# Kyoto Operator Algebra Seminar

## Fall/Winter 2011 (Collateral to Operator Algebras and their Applications)

 Sep. 02 (Fri)Caution! 3:00 - 4:30RIMS 110 Sriwulan Adji (Univesiti Sains Malaysia) The partial-isometric crossed products of systems by semigroups of endomorphisms The partial-isometric crossed product is the $\mathrm{C}^*$-algebra constructed from the system in which the endomorphisms are implemented covariantly by partial isometries. This class of $\mathrm{C}^*$-algebras provides a good example of the Toeplitz algebras ${\mathcal T}_X$ associated to a product system of Hilbert bimodules $X$ over a semigroup, which was introduced by Fowler (2002). I am interested in particularly getting a description about invariant ideals of the crossed product. In this talk, I will first discuss about some basic known results from the paper of Lindiarni and Raeburn. Then I will explain some various connections to the isometric crossed products and hence to the usual crossed product by groups of automorphisms. Sep. 02 (Fri)Caution! 4:45 - 6:15RIMS 110 Ilan Hirshberg (Ben Gurion University) Rokhlin dimension for automorphisms of $\mathrm{C}^*$-algebras I will discuss a generalization of the Rokhlin property for an automorphism, which can be thought of as a higher dimensional analogue. This property is more common than the Rokhlin property, and is still strong enough to establish permanence properties for finite nuclear dimension and ${\mathcal Z}$-stability. This is joint work with Wilhelm Winter and Joachim Zacharias. Sep. 05 - 09 RIMS Conference on $\mathrm{C}^*$-Algebras and Related Topics Oct. 05 2:45 - 4:15RIMS 006 Chris Phillips (University of Oregon/RIMS) Equivariant semiprojectivity Semiprojectivity of a $\mathrm{C}^*$-algebra $A$ is the most useful formulation of the idea that approximate homomorphisms from $A$ should be close to exact homomorphisms. For example, the algebra of continuous functions on the circle is semiprojective, essentially because an approximate homomorphism to $B$ corresponds to an approximately unitary element in $B$, and the nearby exact homomorphism comes from the unitary part of its polar decomposition. We investigate a version of semiprojectivity which requires equivariance with respect to an action of a compact group. Unconventional methods seem to be required. A number of open problems will be stated. The talk should be accessible without background in the area. Oct. 12 2:45 - 4:15RIMS 006 Narutaka Ozawa (RIMS) On Simple Finitely Generated Amenable Groups (after Grigorchuk and Medynets) Grigorchuk and Medynets discovered the first examples of infinite finitely generated groups that are simple and amenable. More precisely, they proved that the topological full groups of any minimal transformation on the Cantor space is amenable. It was previously known that the commutator subgroup of such a group is simple (Bezuglyi--Medynets and Matui) and sometimes finitely generated (Matui). I will discuss these results and their background. arXiv:1105.0719 NB: There is a flaw in the proof of Lemma 4.4 (v2). Oct. 12Colloquium 4:30 - 5:30MATH 110 Benoît Collins (University of Ottawa/RIMS) New results in Free Probability theory and applications to Quantum Information theory Motivated by Operator Algebraic questions, Voiculescu introduced Free Probability in the early eighties. Ten years later, he discovered that Free Probability explains some behaviours of eigenvalues of Random Matrices in the limit of large dimension. In this talk we will first describe new convergence results for norms for random matrices with unitary invariance, hereby solving conjectures by Pisier and Haagerup. Then, we will describe some applications to Quantum Information theory. This talk is based on results obtained through collaborations with Belinschi, Nechita, Fukuda and Male. Oct. 19 2:45 - 4:15RIMS 006 Chris Phillips (University of Oregon/RIMS) Analogs of Cuntz algebras on $L^p$ spaces The Cuntz algebras are among the most important examples of $\mathrm{C}^*$-algebras. The algebra ${\mathcal O}_d$ is generated by $d$ isometries on a Hilbert space with orthogonal ranges which add up to the whole space, and their adjoints. It does not depend on the choice of the isometries. It turns out that for $p > 1$ and finite, there is a very similar statement in which the Hilbert space is replaced by $L^p (X)$. The resulting algebras are purely infinite and simple, and have the same K-theory as the usual Cuntz algebras. For different values of $p$ they are strongly incomparable: there is no nonzero continuous homomorphism from one to the other. The subject is just getting started: so far, there is little complicated machinery, and there are many open questions. Oct. 25 - 28 ShuGakuIn Operator Algebras and Mathematical Physics at Kansai Seminar House Nov. 02 Okinawa No Talk (Annual Workshop on Operator Theory and Opretaor Algebras, Ryukyu University, Nov. 03-06.) Nov. 08 (Tue)Caution! 1:15 - 2:45RIMS 110 Étienne Blanchard (Institut de Mathématiques de Jussieu) Fiberwise products of $\mathrm{C}^*$-bundles We describe which classical amalgamated products of continuous $\mathrm{C}^*$-bundles are continuous $\mathrm{C}^*$-bundles and we analyse the involved extension problems for continuous $\mathrm{C}^*$-bundles. Nov. 16 2:45 - 4:15RIMS 006 Yoshikata Kida (Kyoto University) Invariants for orbit equivalence relations of Baumslag-Solitar groups For non-zero integers $p, q$, the Baumslag-Solitar group $\mathrm{BS}(p,q)$ is defined by the presentation $\langle a, t \mid ta^pt^{-1}=a^q\rangle$. We consider ergodic free probability-measure-preserving actions of $\mathrm{BS}(p,q)$, and introduce an invariant of them under orbit equivalence. We also present results on groups which are measure equivalent to $\mathrm{BS}(p,q)$. Nov. 23 Holiday (Labor Thanksgiving Day) Nov. 30 2:45 - 4:15RIMS 006 Masaki Izumi (Kyoto University) A weak homotopy equivalence type result related to Kirchberg algebras Dec. 06 (Tue)Caution! 2:45 - 4:15RIMS 111 Snigdhayan Mahanta (University of Adelaide) On the integral algebraic K-theoretic Novikov conjecture The Novikov conjecture in topology is known to follow from the (rational) injectivity of the Baum-Connes assembly map. For a discrete and torsion free group $G$ the integral algebraic K-theoretic Novikov conjecture asserts the (split) injectivity of a similar Loday assembly map. After recalling briefly some background material, in this talk I shall show how any countable discrete and torsion free subgroup of a general linear group over any field satisfies the integral algebraic K-theoretic Novikov conjecture with coefficients in certain operator algebras. Dec. 07 - 16 RIMS Winter School on Operator Algebras Dec. 12 - 16 RIMS Analysis and Geometry of Discrete Groups and Hyperbolic Spaces (Program in Japanese) Dec. 17 - 18 Kinosaki Kansai Operator Algebra Seminar Dec. 19 - 21 RIMS Mathematical Studies on Independence and Dependence Structure Winter Break BouNenKai (Forget Year Party): Dec. 21 Jan. 09 - 13 RIMS Conference on von Neumann Algebras and Related Topics Jan. 11Colloquium 4:30 - 5:30RIMS 110 Chris Phillips (University of Oregon/RIMS) Existence of outer automorphisms of the Calkin algebra is undecidable Let $H$ be a separable infinite dimensional Hilbert space, for example, the space $\ell^2$ of all square summable sequences. Let $L (H)$ be the algebra of all continuous linear maps from $H$ to $H,$ and let $K (H)$ be the closure in $L (H)$ of the set of continuous linear maps which have finite rank. Then $K (H)$ is an ideal in $L (H),$ and we can form the quotient algebra $Q = L (H) / K (H)$. It is called the Calkin algebra, and is an example of a $\mathrm{C}^*$-algebra. Question: Does the Calkin algebra have outer automorphisms, that is, automorphisms not of the form $a \mapsto u a u^{-1}$ for suitable $u \in Q$ ? It turns out that this question is undecidable in ZFC. If one assumes the Continuum Hypothesis, then outer automorphisms exist. In fact, there are more automorphisms than there are possible choices for $u$. (This is joint work with Nik Weaver.) However, Ilijas Farah proved that it is consistent with ZFC that $Q$ has no outer automorphisms. This talk is intended for a general audience. I will describe some of the background, and give several of the ideas of the proof that the Continuum Hypothesis implies the existence of outer automorphisms of the Calkin algebra. Jan. 18 2:45 - 4:15RIMS 006 Volodymyr Nekrashevych (Texas A&M University) $\mathrm{C}^*$-algebras of hyperbolic groupoids Hyperbolic groupoids is a generalization of such notions as Gromov hyperbolic group actions on their boundaries, groupoids generated by expanding maps, Ruelle groupoids of Smale spaces, etc. An interesting aspect of the theory of hyperbolic groupoids is a duality theory: rather than a groupoid acting on its boundary, we get a pair of groupoids acting on each others boundaries. We will discuss $\mathrm{C}^*$-algebras associated with hyperbolic groupoids (e.g., with hyperbolic complex rational functions), the corresponding duality theory, and applications to dynamics. Feb. 22Colloquium 4:30 - 5:30MATH 110 Narutaka Ozawa (RIMS) Is an irng singly generated as an ideal? I will address the problem whether every finitely generated idempotent ideal of a ring is singly generated as an ideal, and discuss its relationship to the Wiegold problem. Joint work with N. Monod and A. Thom. Spring Break Semester End

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