of Olsen s questions and also allows to construct new examples of semiprojective C -algebras. Coauthor: Terry Loring Contact address: shulman@math.ku.dk Mitsuharu Takeori Application of E-theory to the classification of C -algebras. Abstract: In many classification results for nuclear C -algebras, one of the most remarkable results is Kirchberg s classification of non-simple nuclear C -algebras by the generalization of Kasparov KK-theory. His result shows that stable, separable, nuclear, strongly purely infinite C -algebras with the primitive ideal space homeomorphic to a topological space X are classified up to isomorphism by KK(X)-theory. As in the Kasparov KK-theory, however, KK(X)-theory has some technical restrictions. For instance, the bivariant functor KK(X) in general fails to be half-exact. To resolve these technical problems, M. Dadarlat and R. Meyer recently introduced the E(X)-theory which is the generalization of Connes and Higson s E-theory. This theory was formulated based on the theory of C -algebras over topological spaces which extends the theory of C -algebra

27 bundles. Here a C -algebra over a (possibly non-hausdorff) topological space X is defined as a C -algebra that possesses a continuous map from its primitive ideal space to X, and if X is a locally compact Hausdorff space, a C -algebra over X is in fact equivalent to a C 0 (X)-algebra. Using a new technique with E(X)-theory, the author of this paper has reached to a similar classification result to Kirchberg s that for tight separable nuclear C -algebras A and B over X, A and B are E(X)-equivalent if and only if A O K B O K. Here a tight C -algebra over X is a C -algebra with the primitive ideal space homeomorphic to X. This implies that stable Kirchberg algebras over a topological space X are classified up to X-equivariant isomorphism by their E(X)-theory; a Kirchberg algebra over X is a tight, separable, nuclear, strongly purely infinite C -algebra over X. If X is a one-point space, a Kirchberg algebra over X is nothing but a Kirchberg algebra, that is, a simple, separable, nuclear, purely infinite C -algebra. In this short communication, I shall briefly explain technical advantages of E(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 algebras over X in the generalized bootstrap class N X, they are X-equivariantly isomorphic if and only if there is an element that induces invertible maps K (A(U)) K (B(U)) for all open subsets U in X. Here A(U) is the closed ideal 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 separable nuclear 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 classifying continuous separable nuclear C -algebras over X up to E(X)-equivalence. The latter class can be shown to form a triangulated subcategory in the category E(X), closed under direct sums. In this paper, I shall also introduce a new relation between KK(X)-theory and E(X)-theory. Specifically, for separable C -algebras A and B over X, there exists an isomorphism where SA is a suspension of A. E(X; A, B) KK(X; SA, Q(B K)), Contact address: mitsuhar@uni-math.gwdg.de 27 Hannes Thiel Inductive limits of projective C -algebras. Abstract: We show that the class of inductive limits of projective C -algebras consists precisely of the C -algebras with trivial shape. It follows that all contractible C -algebras are inductive limits of projective C -algebras. This is the noncommutative analogue of a classical result about contractible spaces.

28 28 We deduce from our findings that a C*-algebra is (weakly) projective if and only if it is (weakly) semiprojective and has trivial shape. Moreover, we confirm a conjecture of Loring by showing that a C -algebra is projective if and only if it is semiprojective and contractible. These results are also noncommutative analogues of classical results in shape theory. If time permits, we indicate how to extend the above results to the study of inductive limits of semiprojective C -algebras. We show that this class has nice closure properties and is thus quite large. This is a step forward in confirming a conjecture of Blackadar, which predicts that every C -algebra is an inductive limit of semiprojective C -algebras. Contact address: thiel@math.ku.dk Stuart White Perturbations of some crossed product algebras. Abstract: Kadison and Kastler equipped the collection of all operator algebras on a Hilbert space with a metric and asked whether sufficiently close algebras are necessarily (spatially) isomorphic. This was established for injective von Neumann algebras in the late 70 s. In this talk I ll discuss the situation for certain non-injective II 1 factors arising from crossed products. This is joint work with: Cameron, Christensen, Sinclair, Smith and Wiggins. Contact address: stuart.white@glasgow.ac.uk

29 29 5. List of Participants Name Ramon Antoine Pere Ara Dawn Archey Selcuk Barlak Etienne Blanchard Joan Bosa Nathanial P. Brown Jose Carrion Christina Cerny Chi Weng Cheong Marius Dadarlat Liam Dickson Caleb Eckhardt George A. Elliott Dominic Enders Martin Engbers Amaury Freslon Thierry Giordano Kenneth Goodearl Taylor Hines Ilan Hirshberg Mitchell Hitchcock Siri-Malén Høynes Bhishan Jacelon Mirka Johanesova Takeshi Katsura David Kerr Sunho Kim Eberhard Kirchberg Alla Kuznetsova Hyun Ho Lee Jae Hyup Lee Institution Universitat Autònoma de Barcelona Universitat Autònoma de Barcelona Marymount Manhattan College University of Münster Institut Math. de Jussieu Universitat Autònoma de Barcelona The Pennsylvania State University Purdue University University of Nottingham Purdue University Purdue University University of Glasgow Purdue University University of Toronto University of Münster Westfälische Wilhelms Universität Münster Université de Paris VII University of Ottawa University of California at Santa Barbara Purdue University Ben Gurion University of The Negev Texas A&M University Norwegian University of Science and Technology The University of Glasgow University of Nottingham University of Copenhagen Texas A&M University Seoul National University Humboldt Universität zu Berlin Kazan Federal University University of Ulsan Seoul National University

30 30 Name Xin Li Huaxin Lin José Luis Lugo Hiroki Matui Yasushi Nagai Ping Wong Ng Rui Okayasu Eduard Ortega Rui Palma Cornel Pasnicu Francesc Perera Christopher N. Phillips Leonel Robert Mikael Rørdam Efren Ruiz Luis Santiago Yasuhiko Sato David Sherman Tatiana Shulman Nicolai Stammeier Karen Strung Mitsuharu Takeori Hannes Thiel Mark Tomforde Andrew Toms Maria GraziaViola Stuart White Wilhelm Winter Joachim Zacharias Aleksey Zelenberg Institution University of Münster University of Oregon Purdue University Chiba University Keio University University of Louisiana at Lafayette Osaka Kyoiku University Norwegian University of Science and Technology University of Oslo University of Puerto Rico Universitat Autònoma de Barcelona University of Oregon University of Copenhagen University of Copenhagen University of Hawaii at Hilo Universitat Autònoma de Barcelona Kyoto University University of Virginia University of Copenhagen Universität Münster University of Nottingham University of Goettingen University of Copenhagen University of Houston Purdue University Lakehead University University of Glasgow University of Nottingham University of Nottingham The Pennsylvania State University