WORKSHOP TALKS. Hong, Guixiang. Kwaśniewski, Bartosz. Li, Zhiqiang. Pardo, Enrique. Petzka, Henning. Savchuk, Yurik. Thiel, Hannes

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1 WORKSHOP TALKS Aula Naranja (ICMat, Campus de Cantoblanco) Abstracts (in alphabetical order): Albandik, Suliman Antoine, Ramon Argerami, Martín Aukhadiev, Marat Bédos, Erik Bohata, Martin Bosa, Joan Fernandes, Cláudio Gardella, Eusebio Gillaspy, Elizabeth Hamhalter, Jan Hong, Guixiang Kuznetsova, Alla Kwaśniewski, Bartosz Li, Zhiqiang Palazuelos, Carlos Pardo, Enrique Petzka, Henning Savchuk, Yurik Thiel, Hannes CROSSED PRODUCTS AND CUNTZ-PIMSNER ALGEBRAS AS COLIMITS Suliman Albandik, Georg-August-Universität Göttingen We study colimits of diagrams in the 2-category of correspondences. Colimits for group actions or crossed module actions are the usual crossed products. The CuntzâPimsner algebra is obtained as a colimit of a diagram consisting of a single proper correspondence. We prove that any diagram of proper correspondences has a colimit. (This is a joint work with Ralf Meyer). REGULARITY PROPERTIES OF THE CUNTZ SEMIGROUP Ramon Antoine, Universitat Autònoma de Barcelona / Universidad Autónoma de Barcelona Let D be a strongly self-absorbing C -algebra. We observe that this endows its Cuntz semigroup D = Cu(D) with a semiring structure and also Cu(A) with a D-module structure for any D-stable C -algebra A. We explore this corresponding notions in the category Cu, which leads us to define and study tensor products in this category. We show how some analytical results can be interpreted in this context, with emphasis on the known cases of strongly self absorbing C -algebras and specially for the Jiang-Su algebra Z. This is a joint work with Francesc Perera (Barcelona) and Hannes Thiel (Münster).

2 CLASSIFICATION OF SINGLY-GENERATED OPERATOR SYSTEMS Martín Argerami, University of Regina As a vector space, a singly-generated operator system is necessarily 3-dimensional, and thus all singly-generated operator systems are isomorphic in the category of vector spaces. But in the right category, that is operator systems with unital completely positive maps as morphisms, the situation is very different, even when restricted to operators in a finite-dimensional setting. In this talk we will discuss recent advances in determining the C -envelope of such operator systems in many cases of interest. SEMIGROUP C -ALGEBRA AS A WAY OF QUANTUM DEFORMATION Marat Aukhadiev, Kazan State Power Engineering University Taking a compact abelian group G, we can deform algebra of continuous functions C(G) by means of Pontryagin dual group Γ. In fact, this new C -algebra is generated by an inverse semigroup, and is called a reduced semigroup C -algebra. We show that this C - algebra can be regarded as an algebra of functions on a compact quantum semigroup QS. This quantum semigroup QS is endowed with a natural nonergodic action of G, given by a C -dynamical system. Such compact quantum semigroups form a tensor category, which is embedded in a special category of abelian semigroups.. ON EQUIVARIANT REPRESENTATIONS OF C -DYNAMICAL SYSTEMS Erik Bédos, Matematisk Institutt, Universitetet i Oslo Let Σ = (A, G, α, σ) denote a unital discrete twisted C -dynamical system. In our recent work with Roberto Conti (Rome), it has emerged that the so-called equivariant representations of Σ on Hilbert A-modules play an interesting role, complementing the one played by covariant representations. We will discuss some aspects of this notion and illustrate its usefulness in the study of the C -crossed products associated with Σ. STAR ORDER ON FUNCTION ALGEBRAS AND JBW ALGEBRAS Martin Bohata, Czech Technical University in Prague The star order is a partial order on certain -algebras defined as follows: x y if x x = x y and xx = yx. In the first part of this talk, we look at the infimum problem for the star order on an abelian -algebra C(X) of all continuous complex-valued functions on a Hausdorff topological space X. We show that the infimum of two elements of C(X) exists whenever X is locally connected or extremely disconnected topological space. Moreover, we present an example showing that the infimum need not exist in the case of a Hausdorff topological space which is neither locally connected nor extremely disconnected. In the second part, we generalize the star order to JBW algebras which are nonassociative counterpart of well known von Neumann algebras. The main objective of the presentation is to investigate star order isomorphisms between JBW algebras. We show that certain continuous (not necessarily linear) star order isomorphisms between JBW algebras without Type I 2 direct summand are more or less given by a composition of functional calculus with a Jordan isomorphism. This can be considered as a generalization of our result (joint with J. Hamhalter) for continuous star order isomorphisms between self-adjoint parts of von Neumann algebras. The proof of the result is based on Gleason s type theorem for JBW algebras proved by Bunce and Wright.

3 LOCAL TRIVIALITY FOR CONTINUOUS FIELDS OF C -ALGEBRAS Joan Bosa Puigredon, Universitat Autònoma de Barcelona / Universidad Autónoma de Barcelona Let X be a finite dimensional compact metrizable space. Let A be a separable continuous field C -algebra over X with all fibers isomorphic to the same stable Kirchberg algebra D. We show that if D has finitely generated K-theory and it satisfies the Universal Coefficient Theorem in KK-theory, then there exists a dense open subset U of X such that the ideal A(U) is locally trivial. The assumptions that the space X is finite dimensional and that the K-theory of the fiber is finitely generated are necessary. A C -ALGEBRA OF SINGULAR INTEGRAL OPERATORS WITH SHIFTS ADMITTING DISTINCT FIXED POINTS Cláudio Fernandes, Universidade Nova de Lisboa Representations for a nonlocal C -algebra B of singular integral operators with piecewise slowly oscillating coefficients extended by a group of unitary shift operators are constructed. The group of unitary shift operators U g in the C -algebra B is associated with a discrete amenable group G of orientation-preserving piecewise smooth homeomorphisms g : T T that acts topologically freely on T but admits distinct fixed points for different shifts. Due to this fact, to study the Fredholmness in the C -algebra B, in addition to localtrajectory method we need to use a suitable version the lifting theorem for C -algebras. As a result, a Fredholm symbol calculus for the C -algebra B and a Fredholm criterion for operators B B is established. This talk is based on a joint work with M. Amélia Bastos and Yuri Karlovich. ROKHLIN PROPERTY FOR ACTIONS OF THE CIRCLE ON C -ALGEBRAS Eusebio Gardella, University of Oregon The Rokhlin property for actions of compact groups on C -algebras, as defined by Hirshberg and Winter, is a (strong) noncommutative analog of freeness for actions on compact spaces. The definition for finite groups was introduced earlier and has been extensively studied. On the other hand, very little is known about actions of non finite, compact groups with the Rokhlin property. This talk will focus on the Rokhlin property for actions of the circle. We will introduce the definition, exhibit some examples, and present a number of results, mainly on the classification of these actions. Somewhat surprisingly, the machinery developed by Ruy Exel for partial actions turns out to be crucial in the classification of Rokhlin circle actions on Kirchberg algebras. K-THEORY AND HOMOTOPIES OF TWISTED GROUP(OID) C -ALGEBRAS Elizabeth Gillaspy, Dartmouth College If G is a locally compact Hausdorff group that satisfies the Baum-Connes conjecture with coefficients, and {ω t } t [0,1] is a family of 2-cocycles on G that varies continuously in t, then Echterhoff, Lück, Phillips, and Walter proved in 2010 that this homotopy of cocycles induces an isomorphism of the K-theory groups K (C r (G, ω s )) = K (C r (G, ω t )) of the reduced twisted group C -algebras. In this talk we will present a generalization of this result to the case when G = H X is a transformation group acting on a compact space X, and we will discuss the apparent obstacles to extending this result further into the world of groupoids.

4 QUASI LINEAR MAPS AND THE STRUCTURE OF ABELIAN SUBALGEBRAS Jan Hamhalter, Czech Technical University in Prague Let A be a C -algebra and X a normed linear space. A map T : A X is called quasilinear if it is linear when restricted to any abelian C -subalgebra of A. Even if there is a deep Generalized Mackey Gleason Theorem saying that quasi-linear maps on von Neumann algebras are usually linear, the question of linearity of quasi linear maps on C -algebras is largely open. We show some sufficient conditions for linearity of quasi linear maps, including the fact that quasi linear map that is well behaved with respect to special Jordan product is already linear in some important cases. Further, we shall investigate the ordered structure Abel(A) of all abelian C -subalgebras of A ordered by set inclusion. This structure plays an important role in recent development in foundations of quantum theory. We show that order isomorphisms between such structures are naturally induced by quasi linear Jordan isomorphisms. In case of AW -algebras we generalize celebrated Dye Theorem on orthoisomorphisms of projection lattices and show its connection to newly established invariants for AW -algebras based on the structure Abel(A) and action of symmetry group on it. POINTWISE CONVERGENCE OF TRUNCATED SINGULAR INTEGRAL OPERATORS WITH MATRIX-VALUED CONVOLUTION KERNELS Guixiang Hong, Instituto de Ciencias Matemáticas In this talk, we are concerned with pointwise convergence of truncated singular integrals with matrix-valued convolution kernels. For this purpose, we first find a sufficient condition on the kernel for the singular integral operator to be L (B(l 2 ); L c 2 (Rn ))-bounded; then using ideas from the theory of noncommutative martingale inequalities/operator space, we establish some maximal inequalities associated to truncated singular integral operators; finally applying the method for the individual ergodic theorem, we obtain the pointwise convergence of truncated singular integrals with the family given by ε > 0. We also consider the truncated singular integrals with the family given by E = i,j ε ij e ij which seems completely new since there exists no analogue in the classical setting. For this kind of truncated singular integrals, we prove the pointwise convergence result for the truncations of particular shape. This a joint work with Javier Parcet (ICMAT). TORUS ACTION ON C -ALGEBRA GENERATED BY MAPPING Alla Yurjevna Kuznetsova, Kazan Federal University Algebra C ϕ(x) is generated by a mapping ϕ : X X, where X is a countable set. This mapping gives rise to a family of partial isometries (finite or countable) acting on l 2 (X). These last are the generators of C ϕ(x) satisfying the relations defined by a preassigned mapping on X. So C ϕ(x) can be regarded as a C -algebra generated by partial isometries whose range and initial projections satisfy some conditions. I will equip C ϕ(x) with a semi-saturated circle action as well as a torus action which helps to investigate the structure of C ϕ(x). I will discuss various results on fixed point subalgebras. In particular, I will formulate the condition when these coincide and are commutative.

5 CROSSED PRODUCTS BY ENDOMORPHISMS AND RELATED PROBLEMS Bartosz Kosma Kwaśniewski, University of Białystok Starting from an arbitrary -endomorphism α of a unital C -algebra A, by elaborating special matrix calculus and giving explicit norm formulas, we will construct a crossed product. We will study the relations that define this crossed product as a universal C -algebra and reveal its relationship with other relevant constructions (e.g. Stacey s crossed product or Katsura s construction of a canonical C -correspondence). Also we plan to discuss problems of dilation and topological freeness in this context. (Based on joint work with A. V. Lebedev). A K-THEORETIC CLASSIFICATION OF INDUCTIVE LIMIT Z/pZ ACTIONS ON AF ALGEBRAS Zhiqiang Li, University of Toronto A K-theoretic classification is given of the C -dynamical systems lím (A n, α n, G), where A n is finite dimensional and G is any cyclic group of prime order (namely, G = Z/pZ for some prime number p). Such actions contain natural examples (constructed by N. C. Phillips) of finite group actions on UHF algebras which don t have the tracial Rokhlin property. OPERATOR ALGEBRAS AND QUANTUM CHANNELS Carlos Palazuelos, Instituto de Ciencias Matemáticas In this talk we will explain some connections between operator algebras and the study of quantum channel capacities. In particular, we will show how certain problems from the theory of operator algebras naturally appear in the context of quantum channels. The study of embedding between non-commutative L p -spaces, the study of completely p-summing maps and the multiplicativity properties of certain norms are examples of the interconnections between both fields. Finally, we will discuss how to use these connections in order to compute some capacities of channels. SYMBOLIC DYNAMICS IN THE CLASSIFICATION OF KIRCHBERG ALGEBRAS Enrique Pardo, Universidad de Cádiz In this talk, motivated by the search of an algebraic analog of the Kirchberg-Phillips Classification Theorem, we will study a combinatorial model of Kirchberg algebras, due to Takeshi Katsura, from a point of view connecting combinatorics with symbolic dynamics. We will show how this approach applies to characterize some properties of these algebras in a diferent way to that used by Katsura, by using dynamical properties of actions of inverse semigroups on topological spaces and its associated groupoids. The central part of this talk is contained in a joint work with Ruy Exel (Departamento de Matemática, Universidade Federal de Santa Catarina, Florianópolis, Brazil).

6 GEOMETRIC STRUCTURE OF DIMENSION FUNCTIONS OF CERTAIN CONTINUOUS FIELDS Henning Petzka, Universitat Autònoma de Barcelona / Universidad Autónoma de Barcelona We study structural properties of the Cuntz semigroup and its functionals for continuous fields of C -algebras over finite dimensional spaces. In a variety of cases, this leads to an answer to a conjecture posed by Blackadar and Handelman. Enroute to our results, we determine when the stable rank of continuous fields of C -algebras over one dimensional spaces is one. Joint work with Ramon Antoine, Joan Bosa and Francesc Perera. EXAMPLES OF C -CROSSED PRODUCTS BY PARTIAL ACTIONS Yuriy Savchuk, Universität Erlangen-Nürnberg In my talk I will discuss -algebras coming from the theory of quantum groups. I will show that many of them generate C -algebras (in the sense of S.L.Woronowicz) which are canonically isomorphic to C -crossed prod- ucts by partial group actions on commutative C -algebras. The following examples will be discussed in detail: the q-oscillator algebra, the Podles sphere and the q-deformed enveloping algebra of su(2). EQUIVARIANT SEMIPROJECTIVITY Hannes Thiel, Universität of Münster Semiprojectivity is the basic notion of non-commutative shape theory. It is defined in terms of a certain lifting property, and it has been used successfully in the classification program. Equivariant semiprojectivity is a version of semiprojectivity that takes group actions into account. It was introduced by Phillips with the goal of classifying certain actions on Kirchberg algebras. We will discuss how equivariant semiprojectivity of a C -algebra is related to ordinary semiprojectivity of the same algebra and its fixed point and crossed product algebra. This talk is about joint work with N. Christopher Phillips and Adam Srensen.