14:30♯ - 15:20 | B.V. Rajarama Bhat (Indian Statistical Institute, Bangalore) | |
Bures distance for completely positive maps | ||
Building 58, seminar room -101 (basement) | ||
D. Bures had defined a metric on the set of normal states on a von Neumann algebra using GNS representations of states. This notion has been extended to completely positive maps between C*-algebras by D. Kretschmann, D. Schlingemann and R. F. Werner. They also explored applications of the notion in quantum information. We present a Hilbert C*-module version of this theory. We show that we do get a metric when the completely positive maps under consideration map to a von Neumann algebra. Further, we include several examples and counter examples and also explore a different approach. This is based on joint works with K. Sumesh and Mithun Mukherjee.
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15:30♯ - 16:20 | Ilan Hirshberg (BGU) | |
Simple non-separable nuclear C*-algebras not isomorphic to their opposites | ||
Building 58, seminar room -101 (basement) | ||
Given a C*-algebra $A$, we can construct the opposite algebra as
being the same Banach space with the same involution, but with the
reverse multiplication. In the separable setting, there are known
examples of simple non-nuclear C*-algebras which aren't isomorphic to
their opposite algebras, and there are non-simple nuclear examples.
Whether there exists a separable nuclear example is a difficult open
problem, and is of interest from the perspective of structure and
classification: the Elliott invariant and the Cuntz semigroup cannot
distinguish a C*-algebra from its opposite, so if such an example
exists, it would most likely reveal new phenomena. In the non-separable setting, things become stranger. I'll discuss a recent preprint in which it is shown that it is consistent with ZFC that there is a simple nuclear non-separable C*-algebra which is not isomorphic to its opposite algebra. One can in fact guarantee that this example is an inductive limit of unital copies of the Cuntz algebra $\mathcal{O}_2$, or of the CAR algebra. This is joint work with Ilijas Farah. | ||
16:30♯ - 17:20 | Eli Shamovich (Technion) | |
Multipliers and Varieties in the Noncommutative Ball | ||
Building 58, seminar room -101 (basement) | ||
In this talk we will focus on the algebra of multipliers on the noncommutative $H^2$ of the unit ball, namely the noncommutative set of d-tuples of matrices $(A_1, \ldots , A_d)$, satisfying $\sum_{k=1}^d A_kA_k^* \leq 1$. This algebra is in fact the WOT closed algebra generated by the creation operators and was considered by many including Bunce, Arias and Popescu and Davidson and Pitts. We will consider quotients of this algebra and the corresponding noncommutative varieties in the ball. We will discuss the isomorphism problem between noncommutative varieties in the ball and completely isometric isomorphisms between the corresponding algebras of multipliers. | ||
13:30♯ - 14:20 | Michael Skeide (University of Molise) | |
Subproduct Systems from Coalgebras and the Representations Theorem for Quantum Lévy Processes | ||
Amado building (Math), room 719 | ||
Convolution semigroups of semi-inner products on colagebras give rise to subproduct systems of Hilbert spaces. By a concrete construction, we show that the Arveson systems generated by these coalgebra subproduct systems are type I, that is, Fock spaces. By an application of this result,we reprove Michael Schürmann’s result that every quantum Lévy process posses a representation on the Fock space. This is joint work with Malte Gerhold and part of our seeking for finite-dimensional subproduct systems. The proof is actually inspired by our paper with Michael Schürmann and his former MSc student Sylvia Volkwardt, which in turn is inspired by our joint work with Volkmar Liebscher how to construct units in product systems. | ||
14:30♯ - 15:20 | Guy Salomon (Technion) | |
The noncommutative Choquet boundary of graph tensor-algebras | ||
Amado building (Math), room 619 | ||
The fundamental nonselfadjoint operator-algebra associated with a countable directed graph is its tensor-algebra. Ten years ago, Katsoulis and Kribs showed that its C*-envelope --- the noncommutative counterpart of the Shilov boundary --- is the Cuntz-Krieger algebra of the graph. My aim in this talk is to describe the noncommutative counterpart of points in the Choquet boundary of the tensor-algebra and to provide a full characterization of them. This leads both to a new proof of Katsoulis-Kribs theorem mentioned above and to a characterization --- in terms of the graph itself --- of the tensor-algebra hyperrigidity inside the Cuntz-Krieger algebra. The talk is based on joint work with Adam Dor-On. | ||
12:00♯ - 13:00 | Victor Vinnikov (Ben-Gurion University) | |
Interpolation and relization for nc functions | ||
Shchreiber building (Math), room 008 | ||
I will state a fairly general interpolation and realization theorem for contractive nc functions. The domain of the function can be the ``unit ball'' defined by any nc function on a full nc set. This is a joint work with Joe Ball and Gregory Marx.
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14:10♯ - 16:00 | Ilan Hirshberg (Ben-Gurion University) | |
A survey of recent advances in the classification program of simple nuclear C*-algebras | ||
Ornstein building (Chemistry), room 102 | ||
The Elliott classification program for simple nuclear C*-algebras has seen major recent breakthroughs, by a number of contributors. In this survey talk, I will review the background and history of the Elliott program, and explain the most recent developments and current state of the program. | ||
11:30♯ - 12:30 | Paul Muhly (University of Iowa) | |
Much to do about 2 in Free Analysis | ||
Amado Building, room 719 | ||
In this talk, I will discuss some recent results about operator algebras built from algebras of generic matrices and their trace algebras. For the purpose of illustration, I will focus on the algebra of 2-generic 2 x 2 matrices. Even in this "simple" situation there are surprises.
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14:10♯ - 15:10 | Adam Dor-On (University of Waterloo) | |
Classification of tensor algebras associated to weighted partial systems | ||
Amado Building, room 719 | ||
abstract here (pdf) | ||
11:10♯ - 12:00 | Ilijas Farah (York University) | |
Corona rigidity | ||
Building 58, seminar room -101 (basement) | ||
To a non-unital C*-algebra A one can associate its multiplier algebra M(A)
(this is the non-commutative analogue of the Stone-Cech compactification) and the
corona, M(A)/A. For example, the corona of an abelian algebra $C_0(X)$ is the
algebra $C(\beta X\setminus X)$ of all continuous functions on the Stone-Cech remainder
of X. The corona of the algebra K of compact operators on a given Hilbert space H
is the Calkin algebra.
Assuming there is an isomorphism between coronas M(A)/A and M(B)/B, what
can one say about the relation between algebras A and B? When is it possible
to `lift’ an isomorphism to a reasonably nice map (e.g., a *-homomorphism) from
M(A) to M(B)? I will present current state of the art on this problem and its relation
to the following.
In 1977 Brown, Douglas and Fillmore asked whether there is a K-theory reversing
automorphism of the Calkin algebra. It is not known whether this question has a negative
answer, but it is known that it does not have a positive answer. This is because certain
additional set-theoretic axioms imply that all automorphisms of the Calkin algebra are
inner. | ||
13:30♯ - 14:20 | Kate Juschenko (Northwestern University) | |
Techniques and concepts of amenability of discrete groups | ||
Building 58, seminar room -101 | ||
The subject of amenability essentially begins in 1900's with Lebesgue. He asked whether the properties of his integral are really fundamental and follow from more familiar integral axioms. This led to the study of positive, finitely additive and translation invariant measure on reals as well as on other spaces. In particular the study of isometry-invariant measure led to the Banach-Tarski decomposition theorem in 1924. The class of amenable groups was introduced by von Neumann in 1929, who explained why the paradox appeared only in dimensions greater or equal to three, and does not happen when we would like to decompose the two-dimensional ball. In 1940's, M. Day defined a class of elementary amenable groups as the largest class of groups amenability of which was known to von Neumann. He asked whether there are other groups then that. We will give an introductory to amenability talk, and explain more recent developments in this field. In particular, I will explain how to obtain all known non-elementary amenable groups using only one approach.
Slides (pdf) | ||
14:40♯ - 15:30 | Joav Orovitz (Ben-Gurion University) | |
Strict comparison and crossed products | ||
Building 58, seminar room -101 | ||
I will give an overview of recent work with Chris Phillips and Qingyun Wang. Strict comparison is one of the so called regularity properties studied in Elliott's classification program. It is conjectured that the regularity properties precisely define the largest subclass of simple separable nuclear $C^*$-algebras which can be classified by $K$-theoretic invariants. In recent years, many efforts have been directed towards proving that a regularity property holds for various classes of $C^*$-algebras. In results of this sort, the class of $C^*$-algebras considered is very often made up of crossed product $C^*$-algebras. In this work, we define what it means for an action $\alpha:G\to \mathrm{Aut}(A)$ of a discrete amenable group $G$ on a $C^*$-algebra $A$ to have the weak tracial Rokhlin property. Assuming this to be the case, we show that (under some additional reasonable hypotheses) the crossed product $A\rtimes_\alpha G$ has strict comparison if $A$ does. As a corollary, we show that the same is true for the property of $\mathcal{Z}$-stability. This generalizes work of Hirshberg and Winter from 2005 and 2007, as well as work of Hirshberg and myself from 2012. | ||
14:00♯ - 14:45 | Victor Vinnikov (Ben-Gurion University) | |
Noncommutative reproducing kernel Hilbert spaces | ||
room 719, Amado building | ||
I will discuss the analogue of the classical Aronszajn's theory of reproducing kernel Hilbert spaces in the setting of noncommutative functions and noncommutative completely positive kernels. The completely positive kernels of BBLS [Barreto-Bhat-Liebscher-Skeide] also appear as a special and in some sense commutative version [since the noncommutative set where the kernel is defined consists of diagonal matrices]. In addition to their intrinsic interest, noncommutative completely positive kernels provide a proper framework for the study of classical interpolation problems for noncommutative functions, cf. the work of Muhly-Solel and Agler-McCarthy.
This is a joint work with Joe Ball and Gregory Marx. | ||
14:55♯ - 15:40 | Eli Shamovich (Ben-Gurion University) | |
On Dilation of Certain Commutative Semigroups of Contractions | ||
room 719, Amado building | ||
Given a $d$-tuple of commuting bounded dissipative operators $A_1,\ldots,A_d$ on a separable Hilbert space $\mathcal{H}$, we associate to it an overdetermined i/s/o system of differential equations on $\mathbb{R}^d$. We then introduce the system of compatibility PDEs and list a number of conditions for the latter system to have a solution for every initial condition along one of the axes. We then show how to obtain weak solutions for the system and describe some properties of the solution when the initial condition is an $L^2$ function. Then using the space of initial conditions we show that in fact the commuting semigroup of contractions generated by the operators $e^{i A_j}$ admit a unitary dilation. | ||
11:00♯ - 11:45 | Orr Shalit (Technion) | |
On operator algebras associated with monomial ideals in noncommuting variables | ||
Building 58, seminar room -101 | ||
In this talk I will try to give an overview of a recent joint work with Evgenios Kakariadis.
We study several operator algebras (C* as well as non-selfadjoint) associated with monomial ideals. These algebras are interesting because they are connected to several different active programs in operator algebras (as well as to some dormant ones). Our study leads us to revisit several operator algebra constructions that have been considered before. These include Matsumoto's subshift C*-algebras, and the tensor, the Toeplitz and the Cuntz algebras associated with subproduct systems, as well as the tensor and the Pimsner algebras associated with dynamical systems or topological graphs. We sort out the various relations by giving concrete conditions and counterexamples that orientate the operator algebras of the associated C*-correspondence. Furthermore we identify boundary representations, we calculate their C*-envelopes, and we give criteria for hyperrigidity of the tensor algebras. Finally, we classify the tensor algebras up to algebraic isomorphism in terms of the data derived from the monomial ideals. Slides (pdf) | ||
11:50♯ - 12:35 | Adam Dor-On (Univ. of Waterloo) | |
Tensor Algebras arising from Markov-Feller maps | ||
Building 58, seminar room -101 | ||
Given a compact metric space X, a Feller map is a continuous map $P : X \to Prob(X)$ where $Prob(X)$, imbued with the w*-topology, is the convex compact set of
probability measures on X. It turns out that Feller maps encompass many different classical dynamical objects, such as finite stochastic matrices, topological dynamical systems on X and iterated function systems on X. We associate non-self-adjoint Tensor operator algebras to such Feller maps and classify them up to bounded / isometric isomorphism by using Hilbert bundle theory, effectively providing generalizations
and analogues of results by Davidson, Katsoulis, Kribs, Solel and others. | ||
14:00♯ - 14:45 | Ilan Hirshberg (Ben-Gurion University) | |
Rokhlin dimension for flows | ||
Building 58, seminar room -101 | ||
By a flow I mean a one-parameter point-norm continuous group of automorphisms of a C*-algebra. In 1996, Kishimoto introduced a concept of the Rokhlin property for flows, which is analogous to the Rokhlin property for a single automorphism. I'll discuss a generalization of this, Rokhlin dimension. The results parallel to a great extent results previously obtained in the discrete settings for actions of Z, Z^n, finite groups and certain residually finite groups. The main results are that crossed products by flows with finite Rokhlin dimension preserve finite nuclear dimension and D-absorption (the latter with an additional technical assumption), and that any free flow on a commutative C*-algebra with a (locally connected) finite dimensional spectrum has finite Rokhlin dimension. This is joint work with Szabo, Winter and Wu. | ||
14:50♯ - 15:35 | Ami Viselter (Haifa University) | |
On positive definiteness over locally compact quantum groups, part 2 | ||
Building 58, seminar room -101 | ||
See abstract below in part 1. See also the summary of the previous talk. | ||
11:30 - 12:30 | Ami Viselter (Haifa University) | |
On positive definiteness over locally compact quantum groups, part 1 | ||
Schreiber bldg, room 210 | ||
Positive-definite functions over locally compact groups are a very fundamental tool in abstract harmonic analysis and related areas. They are tightly connected with various aspects of the group, such as representations, group properties (amenability and other approximation properties, property (T), etc.), the algebras associated to the group and many more.
The concept of positive-definite functions in the setting of locally compact quantum groups (LCQGs) has lately appeared in a few papers in relation to other notions. This stimulated a very recent work of Daws and Salmi, in which they examined several ways to define "(completely) positive-definite functions over LCQGs", and proved that, under some mild assumptions, they are all equivalent. We continued this line of research, generalizing some known results about positive definiteness over groups to the quantum setting. These will be presented in the talk, after brief introductions to LCQGs and positive-definite functions. This is joint work with Volker Runde. | ||
14:00 - 15:00 | Joseph Ball (Virginia Tech) | |
Nudelman Nevanlinna-Pick interpolation versus Riesz-Dunford Nevanlinna- Pick interpolation: positive Pick matrices versus completely positive Pick matrices | ||
Schreiber bldg, room 8 | ||
Abstract | ||
15:15 - 16:15 | Claude Schochet (Wayne State University and Technion) | |
Spanier-Whitehead Duality for C*-algebras | ||
Schreiber bldg, room 8 | ||
Spanier-Whitehead duality was developed in the 1950's when it was realized that there is a strong connection between the homology of a finite CW complex and the cohomology of its complement in a sphere. In recent years there have been various efforts (by Paschke and by Kaminker, Putnam and Whittaker most recently) to develop an analogous K-theoretic tool that encompasses the known examples of such duality for C*-algebras coming, particularly, from hyperbolic dynamical systems. We offer a unified framework for these efforts and illustrate the strengthening of KK-theory needed to carry it out. These results (and speculation) are joint work with Jerry Kaminker. | ||
All the talks will be in the Schreiber building (Mathematics), room 209.
11:00 - 12:00 | Alex Vernik (Ben-Gurion University) | |
Dilations of CP maps commuting according to a graph | ||
I'll speak about results from my work on my master's thesis, done under the supervision of Dr. Orr Shalit. Given two commuting CP maps (that is, completeley positive, completely contractive normal maps) on a von Neumann algebra, it is known that one can obtain a joint, commuting *-endomorphic dilation. In my thesis, I try to see what happens with tuples of CP maps given with different commutation relations, and obtain in particular the following result: If a tuple of CP maps has commutation relations encoded in a graph $G$, and $G$ is acyclic, then there is a joint *-endomorphic dilation, which also commutes according to $G$. We also obtain a counterexample in the case of a cyclical graph, generalizing Shalit and Solel's counterexample for a 3-cycle. To do this, we make use of Shalit and Solel's notion of subproduct systems, with their representations and dilation theory. | ||
13:30 - 14:30 | Paul Muhly (University of Iowa) | |
Noncommutative Function Algebras in Free Analysis | ||
14:45 - 15:45 | Ilan Hirshberg (Ben-Gurion University) | |
Rokhlin dimension for finite group actions | ||
I'll discuss a generalization of the Rokhlin property for finite group actions on C*-algebras. I'll try to briefly explain what it is good for, and say a few words about K-theoretic obstructions for the existence of such actions on certain strongly self-absorbing C*-algebras. This is based on joint work with Winter and Zacharias and with Phillips. | ||
11:30 - 12:30 | Aldo Lazar (Tel Aviv University) | |
Continuous fields of postliminal C*-algebras | ||
Schreiber building (Math), room 209 | ||
Let $((A(t))_{t\in T},\Gamma)$ be a continuous field of postliminal C*-algebras and denote by $B(t)$ the greatest liminal ideal of $A(t)$. Dixmier asked (Problem 10.10.11 in the C*-algebras book) if $((B(t)), \Gamma')$, where $\Gamma' := \{x\in \Gamma | x(t)\in B(t), \forall t\in T\}$, is a continuous field of C*-algebras. It is quite easy to answer this question in the negative. In the talk I shall discuss a rather general situation when the answer is positive. On the other hand there is an example of a continuous field of postliminal C*-algebras over [0,1] whose fibers are mutually isomorphic and which has the property that all its restrictions to the relatively open subsets of the interval provide counterexamples to Dixmier's question. | ||
14:00 - 15:00 | Daniel Markiewicz (Ben-Gurion University) | |
Tensor algebras and subproduct systems arising from stochastic matrices | ||
Shenkar building (Physics), room 105 | ||
This is joint work with Adam Dor-On. Stochastic matrices give rise to subproduct systems over abelian von Neumann algebras (in the sense of O. M. Shalit and B. Solel) in a natural way. We will discuss the classification of the non-self-adjoint tensor algebras associated to these subproduct systems, and how much they remember about the matrices. | ||
The talks will take place in Room -101, Math building (58).
11:30 - 12:30 | David Cohen (Ben-Gurion University) | |
Dilations of commuting matrices | ||
We define a new kind of dilation for a k-tuple of commuting operators acting on a finite dimensional space, and study the relation to normal $\partial X$ dilations. | ||
14:00 - 15:00 | Sourav Pal (Ben-Gurion University) | |
Spectral sets and distinguished varieties in the symmetrized bidisc. | ||
This is a joint work with Orr Moshe Shalit. In this talk, we show that for every pair of matrices $(S,P)$, having the closed symmetrized bidisc $\Gamma$ as a spectral set, there is a one dimensional complex algebraic variety $\Lambda$ in $\Gamma$ such that for every matrix valued polynomial $f(z_1,z_2)$, $$\|f(S,P)\|\leq \max_{(z_1,z_2)\in \Lambda}\|f(z_1,z_2)\|.$$ The variety $\Lambda$ is shown to have the determinantal representation $$\Lambda = \{(s,p) \in \Gamma : \det(F + pF^* - sI) = 0\} ,$$ where $F$ is the unique matrix of numerical radius not greater than 1 that satisfies $$S-S^*P=(I-P^*P)^{\frac{1}{2}}F(I-P^*P)^{\frac{1}{2}}.$$ When $(S,P)$ is a strict $\Gamma$-contraction, then $\Lambda$ is a distinguished variety in the symmetrized bidisc, i.e. a one dimensional algebraic variety that exits the symmetrized bidisc through its distinguished boundary. We characterize all distinguished varieties of the symmetrized bidisc by a determinantal representation as above. | ||
The talks will take place in Room 814, Amado building.
11:30 - 12:30 | Leonid Helmer (Technion) |
Reflexivity of noncommutative Hardy algebras and related algebras | |
In this talk we will consider the question of reflexivity of noncommutative Hardy algebras. An operator algebra is reflexive if it is defined by its invariant subspaces. We consider the image of the Hardy algebra under an induced representation and prove some partial results concerning reflexivity for such algebras. Then we define a natural co-invariant subspace and prove the reflexivity of the Hardy algebra when compressed to this sudspace. This generalizes a result of Popescu from 2007. | |
14:00 - 15:00 | Panchugopal Bikram (Ben-Gurion University) |
Extendable Endomorphisms on factors | |
In this talk, we examine what it means to say that certain endomorphisms of a factor (which we call equi-modular) are extendable. We obtain several conditions on an equi-modular endomorphism. We then exhibit the obvious example of endomorphisms satisfying the condition in this theorem. We use our theorem to determine when every endomorphism in an E$_0$-semigroup on a factor is extendable - which property is easily seen to be a cocycle-conjugacy invariant of the E$_0$-semigroup. We conclude by giving examples of extendable E$_0$-semigroups, and by showing that the Clifford flow and free flow on the hyperfinite II$_1$ factor are not extendable, neither is the CAR flow on type III factors. | |
The talks will take place in Room 210, Schreiber building.
14:00 - 14:50 | Evgenios Kakariadis (Ben-Gurion University) |
C*-dynamical systems and Operator Algebras | |
The (usual) C*-crossed product may be seen as the natural
C*-algebra arising from an automorphic C*-dynamical system. When the system
is no longer automorphic, though, there is a signi ficant number of such
possible candidates, i.e., generalized C*-crossed products. Following
Stacey, the main scheme is to relate each one of them to (a full corner of)
a usual C*-crossed product. The first objective of this talk will be to
present several obstructions to this task. Another candidate arises in the context of "Arveson's Program on the C*-envelope". This is the C*-envelope of an appropriate non-selfadjoint operator algebra, e.g., of a semicrossed product, of a tensor algebra in the sense of Muhly and Solel, etc. The second objective of this talk will be to present certain advantages of this approach and of the use of these non-selfadjoint operator algebras. In particular we will comment on their flexibility and on the fact that they capture the structure of the dynamics. (The talk is based on joint works with Ken Davidson, Elias Katsoulis and Justin Peters.) | |
15:00 - 15:50 | David Kimsey (Ben-Gurion University) |
Trace formulas for a class of truncated block Toeplitz operators | |
The strong Szego limit theorem may be formulated in terms of operators of
the form
$$(P_N \mathfrak{G} P_N)^n - P_N \mathfrak{G}^n P_N \quad\quad {\rm for}
\quad n=1,2,\ldots,$$
where $\mathfrak{G}$ denotes the operator of multiplication by a suitably
restricted $d \times d$ mvf (matrix-valued function) acting on the space of
$d \times 1$ vvf's (vector-valued functions) that meet the summability
constraint
$$\int_0^{2\pi} f(e^{i\theta})^* \Delta(e^{i \theta}) f(e^{i \theta})
d\theta < \infty,$$
where $\Delta(e^{i \theta}) = I_d$ and $P_N$ denotes the orthogonal
projection onto the space of $d \times 1$ trigonometric vector polynomials
which meet the same summability constraint as above. We will study these operators for a class of positive definite mvf's $\Delta$ which admit the factorization $\Delta = Q^* Q = R R^*$, where $Q^{\pm 1}, R^{\pm 1}$ belong a certain algebra of mvf's. We shall show that $$\kappa_n(G) = \lim_{N \uparrow \infty} {\rm trace}\{ (P_N \mathfrak{G} P_N)^n - P_N \mathfrak{G}^n P_N \} $$ exists for $n=2,3,\ldots$. An explicit formula for $\kappa_n(G)$ will be obtained when $GQ = QG$ and $G R^* = R^* G$. In addition, a weighted analog of the strong Szego limit theorem will be considered. (Joint work with Harry Dym) | |
The talks will take place in Math building (#58), room -101.
14:30 - 15:30 | David Cohen (Ben-Gurion University) |
Normal dilation of commuting matrices | |
I will show that a tuple of operators acting on a finite dimensional space has a normal dilation supported on a compact set $K$ if and only if it has, for every $d$, a normal $d$-dilation acting on a finite dimensional Hilbert space. | |
All the talks will take place in room 719, Amado building.
11:30 - 12:30 | Paul Muhly (University of Iowa) |
Unrestraining Boundaries for Tensor Algebras | |
In this talk I will discuss some of the work that Baruch Solel and I have done on the representation theory of tensor algebras that was inspired by Bill Arveson's vision of operator algebras | |
12:30 - 14:30 | Lunch Break |
14:30 - 15:30 | Ami Viselter (University of Alberta) |
Ergodic Theory for quantum (semi)groups | |
We shall describe a generalization of recent work on aspects of ergodic theory of semigroup actions on von Neumann algebras to the context of quantum semigroups. A brief introduction to the analytic theory of quantum (semi)groups will be given. | |
15:30 - 16:30 | Claude Schochet (Wayne State University) |
The Unitary Group of a C*-algebra | |
If A is a unital C*-algebra then its group of unitaries UA is a nice topological group, and it holds a rich lode of information about A itself. In this expository talk we recall how UA determines the topological K-theory K_*(A). We then consider other topological information about A that is lost when one passes exclusively to K-theory. In particular, we review what is known about the homotopy groups and the rational homotopy groups of UA. | |