NONCOMMUTATIVE GEOMETRY SEMINAR

Mathematical Institute of the Polish Academy of Sciences

Ul. Śniadeckich 8, room 321, Mondays

1999/2001 2002/2003 2003/2004 2004/2005 2005/2006 2006/2007 2007/2008 2008/2009 2009/2010 2010/2011 2011/2012 2012/2013 2013/2014

3 November 2014, 10:15-12:00 (inauguration lecture)

ODD-DIMENSIONAL MULTI-PULLBACK QUANTUM SPHERES

We construct a noncommutative deformation of odd-dimensional spheres that preserves the natural partition of the (2n+1)-dimensional sphere into (n+1)-many solid tori. This generalizes the case n = 1 referred to as the Heegaard quantum sphere. Our odd-dimensional quantum sphere C*-algebras are given as multi-pullback C*-algebras. We prove that they are isomorphic to the universal C*-algebras generated by certain isometries, and use this result to compute the K-groups of our odd-dimensional quantum spheres. Furthermore, we prove that the fixed-point subalgebras under the diagonal U(1)-action on our quantum sphere C*-algebras yield the independently defined C*-algebras of the quantum complex projective spaces constructed from the Toeplitz cubes. Then, by constructing a strong connection, we show that this U(1)-action is free. This leads to the main result stating that the noncommutative line bundles over the quantum complex projective spaces that are associated to this action via non-trivial representations of U(1) are not stably trivial. (Based on joint work with D. Pask, A. Sims and B. Zieliński.)

PIOTR M. HAJAC (IMPAN)

3 November 2014, 14:15-16:00

INVARIANT MEANS ON COMPACTIFICATIONS OF LOCALLY COMPACT QUANTUM GROUPS

Given a locally compact group G with G^w and Bohr(G) being the weakly almost periodic and Bohr compactification of G respectively, there exists a unique invariant mean on C(G^w) which restricts to C(Bohr(G)) and gives the Haar state on C(Bohr(G)). This mean is crucially used to prove that the space of almost periodic functions is complemented inside the space of weakly almost periodic functions. Recently, these compactifications were generalized within the framework of quantum groups. In this talk, our aim is to show that there is a unique way to construct invariant means on these quantum-group compactifications. We will argue that our invariant means can be used much as in the classical case. Furthermore, we will also show a more general result that unital C*-bialgebras with certain regularities always have invariant means. (Based on joint work with Matthew Daws.)

BISWARUP DAS (IMPAN)

10 November 2014, 10:15-12:00

PIECEWISE PRINCIPAL COACTIONS OF CO-COMMUTATIVE HOPF ALGEBRAS

Principal comodule algebras can be thought of as objects representing compact principal bundles in noncommutative geometry. The principality of a coaction is characterized by the existence of a strong connection. For instance, it is known how to construct a strong connection on a multi-pullback comodule algebra from strong connections on its multi-pullback components. However, this general strong-connection formula can easily become unwieldy. The goal of this talk is to present a much easier to use strong-connection formula for co-commutative Hopf algebras coacting on multi-pullback algebras. We instantiate our construction in the case of the antipodal action on the triple-pullback quantum 2-sphere.

BARTOSZ ZIELIŃSKI (Uniwersytet Łódzki)

10 November 2014, 14:15-16:00

ONE-TO-ONE CORRESPONDENCE BETWEEN GENERATING FUNCTIONALS AND COCYCLES ON QUANTUM GROUPS IN THE PRESENCE OF SYMMETRY

Generating functionals are important from the view point of quantum probability (they are generators of Lévy processes on quantum groups) as well as geometric group theory (the Haagerup property of a discrete group can be characterised via the existence of a special-type generating functional). It was shown by M. Schürmann that to a given generating functional one can canonically associate a cocycle. This brings up the question whether this association is bijective. Although there are certain quantum groups for which such bijectivity holds, one can also give counter-examples. The aim of this talk is to show that the bijectivity question has the affirmative answer if we further assume a certain symmetry in the picture. We will also show how our set-up accommodates some known examples. (Based on joint work with Adam Skalski, Uwe Franz and Anna W. Kula.)

BISWARUP DAS (IMPAN)

17 November 2014, 10:15-12:00

CLASSIFICATION OF C*-ALGEBRAS OF MINIMAL DYNAMICAL SYSTEMS UP TO JIANG-SU STABILITY

For any minimal dynamical system (X,α), where X is an infinite compact metric space, one has an associated separable simple unital nuclear C*-algebra, the crossed product of C(X) with the integers via the action of α. George Elliott conjectured that separable simple unital nuclear C*-algebras should be classified, up to *-isomorphism, by K-theory paired with traces. However, since the construction of the Jiang-Su algebra Z, it is clear that this will not be possible in full generality, as the invariant does not detect tensoring by Z. I will show that, under some restrictions on ergodic measures of the dynamical systems, two such crossed products are *-isomorphic after tensoring with the Jiang-Su algebra Z if and only if there is an isomorphism on K-theory and traces. This is based on joint work with Wilhelm Winter.

KAREN R. STRUNG (IMPAN)

17 November 2014, 14:15-16:00

THE MOYAL SPHERE AND CONE

The two-dimensional plane admits a metric with a constant scalar curvature. In this talk, I will discuss whether this is possible for the Moyal deformation of the plane. I will also present the Moyal cone and sphere, and show that in the deformed case the conical singularity disappears.

ANDRZEJ SITARZ (Uniwersytet Jagielloński / IMPAN)

1 December 2014, 10:15-12:00

MONOIDAL CATEGORY OF C*-ALGEBRAS (PART 1)

We consider the category of C*-algebras equipped with actions of a locally compact quantum group. This category admits a monoidal structure satisfying certain natural conditions if and only if the quantum group is quasitriangular. The monoidal structures are in bijective correspondence with unitary R-matrices.

STANISŁAW L. WORONOWICZ (IMPAN / Uniwersytet Warszawski)

1 December 2014, 14:15-16:00

APPROXIMATELY SUBHOMOGENEOUS C*-ALGEBRAS AND APPLICATIONS TO C*-ALGEBRAS OF MINIMAL DYNAMICAL SYSTEMS

It has been shown that direct limits of subhomogeneous C*-algebras are general enough to exhaust all of the Elliott classification invariant for separable simple unital nuclear C*-algebras. Nevertheless, these algebras remain unclassified in full generality. I will discuss a classification result, obtained using Huaxin Lin's concept of tracial approximation, for a subclass of such C*-algebras which, up to restrictions on the tracial state space and its pairing with K₀, have been shown to exhaust the invariant. I will also discuss the applications of this result to the classification problem for the C*-algebras of minimal dynamical systems. This is based on joint work with Wilhelm Winter.

KAREN R. STRUNG (IMPAN)

8 December 2014, 10:15-12:00

MONOIDAL CATEGORY OF C*-ALGEBRAS (PART 2)

We consider the category of C*-algebras equipped with actions of a locally compact quantum group. This category admits a monoidal structure satisfying certain natural conditions if and only if the quantum group is quasitriangular. The monoidal structures are in bijective correspondence with unitary R-matrices.

STANISŁAW L. WORONOWICZ (IMPAN / Uniwersytet Warszawski)

8 December 2014, 14:15-16:00

COMPACT SEMITOPOLOGICAL QUANTUM SEMIGROUPS

After explaining the notion of a compact semitopological quantum semigroup introduced by M. Daws, we will derive sufficient conditions for a compact semitopological quantum semigroup to be a compact quantum group. (Based on joint work with B. Das.)

COLIN MROZINSKI (IMPAN)

18 May 2015, 14:15-16:00

D-BRANES AND TWISTED K-HOMOLOGY

K-homology (i.e. the dual theory to K-theory) can be defined in three ways: via homotopy theory, via geometric cycles, and (following Atiyah and Kasparov) via functional analysis. The same is true for twisted K-homology. This talk will give the geometric-cycle definition of twisted K-homology. Also explained in the talk, will be the connection to the D-branes of string theory and how twisted K-homology provides a general setting for twisted index theory. Joint work with Alan Carey and Bai-Ling Wang.

PAUL F. BAUM (Penn State / IMPAN)

25 May 2015, 10:15-12:00

QUANTUM AUTOMORPHISM GROUPS AND SO(3)-REPRESENTATIONS

Since its definition by Wang in 1998, the quantum automorphism group of a finite-dimensional C*-algebra endowed with a positive functional has been the target of numerous investigations and generalisations. In particular, Banica showed in 1999 that the quantum automorphism group of a well chosen C*-algebra and functional has the same representation theory as SO(3). In this talk, I will explain the converse of this result: any compact quantum group having the same representation theory as SO(3) is the quantum automorphism group of a finite-dimensional C*-algebra endowed with a positive functional.

COLIN MROZINSKI (IMPAN)

25 May 2015, 14:15-16:00

MAXIMAL GAUSSIAN PARTS OF FUNCTIONALS ON QUANTUM GROUP ALGEBRAS

Given a cocycle on a compact quantum group algebra, it is possible to extract the maximal Gaussian part from the cocycle. Given a conditionally positive functional on a compact quantum group algebra, there is a canonical way of associating with it a Hilbert space and a cocycle, which is known as Schürmann's construction. M. Schürmann in 1933 posed the question: Suppose we extract the maximal Gaussian part from the cocycle obtained from a conditionally positive functional. Then can we find another conditionally positive functional such that the cocycle coming out of the Schürmann construction on this functional will yield the same maximal Gaussian part? If the answer is positive, then any conditionally positive functional can be written as a sum of two conditionally positive functionals, one of them coming from a Gaussian cocycle. This would be a quantum generalization of a deep result in classical probability called the Levy-Khintchine decomposition of generators of the Levy processes. Unfortunately, the question is still open. In this talk, we will answer this question for a special type of functionals. This is based on joint work with Adam Skalski, Anna W. Kula and Uwe Franz.

BISWARUP DAS (IMPAN)