In a certain analogy with the case of the numerical range, one can define a notion of "spatial" matricial range. It turns out that the matricial range is the C$^*$-convex hull of the spatial matricial range. The proof of that fact is a wonderful tour-de-force along several of the most fundamental results about C$^*$-algebras and von Neumann algebras.
Crouzeix's conjecture is an interesting open problem in operator theory. The conjecture states that the numerical range $W(A)$ of any bounded linear operator $A$ in a complex Hilbert space is a (complete) $2$-spectral set for $A$. Two weeks ago, Crouzeix and Palencia proved that $W(A)$ is (complete) $1 + \sqrt(2)$ spectral set. We will discuss this proof and some applications.
Crouzeix's conjecture is an interesting open problem in operator theory. The conjecture states that the numerical range $W(A)$ of any bounded linear operator $A$ in a complex Hilbert space is a (complete) $2$-spectral set for $A$. Two weeks ago, Crouzeix and Palencia proved that $W(A)$ is (complete) $1 + \sqrt(2)$ spectral set. We will discuss this proof and some applications.
The matricial range is a natural generalization of the numerical range studied by Arveson. There is basically a single non-trivial case where it is fully characterized. This is a wrap-up talk of the previous one, where Ando's Theorem will be applied to the matricial range.
The matricial range is a natural generalization of the numerical range studied by Arveson. There is basically a single non-trivial case where it is fully characterized. In this last talk we will talk how Ando used unitary dilations to approach the problem of the matricial range of the $2\times 2$ unilateral shift.
The matricial range is a natural generalization of the numerical range studied by Arveson. There is basically a single non-trivial case where it is fully characterized. This is a consequence of work of Arveson, which we will discuss.
The matricial range is a natural generalization of the numerical range studied by Arveson. We will provide some basic results together with some (not trivial at all) examples.
We construct a class of quantum dynamical semigroups, called quasi-free quantum dynamical semigroups, which act on the von Neumann of an arbitrary Weyl system, with respect to some compatible families of quasi-measures and $C_0$-semigroups. We then describe the subproduct systems of the quasi-free quantum dynamical semigroups corresponding to irreducible Weyl systems.
We construct a class of quantum dynamical semigroups, called quasi-free quantum dynamical semigroups, which act on the von Neumann of an arbitrary Weyl system, with respect to some compatible families of quasi-measures and $C_0$-semigroups. We then describe the subproduct systems of the quasi-free quantum dynamical semigroups corresponding to irreducible Weyl systems.
I shall discuss ongoing work where we study random matrix models suggested by noncommutative geometry and the quest to quantize gravity. As a rule these models are far more complicated than most of the well understood matrix models.
The possibility of a rigorous proof of a phase transition as the coupling constants varies is a very interesting and challenging problem. I shall provide necessary background material in the talk.
The numerical range of an operator A is a remarkable region of the complex plane. It captures information about the spectrum and the behavior of A. In this third lecture, I will introduce spectral sets and discuss some important properties related to special classes of operators and the numerical range of an arbitrary operator. Moreover, we will discuss Crouzeix's conjecture which is that the numerical range of any square matrix A is a 2-spectral set for A.
The numerical range of an operator A is a remarkable region of the complex plane. It captures information about the spectrum and the behavior of A. In this second lecture, we will continue our discussion on the numerical range of special classes of operators. Moreover, recall that the numerical range is also defined by the quantum states (unital positive linear functionals). We will proof the equivalence between the different definitions using the resolvent norm, and discuss some mapping theorems.
The numerical range of an operator $A$ is a remarkable region of the complex plane. It captures information about the spectrum and the behavior of $A$. In this first lecture, we will review the basic properties of the numerical range and discuss some results about its convexity and the geometry of its boundary.
This talk focuses on the basic ideas of what a quantum Levy process is and some of the concepts that are associated with them. We will cover (in outline form) bi-algebras, quantum probability spaces and the quantum Levy processes themselves. Then we will talk about the Schurmann triple, named after M. Schurmann, who in the 1980's, developed the fundamental ideas.
This talk will be aimed at an audience familiar with operator algebras and will be very introductory in nature
Motivated by the properties of geometric operators and their relations with the algebra of smooth functions, spectral triples are introduced to make it possible to define geometry for noncommutative spaces. In this talk we will have an example oriented introduction to this notion.
Despite of very geometric form of the Gauss-Bonnet theorem,
it turns out that the theorem is indeed the index theorem of
a geometric elliptic differential operator that can be defined on
Riemannian manifolds. The goals of this series of lectures are
to see different versions of this theorem, how it has evolved to
be an index theorem and what is the operator whose index is
computed by this theorem. Later on, we will show that the
Gauss-Bonnet can be proved as a corollary of the Atiyah-Singer
index theorem.
The theory of vector bundles over topological spaces
provides a method to study the topology of the base
space. This is made possible by the theory of characteristic
classes. In the case of vector bundles on smooth manifolds,
Chern-Weil theory gives a computational recipe by which the
curvature two form of connections is used to compute these
classes. In this session of our seminar we will review the
Chern-Weil theory. We are going to follow very closely the
appendix C of the J. Milnor & J. Stasheff's fantastic book
"Characteristic Classes".
The theory of vector bundles over topological spaces provides a method
to study the topology of the base. This is made possible by the theory
of characteristic classes. In the case of vector bundles on smooth
manifolds, Chern-Weil theory gives a computational recipe by which the
curvature two form of connections is used to compute these classes. In
this session (and the next one) of our seminar we will review the
Chern-Weil theory. We are going to follow very closely the appendix C of
the J. Milnor & J. Stasheff's fantastic book "Characteristic Classes".
This is a quick introduction to the theory of vector bundles
on topological spaces. We will go over some constructions like
vector bundles defined by clutching functions and hence classify
all vector bundles on spheres. We will also discuss how a proper
notion of vector bundles for algebras and C*-algebras can
be developed.
Please note that the approach to the topic in this series of lectures
can be considered as a complementary approach to the one taken in
the "characteristic classes seminar" and, therefore, the participants
of that seminar can also benefit from these sessions.
This is a quick introduction to the theory of vector bundles
on topological spaces. We will go over some constructions like
vector bundles defined by clutching functions and hence classify
all vector bundles on spheres. We will also discuss how a proper
notion of vector bundles for algebras and C*-algebras can
be developed.
Please note that the approach to the topic in this series of lectures
can be considered as a complementary approach to the one taken in
the "characteristic classes seminar" and, therefore, the participants
of that seminar can also benefit from these sessions.
This is a quick introduction to the theory of vector bundles
on topological spaces. We will go over some constructions like
vector bundles defined by clutching functions and hence classify
all vector bundles on spheres. We will also discuss how a proper
notion of vector bundles for algebras and C*-algebras can
be developed.
Please note that the approach to the topic in this series of lectures
can be considered as a complementary approach to the one taken in
the "characteristic classes seminar" and, therefore, the participants
of that seminar can also benefit from these sessions.
With the Sobolev theory tools in hand, we are now ready to consider the theory of
elliptic differential operators on a compact manifold and show how they define
(bounded) Fredholm operators on carefully chosen Sobolev spaces.
To accomplish that, a pseudo-inverse of an elliptic operator will be constructed
using the theory of pseudodifferential operators. Moreover, we will show that the
index of all Fredholm operators defined by an elliptic operator on different Sobolev
spaces, are equal and it does make sense to look for a formula for the (Fredholm)
index of elliptic operators.
With the Sobolev theory tools in hand, we are now ready to consider the theory of
elliptic differential operators on a compact manifold and show how they define
(bounded) Fredholm operators on carefully chosen Sobolev spaces.
To accomplish that, a pseudo-inverse of an elliptic operator will be constructed
using the theory of pseudodifferential operators. Moreover, we will show that the
index of all Fredholm operators defined by an elliptic operator on different Sobolev
spaces, are equal and it does make sense to look for a formula for the (Fredholm)
index of elliptic operators.
With the Sobolev theory tools in hand, we are now ready to consider the theory of
elliptic differential operators and show how they define (bounded) Fredholm operators
on carefully chosen Sobolev spaces. To accomplish that, a pseudo-inverse of an
elliptic operator will be constructed using the theory of pseudodifferential operators.
Moreover, we will show that the index of all Fredholm operators defined by an elliptic
operator on different Sobolev spaces, are equal and it does make sense to look for
a formula for the (Fredholm) index of elliptic operators.
We introduce Sobolev spaces and discuss their most important properties.
This introductory lecture is accessible to both undergraduate and graduate students.
In the first of this multi-part learning seminar, we will explore some of the defining analytic properties of commutative Levy processes with a focus on the relationship between the induced measure, characteristic function, the process its self and the respective generating triplet. The talk will culminate with a discussion on the famed Levy-Khintchin theorem on infinitely divisible measures.
We will discuss the main machinery used in the proof and some of the consequences of the theorem.
As this is a learning seminar, the only back ground required is basic probability theory and some function theory. If you have an interest in the subject or a desire to learn about a fascinating topic, feel free to join our seminar.
One final note: our background in mainly analysis and the focus of this seminar is on the analytic and not the probabilistic properties of Levy processes. Anyone with a probabilistic background is invited to share their knowledge with us.
The Toeplitz operators form an important class of operators
which has been studied in different settings. The index
theory for the Fredholm Toeplitz operators, which was worked
out in fifties, is a very good example of what an index theorem is.
One can compute the index of a Toeplitz operator by finding the
winding number of its symbol rather than finding the dimension
of kernel and cokernel. We will study the topic in details.
If time permits, we will also try to study main components of elliptic
differential operator theory through an example.
The theory of pseudospectra is a new tool for studying matrices and linear operators. The traditional tool is the spectrum. It reveals information on the behavior of normal matrices or operators. However, it is less informative as the matrix or the operator is non-normal. Pseudospectra have nevertheless proved to be a powerful tool to study them. In this third lecture, we will proof the Kreiss Matrix Theorem for analytic functions on the unit disc. We will discuss also the behavior of a matrix when we assume an exact knowledge of the resolvent norm.
The Toeplitz operators is an important class of operators on
$l^2(S^1)$ which has been studied in different settings. The index
theory for the Fredholm Toeplitz operators, which was worked
out in fifties, is a very good example of what an index theorem is.
One can compute the index of a Toeplitz operator by finding the
winding number of its symbol rather than finding the dimension
of kernel and cokernel. We will study the topic in details.
The theory of pseudospectra is a new tool for studying matrices and linear operators. The traditional tool is the spectrum. It reveals information on the behavior of normal matrices or operators. However, it is less informative as the matrix or the operator is non-normal. Pseudospectra have nevertheless proved to be a powerful tool to study them. In this second lecture, we will discuss the Kreiss Matrix Theorem and its generalization in the space of analytic functions on the unit disc.
This is an introduction to the theory of Fredholm operators and their index theory.
We will cover definition and some examples, important properties of Fredholm
operators (on abstract Hilbert spaces), as well as important properties of the index
map.
The theory of pseudospectra is a new tool for studying matrices and linear operators. The traditional tool is the spectrum. It reveals information on the behavior of normal matrices or operators. However, it is less informative as the matrix or the operator is non-normal. Pseudospectra have nevertheless proved to be a powerful tool to study them. In this first lecture, we present some properties of the pseudospectra. We will discuss also the Kreiss Matrix Theorem and interesting examples.
The Atiyah-Singer index theorem is in the very heart of the
modern mathematics and physics. It was proved by Michael
Atiyah and Isadore Singer at 1963 and was quickly transformed
to a vast research area; different proofs, among which one can
mention the heat kernel proof, were presented and different
new theories were developed and many applications in both
mathematics and physics were discovered. The theorem
provides a formula for the (Fredholm) index of elliptic operators
in terms of topological invariants of the manifolds and vector
bundles.
As S. S. Chern once said “Even there is no research results,
it is worthwhile to study the Atiyah-Singer index theorem”.
This series of learning seminar will be an introduction to
index theory and the related topics from which graduate students
from different fields can benefit.
Integration of measurable operator-valued functions (aka quantum random variables) with respect to operator-valued measures leads to a notions of expected value and variance for integrable operator-valued functions. In this fourth of a series of lectures I will discuss the geometry of the space of all quantum probability measures and characterise the extreme points of this space.
As is well known from basic measure theory, Fubini's Theorem provides a technique to generate a measure on the product space of a finite number of measure spaces, but what can be done if one has an infinite collection measure spaces? For instance, if one takes
\[
X=\prod_{i=1}^{\infty} [0,1],
\]
how does one go about constructing a measure space involving $X$?
One technique, due to Kolmogorov and Daniell involves the so-called Kolmogorov-Daniell extension theorem. In this talk, we will provide a detailed discussion of the statement and proof of this amazing theorem which is of much interest both to probability and in functional analysis and anyone working with stochastic processes. This talk will be (largely) self contained and will only assume a basic knowledge of measure theory.
Integration of measurable operator-valued functions (aka quantum random variables) with respect to operator-valued measures leads to a notions of expected value and variance for integrable operator-valued functions. In this third of a series of lectures I will introduce quantum variance and discuss some of its properties. The use of nontrivial operator inequalities enters the discussion in a rather elegant way.
Integration of measurable operator-valued functions (aka quantum random variables) with respect to operator-valued measures leads to a notions of expected value and variance for integrable operator-valued functions. In this second of a series of lectures I will introduce quantum expectation and discuss some of its properties.
Integration of measurable operator-valued functions (aka quantum random variables) with respect to operator-valued measures leads to a notions of expected value and variance for integrable operator-valued functions. In this first of a series of lectures I will address some of the operator-theoretic concepts (eg, spectra) that are associated to quantum random variables.
We will discuss some quantum information basics and then consider the algebraic bridge between private quantum subsystems and quantum error correction.
Slides
In this presentation we'll review Ando's theorem which states every commuting pair of contractions dilates to a commuting pair of unitaries and how an analogous result does not hold for three or more commuting contractions. We'll move on to the more general result given by Opela, which states that an n-tuple of contractions that commute according to a graph can be dilated to an n-tuple of unitaries that commute to that same graph.
By Ando's theorem, any pair of commuting contractions on a Hilbert
space can be extended to a pair of commuting unitaries on a superspace.
The analogous result does not hold for an n-tuple of commuting
contractions where n > 2; however, the result may be generalized when
the n-tuple of contractions commute according to a graph, extending to
unitaries that commute according to that graph. In this talk we go
through some preliminaries for Ando's theorem, and then Ando's theorem
itself and a counterexample for the general case.
The Baum-Connes conjecture for a group G states that a certain "topological" formula should yield the K-theory groups of the reduced group C*-algebra of G. It is known to be true for large classes of groups and it has several important corollaries, notably in algebra and geometry. It is however quite hard to prove in the known cases, and even to formulate precisely.
In this talk, after some motivation, I will introduce the notion of G-equivariant KK-theory (by its universal property, as discovered by Higson, Thomsen and Meyer) and explain that it forms a tensor triangulated category. The language of triangulated categories will allow us to give a short, precise and conceptual formulation of the conjecture, which is due to Meyer and Nest.
This is an introductory talk about K-theory, as used in the study of C*-algebras. I will begin by defining the K-theory groups of a C*-algebra and recalling their basic but nontheless remarkable properties. Armed with this, we can proceed to describe some early successes of the theory, such as Elliott's classification of AF-algebras, as well as to touch upon one or two more advanced topics like the Baum-Connes conjecture and Kasparov's KK-theory. The goal is to illustrate how K-theory allows for a most fruitful exchange of ideas between operator theory, algebra, and topology.
The generalization of classical random variables to operator valued random variables, or quantum random
variables, has applications to mathematical models of quantum experiments. It also provides an interesting gener-
alization of classical probability since one looses commutativity when working with operators. In order to establish
probabilistic results in the quantum setting the notion of an operator valued integral, a positive operator valued
probability measure, the principal Radon Nikod´ym derivative, and the non-principal Radon Nikod´ym derivative
will be introduced. These results will be used, along with a de?nition of a quantum martingale, to prove a quantum
martingale convergence theorem holds for a speci?c quantum martingale.
Davidson and Kennedy constructed boundary representations by taking maximal dilations of pure ucp maps. Moreover, they also proved the existence of the NC choquet boundary. This talk will focus on these results.
A sufficient condition for the existence of the C*-envelope is the existence of the Non-commutative Choquet boundary. In 2006, Arveson showed that for a separable operator system S, the Choquet boundary exists. In this talk, we will discuss the results from Arveson.
In 1969, Arveson suggested that for an operator algebra A, there should exist the C*-envelope, that is, the smallest C*-algebra that contains A completely isometrically isomorphically. His idea required the existence of a special ideal in C*(A), the Šilov ideal. He couldn't, however, prove its existence; Hamana did some 10 years later, using very different techniques. Dristchell and Mccullough provided a new proof of the existence of the C*-envelope using $\partial$-representations whereas Arveson used maximal dilations. In this talk we will discuss the results from Arveson and Dristchell and Mccullough’s papers.
To each semigroup of isometries on a Hilbert space $H$, we
associate a semigroup of *-endomorphisms acting on the CAR algebra over
$H$ via the second quantization. We then discuss the possibility of
extending this semigroup to an $E_0$-semigroup of the von Neumann
algebra generated by the GNS representation of the CAR algebra with
respect to a quasi-free state.
In 2009, M. Izumi and R. Srinivasan constructed an
uncountable family of type III non-cocycle conjugate $E_0$-semigroups by
using certain representations of the CAR algebra. It is our purpose to
review this construction in a series of lectures. In the first lecture,
we will discuss some basic properties of the CAR algebra and of its
state space.
We discuss some properties of the class of multiplier Hopf algebras, and introduce the concepts of algebraic quantum groups and algebraic compact quantum groups.
We start by presenting several examples of Hopf algebras. We then discuss the main properties of the antipode of a Hopf algebra, and introduce the concepts of involutive Hopf algebras and multiplier Hopf algebras.
Locally compact quantum groups are mathematical structures that generalize, in an operator algebraic framework, the concepts of quantum group, locally compact group and Kac algebra. It is our purpose, in this lecture series, to discuss their main properties. In the first lecture, we will review the concepts of co-algebra, bi-algebra and Hopf algebra, and discuss several examples.
In this final lecture of the series, I will consider operators system on the generators of the n-fold free product of the cyclic group of order two and use their operator system tensor products to recast Tsirelson's problem on quantum correlations as a functional-analytic problem.
A quantum probability measure is said to be clean if it cannot be irreversibly connected to any other quantum probability measure via a quantum channel. The notion of a clean quantum measure was introduced by Buscemi et al (2005) for finite-dimensional Hilbert space, and variations of the original meaning of "clean" were studied subsequently by Kahn (2007) and Pellonpää (2011). In joint work with Remus Floricel and Sarah Plosker, we give new descriptions of clean quantum probability measures in the case of finite-dimensional Hilbert space. For Hilbert spaces of infinite dimension, we introduce the notion of "approximately clean quantum probability measures" and characterise this property for measures whose range determines a finite-dimensional operator system.
These lectures report on joint work with A. Kavruk, V. Paulsen, I.G. Todorov in which we formulate a general framework for the study of operator systems arising from discrete groups. We study in detail the operator system of finitely generated free groups, as well as the operator systems of the free products of finitely many copies of the group with two elements. We express various sets of quantum correlations studied in the theoretical physics literature in terms of different tensor products of operator systems of discrete groups. We thus recover earlier results of Tsirelson and formulate a new approach for the study of quantum correlation boxes. In particular, we show that to settle the bipartite Tsirelson problem, it is sufficient to study the case of three 2-outcome experiments.
In this fourth lecture I will discuss minimal and maximal tensor product
structures on the tensor product of the operator system generated by $n$
universal unitaries with itself
These lectures report on joint work with A. Kavruk, V. Paulsen, I.G. Todorov in which we formulate a general framework for the study of operator systems arising from discrete groups. We study in detail the operator system of finitely generated free groups, as well as the operator systems of the free products of finitely many copies of the group with two elements. We express various sets of quantum correlations studied in the theoretical physics literature in terms of different tensor products of operator systems of discrete groups. We thus recover earlier results of Tsirelson and formulate a new approach for the study of quantum correlation boxes. In particular, we show that to settle the bipartite Tsirelson problem, it is sufficient to study the case of three 2-outcome experiments.
In this third lecture I will introduce a variety of tensor product structure in the category of operator systems, define the basic objects of study (namely operator systems from discrete groups), and state the main problems to be addressed.
These lectures report on joint work with A. Kavruk, V. Paulsen, I.G. Todorov in which we formulate a general framework for the study of operator systems arising from discrete groups. We study in detail the operator system of finitely generated free groups, as well as the operator systems of the free products of finitely many copies of the group with two elements. We express various sets of quantum correlations studied in the theoretical physics literature in terms of different tensor products of operator systems of discrete groups. We thus recover earlier results of Tsirelson and formulate a new approach for the study of quantum correlation boxes. In particular, we show that to settle the bipartite Tsirelson problem, it is sufficient to study the case of three 2-outcome experiments.
In the second lecture, I will further develop the main properties of the category of operator systems and ucp maps by considering: (i) quotients, (ii) duality of finite-dimensional operator systems, and (iii) tensor products.
These lectures report on joint work with A. Kavruk, V. Paulsen, I.G. Todorov in which we formulate a general framework for the study of operator systems arising from discrete groups. We study in detail the operator system of finitely generated free groups, as well as the operator systems of the free products of finitely many copies of the group with two elements. We express various sets of quantum correlations studied in the theoretical physics literature in terms of different tensor products of operator systems of discrete groups. We thus recover earlier results of Tsirelson and formulate a new approach for the study of quantum correlation boxes. In particular, we show that to settle the bipartite Tsirelson problem, it is sufficient to study the case of three 2-outcome experiments.
In the first lecture, we present an introduction to abstract operator systems.
The Schur-Horn Theorem is an interesting result in matrix analysis that relates majorization and diagonals of
selfadjoint operators. In this talk we will explore several infinite-dimensional generalizations of the original result, mostly due
to A. Neumann, R. Kadison, and W. Arverson.
The study of directed graphs plays a pivotal role in the construction and analysis of C*-algebras describing certain C*-algebras in terms of easily manageable and computable relations. Much is already known of graph C*-algebras, but as graphs are finite, this class of C*-algebras lacks the ability to capture many classical C*-algebras; for instance, continuous function on a given compact Hausdorff space. In 2005, Muhly and Tomforde defined the notion of topological quivers which
generalize the notion of directed graphs in which the sets of vertices and edges are locally compact (second countable) Hausdorff spaces. Associated to a topological quiver Q is a C*-correspondence, and in turn, a Cuntz-Pimsner algebra C*(Q). Given a locally compact group, G, and $\alpha$ and $\beta$ endomorphisms on said group, one may construct a topological quiver with vertex set G, and edge set
$$\Omega_{\alpha,\beta}(G)=\{(x,y)\in G\times G:\ \alpha(y)=\beta(x)\}.$$ We shall examines the Cuntz-Pimsner algebra
$\mathcal O_{\alpha,\beta}(G)$ generated by this topological quiver. In his 2012 PhD thesis, the speaker investigated a notion for topological quiver isomorphisms, generators (and their relations) of the C*-algebras $\mathcal O_{\alpha,\beta}(G)$, its spatial structure, K-groups, simplicity and lattice structure. In this series of talks, these results will be shared with emphasis on the background material.
Talk #3: We explore the properties of C$^*$-algebras associated with topological d-torus quivers
The study of directed graphs plays a pivotal role in the construction and analysis of C*-algebras describing certain C*-algebras in terms of easily manageable and computable relations. Much is already known of graph C*-algebras, but as graphs are finite, this class of C*-algebras lacks the ability to capture many classical C*-algebras; for instance, continuous function on a given compact Hausdorff space. In 2005, Muhly and Tomforde defined the notion of topological quivers which
generalize the notion of directed graphs in which the sets of vertices and edges are locally compact (second countable) Hausdorff spaces. Associated to a topological quiver Q is a C*-correspondence, and in turn, a Cuntz-Pimsner algebra C*(Q). Given a locally compact group, G, and $\alpha$ and $\beta$ endomorphisms on said group, one may construct a topological quiver with vertex set G, and edge set
$$\Omega_{\alpha,\beta}(G)=\{(x,y)\in G\times G:\ \alpha(y)=\beta(x)\}.$$ We shall examines the Cuntz-Pimsner algebra
$\mathcal O_{\alpha,\beta}(G)$ generated by this topological quiver. In his 2012 PhD thesis, the speaker investigated a notion for topological quiver isomorphisms, generators (and their relations) of the C*-algebras $\mathcal O_{\alpha,\beta}(G)$, its spatial structure, K-groups, simplicity and lattice structure. In this series of talks, these results will be shared with emphasis on the background material.
Talk #2: We define topological group quivers and explore many interesting examples including crossed products.
The study of directed graphs plays a pivotal role in the construction and analysis of C*-algebras describing certain C*-algebras in terms of easily manageable and computable relations. Much is already known of graph C*-algebras, but as graphs are finite, this class of C*-algebras lacks the ability to capture many classical C*-algebras; for instance, continuous function on a given compact Hausdorff space. In 2005, Muhly and Tomforde defined the notion of topological quivers which
generalize the notion of directed graphs in which the sets of vertices and edges are locally compact (second countable) Hausdorff spaces. Associated to a topological quiver Q is a C*-correspondence, and in turn, a Cuntz-Pimsner algebra C*(Q). Given a locally compact group, G, and $\alpha$ and $\beta$ endomorphisms on said group, one may construct a topological quiver with vertex set G, and edge set
$$\Omega_{\alpha,\beta}(G)=\{(x,y)\in G\times G:\ \alpha(y)=\beta(x)\}.$$ We shall examines the Cuntz-Pimsner algebra
$\mathcal O_{\alpha,\beta}(G)$ generated by this topological quiver. In his 2012 PhD thesis, the speaker investigated a notion for topological quiver isomorphisms, generators (and their relations) of the C*-algebras $\mathcal O_{\alpha,\beta}(G)$, its spatial structure, K-groups, simplicity and lattice structure. In this series of talks, these results will be shared with emphasis on the background material.
Talk #1: We will discuss background material including graph C$^*$-algebras, Hilbert modules, and topological quivers.
Some 40 years ago, W.B. Arveson
introduced notions of Šilov boundary and Choquet boundary for algebras of
operators acting on complex Hilbert spaces. These ideas were
inspired by the use of Šilov and Choquet boundaries in function theory,
and Arveson sought to develop a theory of similar importance for operator algebras.
In the forty years since the initiation of this program, there has been continued interest in and applications of
this line of thought.
Today, Arveson's notions of noncommutative Šilov boundary (and the related C$^*$-envelope) and noncommutative Choquet boundary
are commonplace in nonselfadjoint operator algebras and in the theory of operator spaces and operator systems.
Fifth lecture: We will discuss concrete examples of C$^*$-envelopes in the abelian and finite-dimensional cases.
Some 40 years ago, W.B. Arveson
introduced notions of Šilov boundary and Choquet boundary for algebras of
operators acting on complex Hilbert spaces. These ideas were
inspired by the use of Šilov and Choquet boundaries in function theory,
and Arveson sought to develop a theory of similar importance for operator algebras.
In the forty years since the initiation of this program, there has been continued interest in and applications of
this line of thought.
Today, Arveson's notions of noncommutative Šilov boundary (and the related C$^*$-envelope) and noncommutative Choquet boundary
are commonplace in nonselfadjoint operator algebras and in the theory of operator spaces and operator systems.
Fourth lecture: We will discuss concrete examples of C$^*$-envelopes.
Some 40 years ago, W.B. Arveson
introduced notions of Šilov boundary and Choquet boundary for algebras of
operators acting on complex Hilbert spaces. These ideas were
inspired by the use of Šilov and Choquet boundaries in function theory,
and Arveson sought to develop a theory of similar importance for operator algebras.
In the forty years since the initiation of this program, there has been continued interest in and applications of
this line of thought.
Today, Arveson's notions of noncommutative Šilov boundary (and the related C$^*$-envelope) and noncommutative Choquet boundary
are commonplace in nonselfadjoint operator algebras and in the theory of operator spaces and operator systems.
Third lecture: we will discuss C$^*$-envelopes and Arveson's Šilov theory.
Some 40 years ago, W.B. Arveson
introduced notions of Šilov boundary and Choquet boundary for algebras of
operators acting on complex Hilbert spaces. These ideas were
inspired by the use of Šilov and Choquet boundaries in function theory,
and Arveson sought to develop a theory of similar importance for operator algebras.
In the forty years since the initiation of this program, there has been continued interest in and applications of
this line of thought.
Today, Arveson's notions of noncommutative Šilov boundary (and the related C$^*$-envelope) and noncommutative Choquet boundary
are commonplace in nonselfadjoint operator algebras and in the theory of operator spaces and operator systems.
Second lecture: we will discuss the basics of Arveson's Choquet theory.
Some 40 years ago, W.B. Arveson
introduced notions of Šilov boundary and Choquet boundary for algebras of
operators acting on complex Hilbert spaces. These ideas were
inspired by the use of Šilov and Choquet boundaries in function theory,
and Arveson sought to develop a theory of similar importance for operator algebras.
In the forty years since the initiation of this program, there has been continued interest in and applications of
this line of thought.
Today, Arveson's notions of noncommutative Šilov boundary (and the related C$^*$-envelope) and noncommutative Choquet boundary
are commonplace in nonselfadjoint operator algebras and in the theory of operator spaces and operator systems.
First lecture: we will discuss the commutative (i.e. classic) case and basic definitions and results for the non-commutative case.
Private quantum channels are a basic tool in quantum key distribution and quantum cryptography. We will review their mathematical definition and explore conjugate quantum channels. Also called complementary channels, these channels allow us to set out algebraic conditions that characterize when a general quantum code is private for a quantum channel. These conditions can be regarded as the private analogue of the Knill-Laflamme conditions for quantum error correction.
This lecture will introduce the notion of Markov map and study examples and properties of such maps. I will also survey some results of Musat and Haggerup that explain the relationship between Markov maps and the Connes Embedding Problem and the Asymptotic Birkhoff Conjecture.
The study of unit-preserving *-endomorphisms of the von Neumann algebra $\mathcal{B}(H)$ of all bounded linear operators on a Hilbert space $H$ has seen substantial progress over the last 20 years, in connection with many other fields of research, particularly with the representation theory of the Cuntz algebras. The work of Arveson, Powers, Laca, Bratteli, Jorgensen and others has emphasized the importance of the class of shift endomorphisms, i.e., endomorphisms with trivial tail algebras. It is our purpose, in this presentation, to introduce a larger class of endomorphisms, called quasi-shifts, and to discuss its relation with the class of shift endmorphisms, as well as various asymptotic properties. This is joint work with R. Floricel.
Quantum dynamical semigroups are pointwise $\sigma$-weak continuous semigroups of unital completely positive maps acting on von Neumann algebras. In this presentation, we discuss a general procedure for dilating a quantum dynamical semigroup to a group of *-automorphisms. Joint work with R. Floricel and T. Zhang.
(click here for the slides)
While operator spaces are determined by their matrix norm structures,
operator systems are determined by their order structures. This leads to
some significant differences. For example, the matrices are not self-dual as
normed objects, but they are self-dual as an operator system. We will then
explain quotients and how to use quotients to describe the duals of the
operator systems of graphs considered by Winter, et al. We will then
introduce tensor products of operator systems and show their relationship to
understanding duals. Finally, we will present recent results that show that
Connes embedding problem is equivalent to deciding whether or not two
operator system structures on a 16 dimensional space are equal.
In a series of lectures, we look at the structure of operator systems on finite-dimensional Hilbert spaces, using arveson's noncommutative choquet
theory, and give some examples. Fourth Lecture: 2-dimensional and 3-dimensional examples, plus
anything not covered in the second lecture.
In a series of lectures, we look at the structure of operator systems on finite-dimensional Hilbert spaces, using arveson's noncommutative choquet
theory, and give some examples. Third Lecture: Parametric representation of reduced and non-reduced operator systems,
Invariance of structural triples and completeness of the invariant
In a series of lectures, we look at the structure of operator systems on finite-dimensional Hilbert spaces, using arveson's noncommutative choquet
theory, and give some examples. Second lecture: Irreducible operator systems: proof of the Boundary Theorem.
.
In a series of lectures, we look at the structure of operator systems on finite-dimensional Hilbert spaces, using arveson's noncommutative choquet
theory, and give some examples. First lecture: classical Šilov and
Choquet boundaries, definition of noncommutative Šilov and Choquet
boundaries, and C*-envelope; the
structure of complete order isomorphisms of operator systems in the
finite-dimensional case.
In a trio of seminal papers in the early 1990s, Kirchberg demonstrated that the Connes Embedding Problem, which is possibly the most important open problem in operator algebra at present, is logically equivalent to what we now call the Kirchberg Problem: does the C*-algebra of a free group have the weak expectation property? In this lecture, based on joint work with Vern Paulsen (Houston), I will explain some new approaches to the Kirchberg Problem using the theory of operator systems.
We describe the reversible part of an abelian relatively weakly
compact semitopological semigroup consisting of bounded linear operators
on a Banach space in terms of the eigenspace of the semigroup.
We show that every compact semitopological semigroup admits a
minimal ideal, referred to as the Sushkevich kernel of the semigroup.
This ideal has the structure of a compact topological group. We use this
theorem in the particular case of an abelian relatively weakly compact
semigroup of operators on a Banach space to obtain a natural decomposition
of the space, and of the semigroup.
(click here for the slides)
It has been unknown to this day if $\mathrm{AP}(\hat{G})$
and $\mathrm{WAP}(\hat{G})$, the spaces of all almost and weakly
almost periodic functionals, respectively, on the Fourier algebra
$A(G)$ of a locally compact group $G$ is a $C^\ast$-subalgebrebra of
the group von Neumann algebra $\mathrm{VN}(G)$, except in a few cases
such as if $G$ is almost abelian or both discrete and amenable.
In his Diplomarbeit of 1982, under the supervision of G. Wittstock,
H. Saar introduced the notion of complete compactness---a variant of
compactness that takes operator space structures into account---,
which, in turn, enables us to define the notion of completely almost
periodic functionals on completely contractive Banach algebras.
We will show that, for a Hopf-von Neumann algebra $(M,\Gamma)$ with
$M$ injective, the space of all completely almost periodic
functionals on the completely contractive Banach algebra $M_\ast$
forms a $C^\ast$-subalgebra of $M$.
We also discuss whether a similar result might hold for the weakly
almost periodic functionals.
I will introduce a notion of operator-valued integral that induces a completely positive linear map on an n-homogeneous C*-algebra A with values in a factor of type I_n. In so doing, one obtains a noncommutative Riesz-type Representation Theorem for certain ucp maps on A. My lecture will begin with a review of the Riesz Theorem and measure-theoretic probability.
In the final installment of this presentation we shall see
the proof of the equality of the Haagerup constants for a C*-algebra
and its crossed product by a discrete amenable group.
The aim is to present the paper by Sinclair-Smith with the
above mentioned title. I will introduce the concept of Crossed
products first and then proceed with the contents of the paper.
The purpose of this presentation is to give a brief introduction
to the relationship between representations of Cuntz algebras and
*-endomorphisms of the algebra of all bounded linear operators on a
separable Hilbert space. Detailed proofs are omitted and focus is placed
on the main results.
The purpose of this presentation is to give a brief introduction
to the relationship between representations of Cuntz algebras and
*-endomorphisms of the algebra of all bounded linear operators on a
separable Hilbert space. Detailed proofs are omitted and focus is placed
on the main results.
While the C* and W* theories usually run their separate lives, some of the deepest results on C*-algebras use W*-techniques in an essential way. We will review some of these results and a few of these techniques. In this third talk we will consider the special case of group algebras.
While the C* and W* theories usually run their separate lives, some of the deepest results on C*-algebras use W*-techniques in an essential way. We will review some of these results and a few of these techniques. In this second talk we will continue exploring the relation between tensor norms, nuclearity, complete positivity, and injectivity.
While the C* and W* theories usually run their separate lives, some of the deepest results on C*-algebras use W*-techniques in an essential way. We will review some of these results and a few of these techniques. In the first talk we will explore the connection between nuclearity and complete positivity.
In this talk we will define the projective and injective C*-norms for the algebraic tensor products of two C*-algebras and then show that they coincide when one of the C*-algebras in consideration is commutative.
In this talk we will study the similarity problem for locally compact groups i.e. is every uniformly bounded continuous representation on a locally compact group $G$ similar to a unitary continuous representation on $G$? We show the Dixmier's theorem which proves that for amenable locally compact groups the answer is yes. On the other hand, we prove that there exists a uniformly bounded continuous representation on $\Bbb{F}_\infty$ (non-abelian free group generated with infinitely countable generators) which is not similar to a unitary representation. This is an interesting contrast because $\Bbb{F}_\infty$ is not amenable.
After recalling some properties of the injective envelope of an operator system, we will show the existence of the C*-envelope, which is the minimal (in the appropriate sense) C*-algebra generated by the operator system. Afterwards we will mention briefly whatever little is known about injective envelopes of C*-algebras.
We will recall some basic properties of injectivity, and then we will prove the existence of the injective envelope. As a particular consequence we will deduce the existence of the C*-envelope of an operator system.
Following on the previous material, we will prove Arveson's Extension Theorem. Motivated by it, we will define and begin a discussion on Injectivity, where the first goals are to consider the injective envelope and the C*-envelope of an operator system.
We will first recap the results on complete positivity from the previous talk a few weeks ago. Then we will consider completely positive maps in the finite-dimensional setting, and use the information gathered to prove Arveson's Extension Theorem. This will lead us into Injectivity, which is the topic of the coming talks.
The notion of complete positivity in C*-algebras plays a crucial role in many parts of the theory. We will introduce the notion of complete positivity and prove several of its main properties.
In the early 1990s, Elliott launched an ambitious program to use K-theory to classify, up to isomorphism, all unital, simple, amenable, separable C*-algebras. I will indicate what the K-theoretic invariants are and describe what is perhaps the most profound contribution in the classification program: a theorem of Kirchberg and Phillips. Lastly, I will remark on the present status of the program.
I will explain how homotopy equivalence of unitaries in matrix algebras over a C$^*$-algebra $A$ translates into equivalence of projections in matrix algebras over the suspension of $A$, leading to a proof that the $K$-groups $K_1(A)$ and $K_0(SA)$ are isomorphic.
We show that the scaled ordered $K_0$-group is a complete isomorphism invariant among AF algebras, and discuss the Effros-Handelman-Shen characterization of this invariant.
The Elliot classification progam is an attempt to find a complete isomorphism invariant for a class of simple nuclear $C^*$-algebras. The purpose of this expository lecture is to introduce and describe the Elliott invariant of a $C^*$-algebra $A$, which is the triple $(K_0(A), K_1(A), T(A))$, where $K_*(A)$ are the $K$-groups of $A$, and $T(A)$ is the set of tracial states of $A$.
We show that the conjugacy class of a pure $E_0$-semigroup $\rho=\{\rho_t\,|\,t\geq0\}$ acting on $\mathfrak{B}(H)$ can be completely described by a class of unitary $\rho$-cocycles, called balanced cocycles.
Introduced by Greenberg, and Bozejko and Speicher, the $q$-commutation relation over a Hilbert space $H$, $$c(f)c(g)^*-qc(g)^*c(f)=f, g>1,\;\;f,\,g\in H,$$ where $-1 \leq q\leq 1$, provides an interpolation between the fermionic($q=-1$) and bosonic ($q=1$) relations via the Cuntz relation $(q=0)$. Our purpose in this talk is to describe the Bozejko-Speicher realization of the $q$-commutation relation on $q$-Fock spaces.
We will define and characterize completely positive maps into $M_n$. Thischaracterization allows us to prove several theorems, including Arveson's Extension Theorem for completely positive maps. We will also look at several properties of positive maps on operator systems. The theorems presented are consequences of a duality between maps into $M_n$ and linear functionals, which will be explained.
Thursday, Dec 17 2009 10:30-12:00
CL508
Michael Skeide Universita degli studi del Molise, Campobasso
In (classical and) quantum dynamics dilation of an irreversible evolution to a reversible one means understanding macroscopic behaviour as a mean (due to incomplete knowledge, say) of an evolution on a microscopic level. Quantum dynamics models such evolutions on C* or von Neumann algebras. In the classification of irreversible and reversible quantum dynamical systems and in the construction of dilations, product systems of Hilbert (bi)modules play a more and more outstanding role on all stages. The classical cases are contained in the descritpion when the algebras are commutative. In particular, bimodules over commutative algebras are an unlimited source for counter expamples for what one expects from the Hilbert space theory of 'standard' quantum mechanics. We will try, in a relaxed way, to explain some of these issues.
Let $(\Omega,\Sigma,\mu)$ be a probability measure space, and if $\nu$ be an operator convex function. We show that, under suitable conditions, $\nu\big(\int_{\Omega} g^*fg\ d\mu\big)\leq \int_{\Omega}g^*\nu \circ f g\ d\mu$, where $f:\Omega\rightarrow B(H)^{sa}$ is assumed to be Bochner integrable and $g:\Omega\rightarrow B(H)$ is a measurable function with $\int_{\Omega}g^*g\ d\mu={\bf 1}$. * Joint work with F. Bahrami and Ali Bayati.
In this talk we introduce a Jensen's type inequality for operator-valued integrable functions which generalizes some of the previous results in this regard. More precisely, if $(\Omega,\Sigma,\mu)$ is a probability measure space and if $\nu$ is an operator convex function then, under suitable conditions, we show that $\nu\big(\int_{\Omega} g^*fg\ d\mu\big)\leq \int_{\Omega}g^*\nu\circ f g\ d\mu$, where $f:\Omega\rightarrow B(H)^{sa}$ is assumed to be Bochner integrable and $g:\Omega\rightarrow B(H)$ is a measurable function with $\int_{\Omega}g^*g\ d\mu={\bf 1}$. * Joint work with F. Bahrami and Ali Bayati.
Assume that A is a self-adjoint algebra, not necessarily norm closed, of Hilbert space operators, and that M is a semifinite von Neumann algebra equipped with faithful normal trace. Two tracial representations \rho and \pi of A in M are said to be tracially equivalent if \rho (a) and \pi (a) have the same trace for all a in A. A notion of rank equivalence in this framework is introduced. This talk examines these notions of equivalence and several results are obtained. In particular, we prove that the notions of tracial equivalence and rank equivalence coincide if M is a factor of type I_\infty or a factor of type II_1. Finally, if we assume that M is a finite von Neumann algebra acting on a separable Hilbert space, with normal faithful center-valued trace, then we prove that tracial equivalence implies rank equivalence. The converse is also true under the assumption that A is a separable, unital C*-algebra.
We will show that Wigner matrices are asymptotically free from constant matrices. The proof relies on an upper bound for the norm of a linear functional on matrices which is given by a graph. This is joint work with Roland Speicher.
In the last lecture I described how continuous and weakly continuous Hilbert bundles over Stonean spaces lead to Hilbert modules over abelian algebras. I closed with Kasparov's Theorem, which computes the multiplier algebra of $K(E)$, for any Hilbert C$^*$-module $E$. In this week's lecture I will explain what the multiplier and local multiplier algebras of $K(\Omega)$ are, and explain why these computations also apply to spatially defined continuous trace C$^*$-algebras $A$ arising from the underlying continuous Hilbert bundle $\Omega$. This is joint work with M Argerami and P Massey.
In the previous semester I explained in some detail how continuous Hilbert bundles over Stonean spaces lead to what are known as Kaplansky-Hilbert modules. In this lecture, I will summarise the results of this study, which culminate in the computation of the injective envelope and local mutliplier algebras of spatially defined continuous trace C*-algebras with Stonean spectrum. This is work was carried out in collaboration with M Argerami and P Massey.
The goal is to describe all pure $E_0$-semigroups that are cocycle conjugate to a given one in terms of the associated spectral C*-algebra. In the first lecture, I will show that any $E_0$-semigroup is cocycle conjugate to a pure $E_0$-semigroup. For this purpose, I will introduce a continuous version of the Wold-von Neumann decomposition.
Amenability is a concept imported from harmonic analysis to functional analysis, and has special implications in the theory of operator algebras. These three lectures will treat amenability from an operator theoretic perspective.
Amenability is a concept imported from harmonic analysis to functional analysis, and has special implications in the theory of operator algebras. These three lectures will treat amenability from an operator theoretic perspective. The outline for the lectures is: Part I: Definitions and properties Part II: The case of C$^*$-algebras Part III: The similarity problem
The goal is to describe some recent results on the so called "radial masa" in a free group factor (masa stands for "maximal abelian subalgebra"). Fourth lecture: I will recap (after almost two months!) the definition and some basic properties of the Puckanszky invariant defined in lecture three, and we will see how to use it to get information about singular masas.
The goal is to describe some recent results on the so called "radial masa" in a free group factor (masa stands for "maximal abelian subalgebra"). If we want to understand the masas in a II_1 factor better, we should have tools to distinguish among them. This is often not a trivial task. After all, we have proven in lecture 1 that all masas in all separable II_1 factors are isomorphic. But they are not necessarily preserved under automorphisms of the whole factor (this we saw in lecture 2, where we saw examples of regular and singular masas). The distinction among masas is usually done by means of invariants, i.e. objects that are invariant under automorphisms. In the third lecture I will cover the most successful invariant, namely the Puckanszky invariant.
The goal is to describe some recent results on the so called "radial masa" in a free group factor (masa stands for "maximal abelian subalgebra"). In the second lecture I will review the classical Dixmier examples using groups, and I will start to describe Pukanszky's invariant. These will later apply to the analysis of the radial masa.
The goal is to describe some recent results on the so called "radial masa" in a free group factor (masa stands for "maximal abelian subalgebra"). In the first lecture I will recap basic knowledge about masas in II_1 factors, from the work of Dixmier in the 50s and Popa in the 80s, to more recent results.
This week's lecture concludes the proof that the continuous trace C$^*$-algebra arising from a continuous Hilbert bundle over a Stonean space embeds rigidly into the injective C$^*$-algebra of all endomorphisms of the Kaplansky-Hilbert module of weakly continuous vector fields.
In the last lecture I constructed a faithful AW$^*$-module from a weakly continuous Hilbert bundle over a Stonean space. This week I will consider the C$^*$-algebra of bounded endomorphisms of this module, which turns out to be an injective C$^*$-algebra. Further, I will embed Fell's continuous trace C$^*$-algebra (arising from the original Hilbert bundle) into this injective C$^*$-algebra. Our ultimate goal is to prove that this embedding is rigid.
In the last lecture, the notion of a weakly continuous Hilbert bundle was introduced and it was shown that if the base space $\Delta$ is Stonean (ie., compact, Hausdorff, extremely disconnected), then the fibre-wise inner product of weakly continuous vector fields is a continuous function off a meagre subset of $\Delta$. This lecture will pursue the consequences of this fact. The main result will be the construction of a faithful AW$^*$-module over $C(\Delta)$ from weakly continuous vector fields.
This, the first of two lectures, introduces the notion of continuous and weakly continuous Hilbert bundle. Such bundles give rise to Hilbert C*-modules; in fact, a faithful AW*-module results, provided the underlying locally compact Hausdorff space is extremely disconnected. The second lecture will consider (adjointable) endomorphisms of these modules.
To this point in the lectures we have seen that quantum operations admit a Kraus representation; the operators that arise in this representation are called noise. To each operation one associates a "noise commutant." The noise commutant has the property that each of its elements is left fixed by the quantum operation. Does every fixed point arise in this way? This lecture will address this question.
In this lecture I will describe what is meant by one quantum operation being absolutely continuous with respect to another. This notion is characterised by the Noncommutative Radon-Nikodym Theorem. I will give a proof of this general proposition and discuss its implications for quantum operations.
In previous lectures by Remus Floricel, the notions of quantum operations and quantum channels were shown to be at the foundation of quantum information theory. As mentioned last week, quantum operations arise via compressions of representations of certain operator algebras. In this lecture I will pursue this further, and obtain the so-called Kraus decomposition of a completely positive map acting on B(H). Subsequent lectures will address the Radon-Nikodym theorem and the Fixed Point theorem for quantum channels.
We shall discuss various properties of the Holevo capacity of a unital completely positive map (also known as quantum channel in quantum information theory).
In quantum information theory, the noisy quantum channels are described by unital completely positive maps acting on matrix algebras. Our purpose in this lecture is to present several definitions of channel capacity, and to discuss their main properties.
"Can entanglement between input states help to send classical information through quantum channels?" This question, called the additivity conjecture, is arguably one of the most important open problems in quantum information theory. Our purpose is to discuss this conjecture from the perspective of operator algebras.
a long-standing question in the area is whether every separable
von Neumann algebra is singly generated (i.e., generated by a single
operator). More or less elementary considerations reduce the problem to
II$_1$ factors. Recently, many II$_1$ factors have been proven to be singly
generated. The plan is to try to give a good view of the old results
together with new developments that cover most cases of interest. SIXTH LECTURE: Some technical properties of Shen's invariant will be used to
prove that some II_1 factors (namely tensor products, gamma-factors,
factors with cartan subalgebras, crossed products of finitely generated
factors) are singly generated.
a long-standing question in the area is whether every separable
von Neumann algebra is singly generated (i.e., generated by a single
operator). More or less elementary considerations reduce the problem to
II$_1$ factors. Recently, many II$_1$ factors have been proven to be singly
generated. The plan is to try to give a good view of the old results
together with new developments that cover most cases of interest. FIFTH LECTURE: After showing that Shen's invariant has a scaling property, we
will discuss the fundamental group of a II_1 factor and its relation with the
generation problem. Very much related with the fundamental group are the
free group factors: we will discuss them briefly and see what Shen's
invariant has to say about them.
a long-standing question in the area is whether every separable
von Neumann algebra is singly generated (i.e., generated by a single
operator). More or less elementary considerations reduce the problem to
II$_1$ factors. Recently, many II$_1$ factors have been proven to be singly
generated. The plan is to try to give a good view of the old results
together with new developments that cover most cases of interest. FOURTH LECTURE: In lecture 3 we introduced Shen's invariant G(M) and showed that
it can be used to determine that a given II_1 factor is singly generated
(provided that we can calculate it!). In this lecture we will focus on
calculating G(M) for several kinds of II_1 factors.
a long-standing question in the area is whether every separable
von Neumann algebra is singly generated (i.e., generated by a single
operator). More or less elementary considerations reduce the problem to
II$_1$ factors. Recently, many II$_1$ factors have been proven to be singly
generated. The plan is to try to give a good view of the old results
together with new developments that cover most cases of interest. THIRD LECTURE: having reduced the general problem is generation to the case
of II_1 factors, we will proceed to develop the theory of Shen's invariant.
Clever manipulations of this invariant have been used recently by Shen and
others to establish that many well-known II_1 factors are singly generated.
a long-standing question in the area is whether every separable
von Neumann algebra is singly generated (i.e., generated by a single
operator). More or less elementary considerations reduce the problem to
II$_1$ factors. Recently, many II$_1$ factors have been proven to be singly
generated. The plan is to try to give a good view of the old results
together with new developments that cover most cases of interest. SECOND LECTURE: at the beginning we will finish the proof
(started on lecture 1) that any non-type II_1 separable von Neumann algebra
is singly generated. Then we will reduce to II_1 factors, and start
analyzing the specifics of the single generator problem within this context.
a long-standing question in the area is whether every separable
von Neumann algebra is singly generated (i.e., generated by a single
operator). More or less elementary considerations reduce the problem to
II$_1$ factors. Recently, many II$_1$ factors have been proven to be singly
generated. The plan is to try to give a good view of the old results
together with new developments that cover most cases of interest.