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.
