# Seminar in C*-algebras

The C*-seminar is usually held on Wednesdays at 12:15-14:00 in room B71 (though B70 during the spring 2011), in the Niels Henrik Abel Building.

Talks in the seminar are announced on this site and via e-mail. If you want to be included in or deleted from the mailing list, please contact Nadia Larsen.

## Past talks (2011).

• Wednesday, January 5, 12:15-14:00, room B70

Anders C. Hansen (University of Cambridge)

Title: Generalized Sampling and the Resolution of the Gibbs Phenomenon.

Abstract: I will discuss a generalization of the Shannon Sampling Theorem that allows for reconstruction of signals in arbitrary bases. Not only can one reconstruct in arbitrary bases, but this can also be done in a completely stable way. The results that I will present can also be interpreted, from a harmonic analysis point of view, as a resolution of the Gibbs phenomenon. In particular, given the Fourier coefficients of a piecewise continuous function, the function can be reconstructed without the usual Gibbs oscillations.

• Wednesday, February 23, 12:15-14:00, room B70

Jyotishman Bhowmick (University of Oslo)

Title: Quantum isometry groups

• Wednesday, March 9, 12:15-14:00, room B70

Eugene Ha

Title: On the problem of cohomology for an Arakelov divisor

Abstract: We shall discuss Tate's Riemann-Roch theorem for number fields and the peculiar nature that it implies for the cohomology for an Arakelov divisor (that is, a divisor for the spectrum of a number field formally complete at archimedean infinity). Examples will be given. Though no definitive results are available, we shall argue that operator algebras provide a natural setting in which to study arithmetic cohomology.

• Wednesday, March 16, 12:15-14:00, room B70

Eugene Ha

Title: On the problem of cohomology for an Arakelov divisor, II

Abstract: We shall discuss Tate's Riemann-Roch theorem for number fields and the peculiar nature that it implies for the cohomology for an Arakelov divisor (that is, a divisor for the spectrum of a number field formally complete at archimedean infinity). Examples will be given. Though no definitive results are available, we shall argue that operator algebras provide a natural setting in which to study arithmetic cohomology.

• Tuesday, March 22, 12:15-14:00, room B534 (NOTE CHANGE OF ROOM THIS DAY!)

Lukasz Skowronek

Title:Completely Entangled Subspaces, or the linear subspaces that do not intersect the Segre variety

Abstract:A question recently raised in the Quantum Information community will be discussed on the characterization of orthogonal complements of subspaces with no product vector in them, i.e. the Completely Entangled Subspaces. In a number of cases, it turns out that the complements admit a basis, consisting of product states. While we prove that this is not a general phenomenon, we also give strong evidence that it holds for a number of low-dimensional cases. We tackle the problem both by hand and using exact computer algebra. The last method gives us sort of automated proofs to our examples and counterexamples. Relations to PPT states and positive maps will be discussed. People with an Algebraic Geometry background welcome!

• Wednesday, March 30, 12:15-14:00, room B70

Andrew Toms (Purdue University)

Obs: Talk cancelled!

• Wednesday, April 6, 12:15-14:00, room B70

Takeshi Katsura (pt University of Copenhagen)

Title: Application of graph algebras to problems on semiprojectivity

Abstract: Semiprojectivity is an important notion of the theory of C*-algebras. To study semiprojectivity, a graph algebra is a very powerful tool. I will talk about an ongoing joint work with Eilers on the characterization of semiprojective graph algebras, and its application to several open problems.

• Wednesday, May 11, 12:30-14:00, room B62

Title: Ramanujan-Petersson conjecture and operator algebras methods

## Past talks (2010).

• Wednesday January 13, 2010, 12.15-14.00 in B71.

Lukasz Skowronek , (University of Krakow)

"A strong duality relation between certain cones of maps (and why it is important in the theory of quantum information)."

Abstract: "It will be shown that for certain convex cones of linear maps there exists a convex duality relation that is stronger than the standard relation between a convex cone and its dual. In particular, it will be explained why a result by Erling Størmer became very well known among physicists."

• Wednesday March 3, 2010, 12.15-14.00 in B71.

Wojciech Szymanski, (Syddansk Universitet, Odense)

"Endomorphisms of the Cuntz algebras"

Abstract: I will review the results of my recent work on endomorphisms of the Cuntz algebras which preserve either the canonical UHF subalgebra or the diagonal MASA (or both).

• Wednesday March 24, 2010, 14.15-16.00 in B71.

Andrzej Zuk , (University of Paris 7)

"L^2 Betti numbers of closed manifolds"

• Wednesday April 28, 2010, 12.15-14.00 in B71.

Xin Li , (University of Münster)

"Ring C*-algebras - new constructions in operator algebras"

Abstract: I report on recent work with Joachim Cuntz on ring C*-algebras. We study the inner structure of these new constructions, obtain alternative descriptions and discover links to number theory. Moreover, our attempt of computing K-theory for certain ring C*-algebras leads us to unexpected duality theorems for crossed products attached to affine transformations on adele spaces.

• May 31 - June 4, 2010, Summer school C*-algebras and their interplay with dynamical systems, The Sophus Lie Conference Centre, Nordfjordeid, Norway.

• Wednesday September 8, 2010, 12.15-14.00 in B71.

Erling Størmer (Oslo):

"On positive maps"

• OBS: AVLYST! Friday (NB!) October 1, 2010, 12.15-14.00 in B71.

Florin Radulescu , University of Rome "Tor Vergata":

Title:Quantization of Hecke operators, Ramanujan Peterson conjectures and von Neumann algebras

Abstract: Let $\Gamma = \PSL_2(\Z)$, $G = \PGL_2(\Q)$. Let $\L(\Gamma)$ be the associated type II$_1$ factor. We prove that there exists a unitarily equivalent model for the classical Hecke operators acting on Maass forms, as completely positive maps on $\L(\Gamma)$. In this model, the quantized Hecke operators act on the Berezin's quantization deformation of the fundamental domain of $PSL(2,\Z)$ in the upper halfplane. We construct a Stinespring's dilation of the hypergroup of completely positive maps given by the family of Hecke operators. Then the application mapping a double coset into the image of the corresponding Hecke operator in the Calkin algebra extends to a continuous isomorphism from $\H$ into $Q(\ell^2(\Gamma))$. In particular, the Ramanujan-Peterson conjectures hold true for all eigenvectors, with the exception of a finite number.

• NOTE: Colloquium talk, Tuesday October 5, 2010,

Derek Robinson :

• Wednesday October 6, 2010, 12.15-14.00 in B71.

Akitaka Kishimoto :

"Quasi diagonal flows"

Abstract: "We talk on quasi-diagonal flows and MF flows etc. which were introduced following the corresponding notion for C*-algebras. Many results for C*-algebras (due to Voiculescu and Blackadar-Kirchberg) are extended to this case. Approximately inner flows on a quasi-diagonal C*-algebra are quasi-diagonal and quasi-diagonal flows are MF."

• Wednesday October 13, 2010, 12.15-14.00 in B71.

Erik Bedos (Oslo):

"On Doeblin's theorem and semiregular stochastic matrices"

• Wednesday November 3, 2010, 12.15-14.00 in B71.

Ian Putnam (Victoria):

"C*-algebra from projection method tilings"

• Wednesday November 17, 2010, 12.15-14.00 in B71.

Sergey Neshveyev (Oslo):

"Bost-Connes systems, Hecke algebras, and induction"

• Wednesday December 1, 2010, 12.30-14.00 in B71.

Anilesh Mohari (Chennai, p.t. The Mittag-Leffler Institute)

Translation invariant pure state and it's split property

Abstract: We will prove that a refined Kolmogorov's property is a necessary and sufficient condition for an inductive limit state to be pure. In particular we will investigate how this criteria is related with sufficient condition [Bratteli etal Pure states on ${\cal O}_d$, J. of operator theory, 1997] appears in terms of support projection and it's relation with Haag duality property of a translation invariant pure state in one lattice dimensional quantum spin chain.

• Friday December 3, 2010, 12.30-14.00 in B81 (NOTE TIME AND PLACE!).

Eduard Ortega (NTNU)

"The Cuntz semigroup as invariant for C*-algebras"

Abstract: We will make an overview of the Cuntz semigroup and its importance for the classification of nuclear C*-algebras. We also will define some comparison properties of the Cuntz semigroup and how do they influence in the structure of the corresponding C*-algebra.

• Wednesday December 8, 2010, 12.30-14.00 in B71.

Makoto Yamashita (University of Tokyo)

Title: Connes-Landi deformation of spectral triples

Abstract: We consider deformation of spectral triples with 2-torus action, motivated by deformation of manifolds by Connes-Landi. The continuous crossed product presentation of such deformations allows us to describe how the cyclic cocycles are related on the deformed algebras corresponding to different values of the deformation parameter. The $K$-theoretic invariants such as the Chern-Connes character of the equivariant spectral triples exhibit invariance under the deformation.

• Wednesday December 15, 2010, "Oslo-Trondheim seminar", Auditorium 4 (Vilhelm Bjerknes Building) (NB: CHANGE OF ROOM!)

• 11:00-11:50, John Quigg (Arizona State University)

Title: Topological graphs and principal bundles

Abstract: If a group G acts freely on a directed graph E, then the quotient is also a directed graph, and the Gross-Tucker theorem shows how to reconstruct E from E/G. This has implications for the C*-algebras: the C*-algebra of E/G is Morita equivalent to the crossed product of C*(E) by the corresponding action of G. This is ancient history, and can be proved with either noncommutative duality (joint work with Kaliszewski and Raeburn) or groupoid techniques (Kumjian and Pask). In the topological case, the Gross-Tucker theorem must be replaced by a result of Palais on principal G-bundles, and we must appeal to Rieffel's theory of generalized fixed-point algebras. This is joint work with Deaconu and Kumjian.

12:00-13:20 Lunch

• 13:20-14:10, Roberto Conti (Universita G. D'Annunzio di Chieti-Pescara)

Title: Automorphisms of the Cuntz algebra

Abstract: Long time ago Cuntz noticed that the structure of the automorphism group of the Cuntz algebra resembles that of semisimple Lie groups and suggested a definition of the Weyl group in this context. However, not much had been known about the explicit structure of such groups until recently. I'll report on joint work (mainly with W. Szymanski) on this and related issues.

• 14:20-15:10, Toke M. Carlsen (NTNU)

Title: The C*-algebra of a locally injective surjection

Abstract: Klaus Thomsen has introduced the notion of a semi etale groupoid and has constructed a C*-algebra from any semi etale groupoid. One motivation for this notion and construction was to construct a C*-algebra from a locally injective surjection. Klaus and I have studied the structure of the C*-algebra of a locally injective surjection and I shall in this talk present some of our results. These include a description of the gauge invariant ideals, the primitive ideals, the maximal ideals and the corresponding simple quotients. The main result is that a simple quotient of the C*-algebra of a locally injective surjection on a compact metric space of finite covering dimension is either a full matrix algebra, a crossed product of a minimal homeomorphism of a compact metric space of finite covering dimension, or a unital, purely infinite, simple, nuclear and separable C*-algebra. It follows that if the C*-algebra of a locally injective surjection on a compact metric space of finite covering dimension is simple, then it automatically purely infinite unless the the map in question is a homeomorphism. A corollary of this result is that if the C*-algebra of a one-sided subshift is simple, the it is also purely infinite.

15:10-15:40 Coffee break

• 15:40-16:30, Anders C. Hansen (University of Cambridge)

Title: The Solvability Complexity Index in Spectral Theory

Abstract: The importance of determining spectra of linear operators does not need much explanation as such spectra are essential in quantum mechanics, both relativistic and non-relativistic, and in general in mathematical physics. However, the question on how to determine spectra of operators in a constructive way has troubled mathematicians for a long time. It is well known that for compact operators on Hilbert spaces, this is certainly possible given the matrix elements, but in the general case it has until recently been unknown. I will in this talk discuss this fundamental problem and introduce the Solvability Complexity Index as the main tool for solving the problem.

• 16:40-17:30, Steve Kaliszewski (Arizona State University)

Title: Fell bundles and a universal fixed-point algebra

Abstract: An imprimitivity theorem for Fell bundles over groupoids, originally due to Yamagami, leads to a full-crossed-product version of Rieffel's imprimitivity theorem for generalized fixed-point algebras.

17:30-21:00 Dinner in the meeting room on the 12th floor of Niels Henrik Abel Building

• Friday December 17, 2010, 14.15-16.00 in B71.

Christian Voigt (University of Munster)

"Bott periodicity, vertex algebras and modular forms"

Abstract: In this talk I will discuss a certain analogue of the Dirac-dual Dirac method of Atiyah-Kasparov for proving Bott periodicity. This is motivated from considerations in two-dimensional quantum field theory. In the constructions vertex algebras play an important role.