A C^*-algebra is called primitive if it admits a faithful and irreducible
*--representation. We show that the unital C^*-algebra full free product,
A=A_1*A_2, of nontrivial finite dimensional C^*-algebras A_1 and A_2 is
primitive except when A_1 and A_2 are both two dimensional. It follows that A
is antiliminal and the set of pure states is w*-dense in the state space.

The goal of this notice is to establish Not-commutative Point- wise Ergodic
Theorems for actions of the Hyperbolic Groups. Similar non-commutative results
were done by Bufetov, Khristoforov and Kli- menko, and later by Pollicott and
Sharp. We were interested to expand short notice in Policott and Sharp's paper
about non-commutative er- godic theorems.

We establish conditions under which the universal and reduced norms coincide
for a Fell bundle over a groupoid. Specifically, we prove that the full and
reduced C*-algebras of any Fell bundle over a measurewise amenable groupoid
coincide, and also that for a groupoid G whose orbit space is T_0, the full and
reduced algebras of a Fell bundle over G coincide if the full and reduced
algebras of the restriction of the bundle to each isotropy group coincide.

Let p be a polynomial in one variable. It is shown that the universal
C*-algebra of the relation p(x)=0, \|x\| \le C is semiprojective, residually
finite-dimensional and has trivial extension group.

In this paper, we develop the theory of bimodules over von Neumann algebras,
with an emphasis on categorical aspects. We clarify the relationship between
dualizability and finite index. We also show that, for von Neumann algebras
with finite-dimensional centers, the Haagerup L^2-space and Connes fusion are
functorial with respect to homorphisms of finite index. Along the way, we
describe a string diagram notation for maps between bimodules that are not
necessarily bilinear.

Let R be a finite Blaschke product. We study the C*-algebra TC_R generated by
both the composition operator C_R and the Toeplitz operator T_z on the Hardy
space. We show that the simplicity of the quotient algebra OC_R by the ideal of
the compact operators can be characterized by the dynamics near the
Denjoy-Wolff point of R if the degree of R is at least two.

From a non-constant holomorphic map on a connected Riemann surface we
construct an 'etale second countable locally compact Hausdorff groupoid whose
associated groupoid C*-algebra admits a one-parameter group of automorphisms
with the property that its KMS states corresponds to conformal measures in the
sense of Sullivan. In this way certain quadratic polynomials give rise to
quantum statistical models with a phase transition arising from spontaneous
symmetry breaking.

We give an equivalent expression for the $K$-functional associated to the
pair of operator spaces $(R,C)$ formed by the rows and columns respectively.
This yields a description of the real interpolation spaces for the pair
$(M_n(R), M_n(C))$ (uniformly over $n$). More generally, the same result is
valid when $M_n$ (or $B(\ell_2)$) is replaced by any semi-finite von Neumann
algebra.

We construct analogs of the embedding of orthogonal and symplectic groups
into unitary groups in the context of fusion categories. At least some of the
resulting module categories also appear in boundary conformal field theory. We
determine when these categories are unitarizable, and explicitly calculate the
index and principal graph of the resulting subfactors.

We consider three notions of divisibility in the Cuntz semigroup of a
C*-algebra, and show how they reflect properties of the C*-algebra. We develop
methods to construct (simple and non-simple) C*-algebras with specific
divisibility behaviour. As a byproduct of our investigations, we show that
there exists a sequence $(A_n)$ of simple unital infinite dimensional
C*-algebras such that the product $\prod_{n=1}^\infty A_n$ has a character.

The notion of index for arbitrary real factors is introduced and
investigated. The main tool in our approach is the reduction of real factors to
involutive *-anti-automorphisms of their complex enveloping von Neumann
algebras. Similar to the complex case the values of the index for real factors
are calculated.

We construct a free-probability quantum Yang-Mills theory on the two
dimensional plane, determine the Wilson loop expectation values, and show that
this theory is the $N=\infty$ limit of U(N) quantum Yang-Mills theory on the
plane.

We study derivations and Fredholm modules on metric spaces with a local
regular conservative Dirichlet form. In particular, on finitely ramified
fractals, we show that there is a non-trivial Fredholm module if and only if
the fractal is not a tree (i.e. not simply connected). This result relates
Fredholm modules and topology, and refines and improves known results on p.c.f.
fractals.

In this paper we study the completely bounded anti-isomorphisms on operator
algebras, that work similarly to the involutions with the exception for the
property of being completely isometric. We elaborate the Blecher's
characterization theorem for operator algebras to make it applicable to the
so-called operator $K$-algebras with completely bounded reflexive
anti-isomorphism. We also establish a connection of this result with the notion
of smooth $C^*$-modules, that play an important role in Mesland's approach to
Baaj-Julg picture of $KK$-theory.

We present an operator-coefficient version of Sato's infinite-dimensional
Grassmann manifold, and tau-function. In this context, the Burchnall-Chaundy
ring of commuting differential operators becomes a C*-algebra, to which we
apply the Brown-Douglas-Fillmore theory, and topological invariants of the
spectral ring become readily available. We construct KK classes of the spectral
curve of the ring and, motivated by the fact that all isospectral
Burchnall-Chaundy rings make up the Jacobian of the curve, we compare the
(degree-1) K-homology of the curve with that of its Jacobian.

Let $\C$ be a $C^*$-algebra and $\alpha:\C\rightarrow \C$ a unital
*-endomorphism. There is a natural way to construct operator algebras which are
called semicrossed products, using a convolution induced by the action of
$\alpha$ on $\C$. We show that the $C^*$-envelope of a semicrossed product is
(a full corner of) a crossed product. As a consequence, we get that, when
$\alpha$ is *-injective, the semicrossed products are completely isometrically
isomorphic and share the same $\ca$-envelope, the crossed product $\C_\infty
\rtimes_{\alpha_\infty} \bbZ$.

This is an introductory paper about our recent merge of a noncommutative de
Finetti type result with representations of the infinite braid and symmetric
group which allows to derive factorization properties from symmetries. We
explain some of the main ideas of this approach and work out a constructive
procedure to use in applications. Finally we illustrate the method by applying
it to the theory of group characters.

We consider the simplicity of the $C^*$-algebra associated to a labelled
space $(E,\CL,\bE)$, where $(E,\CL)$ is a labelled graph and $\bE$ is the
smallest accommodating set containing all generalized vertices. We prove that
if $C^*(E, \CL, \bE)$ is simple, then $(E, \CL, \bE)$ is strongly cofinal, and
if, in addition, $\{v\}\in \bE$ for every vertex $v$, then $(E, \CL, \bE)$ is
disagreeable.

We introduce the concept of a $1$-coaligned $k$-graph and prove that the
shift maps of a $k$-graph pairwise $*$-commute if and only if the $k$-graph is
$1$-coaligned. We then prove that for $2$-graphs $\Lambda$ generated from basic
data $*$-commuting shift maps is equivalent to a condition that implies that
$C^*(\Lambda)$ is simple and purely infinite. We then consider full shift
spaces and introduce a condition on a block map which ensures the associated
sliding block code $*$-commutes with the shift.

Let $\tau$ be a faithful normal semifinite trace on a von Neumann algebra
$\mathcal{M}$. For a continuous nonnegative convex monotone nondecreasing
function $f$ on convex subset $\Omega$ of $\mathbb{R}$ and weight nonnegative
Borel function $w$ we consider weighted monotonicity inequalities of the form
{equation*} \tau(w(A)^{1/2}f(A)w(A)^{1/2}) \le \tau (w(A)^{1/2}f(B)w(A)^{1/2}),
{equation*} where $A$ and $B$ are unbounded operators affiliated with respect
to algebra $\mathcal{M}$.

We define matrix models that converge to the generating functions of a wide
variety of loop models with fugacity taken in sets with an accumulation point.
The latter can also be seen as moments of a non-commutative law on a subfactor
planar algebra. We apply this construction to compute the generating functions
of the Potts model on a random planar map.

We show that for any compact connected group G the second cohomology group
defined by unitary invariant 2-cocycles on \hat G is canonically isomorphic to
H^2(\widehat{Z(G)};T). This implies that the group of autoequivalences of the
C*-tensor category Rep G is isomorphic to H^2(\widehat{Z(G)};T)\rtimes\Out(G).
We also show that a compact connected group G is completely determined by Rep
G. More generally, extending a result of Etingof-Gelaki and Izumi-Kosaki we
describe all pairs of compact separable monoidally equivalent groups.

The Dixmier (or minimal) angle between submodules $M$ and $N$ of a Hilbert
C*-module $E$ is the angle $\alpha_0 (M,N)$ in $[0, \pi /2]$ whose cosine is
defined by $c_0(M,N)= {\rm sup} \{\| <x,y> \| : x \in M, \|x\| \leq 1 \, , y
\in N, \|y\| \leq 1 \}.$ Suppose $T$ and $S$ are bounded adjointable operators
with close range between Hilbert C*-modules, then $TS$ has closed range if and
only if $Ker(T)+Ran(S)$ is an orthogonal summand, if and only if
$Ker(S^*)+Ran(T^*)$ is an orthogonal summand.

Let A be a von Neumann algebra with a finite trace $\tau$, represented in
$H=L^2(A,\tau)$, and let $B_t\subset A$ be sub-algebras, for $t$ in an interval
$I$. Let $E_t:A\to B_t$ be the unique $\tau$-preserving conditional
expectation. We say that the path $t\mapsto E_t$ is smooth if for every $a\in
A$ and $v \in H$, the map $$ I\ni t\mapsto E_t(a)v\in H $$ is continuously
differentiable. This condition implies the existence of the derivative operator
$$ dE_t(a):H\to H, \ dE_t(a)v=\frac{d}{dt}E_t(a)v.

Let $\mu$ be a probability measure on the real line. In this paper we prove
that there exists a decomposition $\mu = \mu_{0} \boxplus \mu_{1} \boxplus \...
\boxplus \mu_{n} \boxplus \...$ such that $\mu_{0}$ is infinitely divisible and
$\mu_{i}$ is indecomposable for $i \geq 1$. Additionally, we prove that the
family of all $\boxplus$-divisors of a measure $\mu$ is compact up to
translation. Analogous results are also proven in the case of multiplicative
convolution.

Two different models for a Hopf-von Neumann algebra of bounded functions on
the quantum permutation group on infinitely many elements are proposed, one
based on projective limits of enveloping von Neumann algebras related to finite
quantum permutation groups, and the second on universal properties with respect
to infinite magic unitaries.

We introduce P-graphs, which are generalisations of directed graphs in which
paths have a degree in a semigroup P rather than a length in N. We focus on
semigroups P arising as part of a quasi-lattice ordered group (G,P) in the
sense of Nica, and on P-graphs which are finitely aligned in the sense of
Raeburn and Sims. We show that each finitely aligned P-graph admits a
C*-algebra C*_{min}(Lambda) which is co-universal for partial-isometric
representations of Lambda which admit a coaction of G compatible with the
P-valued length function.

We study factorization and dilation properties of Markov maps between von
Neumann algebras equipped with normal faithful states, i.e., completely
positive unital maps which preserve the given states and also intertwine their
automorphism groups. The starting point for our investigation has been the
question of existence of non-factorizable Markov maps, as formulated by C.
Anantharaman-Delaroche.

We continue our study of tensor products in the operator system category. We
define operator system quotients and exactness in this setting and refine the
notion of nuclearity by studying operator systems that preserve various pairs
of tensor products. One of our main goals is to relate these refinements of
nuclearity to the Kirchberg conjecture. In particular, we prove that the
Kirchberg conjecture is equivalent to the statement that every operator system
that is (min,er)-nuclear is also (el,c)-nuclear.

Let M be a von Neumann algebra that has a F{\o}lner net. In the present
article we give conditions that guarantee that the von Neumann crossed product
of M with an amenable discrete group has a F{\o}lner net. The F{\o}lner net for
the crossed product is given explicitly and the result is applied to the
rotation algebra.

We study KMS states on finite-graph C*-algebras with sinks and sources. We
compare finite-graph C*-algebras with C*-algebras associated with complex
dynamical systems of rational functions. We show that if the inverse
temperature $\beta$ is large, then the set of extreme $\beta$-KMS states is
parametrized by the set of sinks of the graph. This means that the sinks of a
graph correspond to the branched points of a rational funcition from the point
of KMS states. Since we consider graphs with sinks and sources, left actions of
the associated bimodules are not injective.

The aim of this paper is to present and analyze a new definition of a quantum
homogeneous space of a locally compact quantum group G. It is shown to be an
appropriate quantum counterpart of the classical notion of homogeneity,
providing an operator algebraic characterization of the transitive group
actions. Furthermore our framework covers different classes of examples such as
the quotient of a locally compact quantum group by its closed quantum subgroup
due to S. Vaes and (generically non-quotient) quantum homogeneous spaces of a
compact quantum group studied by P.

We give a numerical characterization of mutual orthogonality (that is,
complementarity) for subalgebras. In order to give such a characterization for
mutually orthogonal subalgebras $A$ and $B$ of the $k \times k$ matrix algebra
$M_k(\mathbb{C})$, where $A$ and $B$ are isomorphic to some $M_n(\mathbb{C})$
$(n \leq k)$, we consider a density matrix which is induced from the pair $\{A,
B\}$.

We give an introduction into the ideal structure and representation theory of
crossed products by actions of locally compact groups on C*-algebras. In
particular, we discuss the Mackey-Rieffel-Green theory of induced
representations of crossed products and groups. Although we do not give
complete proofs of all results, we try at least to explain the main ideas. For
a more detailed exposition of many of the results presented here we refer to
the beautiful recent book by Dana Williams.

Let G be a second-countable locally-compact Hausdorff groupoid with a Haar
system, and let {x_n} be a sequence in the unit space of G. We show that the
notions of strength of convergence of {x_n} in the orbit space and
measure-theoretic accumulation along the orbits are equivalent ways of
realising multiplicity numbers associated to a sequence of induced
representation of the groupoid C*-algebra.

In a two-state free probability space $(A, \phi, \psi)$, we define an
algebraic two-state free Brownian motion to be a process with two-state freely
independent increments whose two-state free cumulant generating function is
quadratic. Note that a priori, the distribution of the process with respect to
the second state $\psi$ is arbitrary. We show, however, that if $A$ is a von
Neumann algebra, the states $\phi, \psi$ are normal, and $\phi$ is faithful,
then there is only a one-parameter family of such processes.

The present paper deals with the question of representability of nets of
C*-algebras whose underlying poset, indexing the net, is nonupward directed. A
particular class of nets, called C*-net bundles, is classified in terms of
C*-dynamical systems having group the fundamental group of the poset. Any net
of C*-algebras embeds into a unique C*-net bundle, the enveloping net bundle,
which generalizes to nonsimply connected posets the notion of universal
C*-algebra given by Fredenhagen.

A theory of Galois co-objects for von Neumann bialgebras is introduced. This
concept is closely related to the notion of comonoidal W*-Morita equivalence
between von Neumann bialgebras, which is a Morita equivalence taking the
comultiplication structure into account. We show that the property of `being a
von Neumann algebraic quantum group' (i.e. `having invariant weights') is
preserved under this equivalence relation.

The class of separable C*-algebras which can be written as inductive limits
of continuous-trace C*-algebras with spectrum homeomorphic to a disjoint union
of trees and trees with a point removed is classified by the Cuntz semigroup.

We describe an explicit finite presentation for a finite depth subfactor
planar algebra. We also show that such planar algebras are singly generated
with the generator subject to finitely many relations.

For any abstract subfactor planar algebra $P$, there exists a finite index
extremal subfactor $M_0 \subset M_1$ with $P$ as its standard invariant. In
this paper, we classify the automorphism group of a bipartite graph planar
algebra, and obtain subfactor planar subalgebras by taking fixed points under
groups of automorphisms. This construction provides both new examples of
subfactors and new descriptions of the planar algebras of previously known
examples.

In this announcement we explain how to extend the von Neumann algebra
definition of fusion to the case of twisted loop groups. This project completes
research started by my student Robert Verrill in his 2001 Cambridge Ph.D.
thesis.

Given an n x n integer matrix A whose eigenvalues are strictly greater than 1
in absolute value, let \sigma_A be the transformation of the n-torus
T^n=R^n/Z^n defined by \sigma_A(e^{2\pi ix})=e^{2\pi iAx} for x\in R^n. We
study the associated crossed-product C*-algebra, which is defined using a
certain transfer operator for \sigma_A, proving it to be simple and purely
infinite and computing its K-theory groups.

We establish several analogues of the classical Lidskii Theorem for some
special classes of singular traces (Dixmier traces and Connes-Dixmier traces)
used in noncommutative geometry.

Given linear matrix inequalities (LMIs) L_1 and L_2, it is natural to ask:
(Q1) when does one dominate the other, that is, does L_1(X) PsD imply L_2(X)
PsD? (Q2) when do they have the same solution set? Such questions can be
NP-hard. This paper describes a natural relaxation of an LMI, based on
substituting matrices for the variables x_j. With this relaxation, the
domination questions (Q1) and (Q2) have elegant answers, indeed reduce to
constructible semidefinite programs. Assume there is an X such that L_1(X) and
L_2(X) are both PD, and suppose the positivity domain of L_1 is bounded.

We introduce a new notion of twisted actions of inverse semigroups and show
that they correspond bijectively to certain regular Fell bundles over inverse
semigroups, yielding in this way a structure classification of such bundles.
These include as special cases all the stable Fell bundles.

This thesis is dedicated to developing a dilation theory for semigroups of
completely positive maps. The first part treats two-parameter semigroups, and
contains also contributions to dilation theory of product system
representations. The second part deals with completely positive semigroups
parameterized by quite general semigroups, where the major technical tool
introduced is subproduct systems and their representations. In the third part
subproduct systems are studied, together with the multivariable operator theory
and operator algebras they give rise to.

In the given article, we discuss the problem of the classification of general
C$^*$-algebras. Also, it was introduced a new notions of C$^*$-algebra of von
Neumann type I, C$^*$-algebras of types II and III. It is proved that any
GCR-algebra is a C$^*$-algebra of von Neumann type I, and any C$^*$-algebra is
a NGCR-algebra if and only if this C$^*$-algebra does not have a nonzero
abelian annihilator.

We construct spaces of quantum increasing sequences, which give quantum
families of maps in the sense of Soltan. We then introduce a notion of quantum
spreadability for a sequence of noncommutative random variables, by requiring
their joint distribution to be invariant under taking quantum subsequences. Our
main result is a free analogue of a theorem of Ryll-Nardzewski: for an infinite
sequence of noncommutative random variables, quantum spreadability is
equivalent to free independence and identical distribution with respect to a
conditional expectation.

We consider spatial discretizations by the finite section method of the
restricted group algebra of a finitely generated discrete group, which is
represented as a concrete operator algebra via its left-regular representation.
Special emphasis is paid to the quasicommutator ideal of the algebra generated
by the finite sections sequences and to the stability of sequences in that
algebra. For both problems, the sequence of the discrete boundaries plays an
essential role.

Let $M$ be a separable compact Hausdorff space with $\dim M\le 2$ and
$\theta\colon M\to M$ be a homeomorphism with prime period $p$ ($p\ge 2$). Set
$M_\theta=\{x\in M| \theta(x)=x\}\not=\varnothing$ and $M_0=M\backslash
M_\theta$. Suppose that $M_0$ is dense in $M$ and $\mathrm
H^2(M_0/\theta,\Z)\cong 0$, $\mathrm H^2(\chi(M_0/\theta),\Z)\cong 0$. Let $M'$
be another separable compact Hausdorff space with $\dim M'\le 2$ and $\theta'$
be the self--homeomorphism of $M'$ with prime period $p$. Suppose that
$M_0'=M'\backslash M_{\theta'}'$ is dense in $M'$.

This paper is devoted to the study of $\Phi$-moment inequalities for
noncommutative martingales. In particular, we prove the noncommutative
$\Phi$-moment analogues of martingale transformations, Stein's inequalities,
Khintchine's inequalities for Rademacher's random variables, and
Burkholder-Gundy's inequalities. The key ingredient is a noncommutative version
of Marcinkiewicz type interpolation theorem for Orlicz spaces which we
establish in this paper.

This article is focused on properties of monotone convolutions. A criterion
for infinite divisibility and time evolution of convolution semigroups are
mainly studied. In particular, we clarify that many analogues of the classical
results of L\'{e}vy processes hold such as characterizations of subordinators
and strictly stable distributions.

For each K-homolgy element of the Sierpinski gasket we construct a spectral
triple which will generate that element. We show that there must be certain
limits on the choice of the K-homology element if the geometric properties of
the gasket shall be recoverable from that spectral triple. For a big subgroup
of the K-homology group we show that our spectral triples will recover the
metric, the dimension and the Hausdorff measure on the gaket.

In this paper we present a conjecture on intermediate subfactors which is a
generalization of Wall's conjecture from the theory of finite groups. Motivated
by this conjecture, we determine all intermediate subfactors of
Goodman-Harpe-Jones subfactors, and as a result we verify that
Goodman-Harpe-Jones subfactors verify our conjecture. Our result also gives a
negative answer to a question motivated by a conjecture of
Aschbacher-Guralnick.

We complete the computation of spectral measures for SU(3) nimrep graphs
arising in subfactor theory, namely the SU(3) ADE graphs associated with SU(3)
modular invariants and the McKay graphs of finite subgroups of SU(3). For the
SU(2) graphs the spectral measures distill onto very special subsets of the
semicircle/circle, whilst for the SU(3) graphs the spectral measures distill
onto very special subsets of the discoid/torus. The theory of nimreps allows us
to compute these measures precisely.

A study of noncommutative topological entropy of gauge invariant
endomorphisms of Cuntz algebras began in our earlier work with Joachim
Zacharias is continued and extended to endomorphisms which are not necessarily
of permutation type. In particular it is shown that if H is an N-dimensional
Hilbert space, V is an irreducible multiplicative unitary on the tensor product
of H with itself and F is the tensor flip, then the Voiculescu entropy of the
Longo's canonical endomorphism associated with the unitary VF is equal to log
N.

It is shown that if a C*-algebra has nuclear dimension $n$ then its Cuntz
semigroup has the property of $n$-comparison. It then follows from results by
Ortega, Perera, and Rordam that $\sigma$-unital C*-algebras of finite nuclear
dimension (and even of nuclear dimension at most $\omega$) are stable if an
only if they have no non-zero unital quotients and no non-zero bounded traces.

The notion of sofic action was introduced by Gabor Elek and Gabor Lippner for
actions of free group with infinite generators. Their technics employs some
graph theory. Here we define this notion for any action in a more operator
algebraic context, starting from Connes' embedding problem. We prove the
equivalence of this definitions in case of actions of free group with infinite
generators. Also, free product of sofic actions is again sofic and a free
action orbit equivalent with a sofic one is sofic. Using this last statement we
propose a new definition for sofic equivalence relations.

Based on the fact that, for a subfactor $N$ of a II$_1$ factor $M,$ the first
non-trivial Jones index is 2 and then $M$ is decomposed as the crossed product
of $N$ by an outer action of ${\mathbb{Z}}_2,$ we study pairs $ \{N, uNu^* \}$
from a view point of entropy for two subalgebras of $M$ with a connection to
the entropy for automorphisms, where the inclusion of II$_1$ factors $ N
\subset M$ is given as $M$ is the crossed product of $N$ by a finite group of
outer automorphisms and $u$ is a unitary in $M.$

We introduce notions of the Rohlin property and the approximate
representability for inclusions of unital $C^*$-algebras. We investigate a dual
relation between the Rohlin property and the approximate representability. We
prove that a number of classes of unital $C^*$-algebras are closed under
inclusions with the Rohlin property, including:

AF algebras, AI algebras, AT algebras, and related classes characterized by
direct limit decomposition using semiprojective building blocks. $C^*$-algebras
with stable rank one. $C^*$-algebras with real rank zero.

In this paper, we study the K0-group of the C?-algebra associated to a
pinwheel tiling. We prove that it is given by the sum of Z + Z^6 with a
cohomological group. The C?-algebra is endowed with a trace that induces a
linear map on its K0-group. We then compute explicitly the image, under this
map, of the summand Z+Z^6, showing that the image of Z is zero and the image of
Z^6 is included in the module of patch frequencies of the pinwheel tiling. We
finally prove that we can apply the measured index theorem due to A.

We give an example of an exact, stably finite, simple. separable C*-algebra D
which is not isomorphic to its opposite algebra. Moreover, D has the following
additional properties. It is stably finite, approximately divisible, has real
rank zero and stable rank one, has a unique tracial state, and the order on
projections over D is determined by traces. It also absorbs the Jiang-Su
algebra Z, and in fact absorbs the 3^{\infty} UHF algebra.

We survey the results required to pass between full and reduced coactions of
locally compact groups on C*-algebras, which say, roughly speaking, that one
can always do so without changing the crossed-product C*-algebra. Wherever
possible we use definitions and constructions that are well-documented and
accessible to non-experts, and otherwise we provide full details. We then give
a series of applications to illustrate the use of these techniques.

The main theorem provides a characterisation of the finite rank operators
lying in a norm closed Lie ideal of a continuous nest algebra. These operators
are charaterised as those finite rank operators in the nest algebra satisfying
a condition determined by a left order continuous homomorphism on the nest. A
crucial fact used in the proof of this theorem is the decomposability of the
finite rank operators. One shows that a finite rank operator in a norm closed
Lie ideal of a continuous nest algebra can be written as a finite sum of rank
one operators lying in the ideal.

The structure of the $C^*$-algebras corresponding to even-dimensional mirror
quantum spheres is investigated. It is shown that they are isomorphic to both
Cuntz-Pimsner algebras of certain $C^*$-correspondences and $C^*$-algebras of
certain labelled graphs. In order to achieve this, categories of labelled
graphs and $C^*$-correspondences are studied. A functor from labelled graphs to
$C^*$-correspondences is constructed, such that the corresponding associated
$C^*$-algebras are isomorphic.

We study the index problem for the d-bar operators subject to Atiyah-
Patodi-Singer boundary conditions on noncommutative disk and annulus.

The problem of existence of standard (i.e. product-type) invariant MASAs for
endomorphisms of the Cuntz algebra O_n is studied. In particular endomorphisms
which preserve the canonical diagonal MASA D_n are investigated. Conditions on
a unitary in O_n equivalent to the fact that the corresponding endomorphism
preserves D_n are found, and it is shown that they may be satisfied by
unitaries which do not normalize D_n. Unitaries giving rise to endomorphisms
which leave all standard MASAs invariant and have identical actions on them are
characterized.

A C*-algebra $A$ is C*-reflexive if any countably generated Hilbert C*-module
$M$ over $A$ is C*-reflexive, i.e. the second dual module $M''$ coincides with
$M$. We show that a commutative C*-algebra $A$ is C*-reflexive if and only if
for any sequence $I_k$ of disjoint non-zero C*-subalgebras, the canonical
inclusion $\oplus_k I_k\subset A$ doesn't extend to an inclusion of $\prod_k
I_k$.

The construction of an infinite tensor product of the C*-algebra C_0(R) is
not obvious, because it is nonunital, and it has no nonzero projection. Based
on a choice of an approximate identity, we construct here an infinite tensor
product of C_0(R), denoted L_V. We use this to construct (partial) group
algebras for the full continuous unitary representation theory of the group
R^(N) = the infinite sequences with real entries, of which only finitely many
entries are nonzero.

We prove that atomic decomposition for the Hardy spaces h_1 and H_1 is valid
for noncommutative martingales. We also establish that the conditioned Hardy
spaces of noncommutative martingales h_p and bmo form interpolation scales with
respect to both complex and real interpolations.

We give a survey of techniques from quantum group theory which can be used to
show that some quantum spaces (objects of the category dual to the category of
$\mathrm{C}^*$-algebras) do not admit any quantum group structure. We also
provide a number of examples which include some very well known quantum spaces.
Our tools include several purely quantum group theoretical results as well as
study of existence of characters and traces on $\mathrm{C}^*$-algebras
describing the considered quantum spaces as well as properties such as
nuclearity.

Jolissaint and Stalder introduced the definitions of mixing and weak mixing
for von Neumann subalgebras of finite von Neumann algebras. In this paper, we
study various algebraic and analytical properties of mixing and weakly mixing
von Neumann subalgebras. We prove some basic results about mixing inclusions of
von Neumann algebras and establish a connection between mixing properties and
normalizers of von Neumann subalgebras. The special case of mixing subalgebras
arising from inclusions of group von Neumann algebras finds applications to
ergodic theory.

The Furstenberg recurrence theorem (or equivalently, Szemer\'edi's theorem)
can be formulated in the language of von Neumann algebras as follows: given an
integer $k \geq 2$, an abelian finite von Neumann algebra $(\M,\tau)$ with an
automorphism $\alpha: \M \to \M$, and a non-negative $a \in \M$ with
$\tau(a)>0$, one has $\liminf_{N \to \infty} \frac{1}{N} \sum_{n=1}^N \Re
\tau(a \alpha^n (a) ... \alpha^{(k-1)n} (a)) > 0$; a subsequent result of Host
and Kra shows that this limit exists. In particular, $\Re \tau(a \alpha^n (a)
>...

We consider a certain class of unital simple stably finite C^*-algebras which
absorb the Jiang-Su algebra Z tensorially. Under a mild assumption, we show
that the crossed product of a C^*-algebra in this class by a strongly outer
action of Z^N or a finite group is Z-stable. As an application, we also prove
that any strongly outer actions of Z^2 on Z are mutually cocycle conjugate.

It shown that an a locally injective surjection on a compact metric space
admits a canonical locally homeomorphic extension such that the associated
C*-algebras are isomorphic. This is then used in a study of the possible
inverse temperatures of KMS states for a generalized gauge action.

We prove directly that if E is a directed graph in which every cycle has an
entrance, then there exists a C*-algebra which is co-universal for
Toeplitz-Cuntz-Krieger E-families. In particular, our proof does not invoke
ideal-structure theory for graph algebras, nor does it involve use of the gauge
action or its fixed point algebra.

For any countable graph $E$, we investigate the relationship between the
Leavitt path algebra $L_{\C}(E)$ and the graph C*-algebra $C^*(E)$. For graphs
$E$ and $F$, we examine ring homomorphisms, ring *-homomorphisms, algebra
homomorphisms, and algebra *-homomorphisms between $L_{\C}(E)$ and $L_{\C}(F)$.
We prove that in certain situations isomorphisms between $L_{\C}(E)$ and
$L_{\C}(F)$ yield *-isomorphisms between the corresponding C*-algebras $C^*(E)$
and $C^*(F)$.

We show how to extend a classic Morita Equivalence Result of Green's to the
\cs-algebras of Fell bundles over transitive groupoids. This also yields an
interesting result for the \cs-algebras of Fell bundles over groups.

Let A be a unital simple separable C*-algebra with strict comparison of
positive elements. We prove that the Cuntz semigroup of A is recovered
functorially from the Murray-von Neumann semigroup and the tracial state space
T(A) whenever the extreme boundary of T(A) is compact and of finite covering
dimension. Combined with a result of Winter, we obtain Z \otimes A isomorphic
to A whenever A moreover has locally finite decomposition rank.

Using the Foelner condition for coamenable quantum groups we derive
information about the ring theoretical structure of the Hopf algebras arising
from such quantum groups, as well as an approximation result concerning the
Murray von Neumann dimension associated with the corresponding the von Neumann
algebra.

We consider the limiting distribution of $U_NA_NU_N^*$ and $B_N$ (and more
general expressions), where $A_N$ and $B_N$ are $N \times N$ matrices with
entries in a unital C$^*$-algebra $\mathcal B$ which have limiting $\mathcal
B$-valued distributions as $N \to \infty$, and $U_N$ is a $N \times N$ Haar
distributed quantum unitary random matrix with entries independent from
$\mathcal B$. Under a boundedness assumption, we show that $U_NA_NU_N^*$ and
$B_N$ are asymptotically free with amalgamation over $\mathcal B$.

A resistance network is a weighted graph $(G,c)$ with intrinsic (resistance)
metric $R$. We embed the resistance network into the Hilbert space ${\mathcal
H}_{\mathcal E}$ of functions of finite energy. We use the resistance metric to
study ${\mathcal H}_{\mathcal E}$, and vice versa and show that the embedded
images of the vertices $\{v_x\}$ form a reproducing kernel for this Hilbert
space.

In this paper we want to apply the notion of product between ultrafilters to
answer several questions which arise around the Connes' embedding problem. For
instance, we will give a simplification and generalization of a theorem by
Radulescu; we will prove that ultraproduct of hyperlinear groups is still
hyperlinear and consequently the von Neumann algebra of the free group with
uncountable many generators is embeddable into $R^{\omega}$. This follows also
from a general construction that allows, starting from an hyperlinear group, to
find a family of hyperlinear groups.

In this paper we show that, for a class of countable graphs, every
representation of the associated graph algebra in a separable Hilbert space is
unitarily equivalent to a representation obtained via branching systems.

We introduce a noncommutative analogue of the Fig\'a-Talamanca-Herz algebra
$A_p(G)$ on the natural predual of the operator space $\frak{M}_{p,cb}$ of
completely bounded Schur multipliers on Schatten space $S_p$. We determine the
isometric Schur multipliers and prove that the space $\frak{M}_{p}$ of bounded
Schur multipliers on Schatten space $S_p$ is the closure in the weak operator
topology of the span of the isometric multipliers.

States on coupled quantum system whose restrictions to each subsystems are
normalized traces are called marginal tracial states. We investigate extremal
marginal tracial states and maximal rank of such states. Diagonal marginal
tracial states are also considered.

In this paper, we study the harmonic analysis of Bernoulli measures. We show
a variety of orthonormal Fourier bases for the L^2 Hilbert spaces corresponding
to certain Bernoulli measures, making use of contractive transfer operators.
For other cases, we exhibit maximal Fourier families that are not orthonormal
bases.

The paper gives an operator algebras model for the conditional monotone
independence, introduced by T. Hasebe. The construction is used to prove an
embedding result for the N. Muraki's monotone product of C*-algebras. Also, the
formulas from the definition of conditional monotone independence are used to
define the monotone product of maps which is shown to preserve complete
positivity, a similar to the results from the case of free products.

We introduce a notion of an algebra of generalized pseudo-differential
operators and prove that a spectral triple is regular if and only if it admits
an algebra of generalized pseudo-differential operators. We also provide a
self-contained proof of the fact that the product of regular spectral triples
is regular.

We show that any countable subgroup of the multiplicative group
$\mathbb{R}_+^{\times}$ of positive real numbers can be realized as the
fundamental group $\mathcal{F}(A)$ of a separable simple unital $C^*$-algebra
$A$ with unique trace. Furthermore for any fixed countable subgroup $G$ of
$\mathbb{R}_+^{\times}$, there exist uncountably many mutually nonisomorphic
such algebras $A$ with $G = \mathcal{F}(A)$.

Convolution semigroups of states on a quantum group form the natural
noncommutative analogue of convolution semigroups of probability measures on a
locally compact group. Here we initiate a theory of weakly continuous
convolution semigroups of functionals on a C*-bialgebra, the noncommutative
counterpart of locally compact semigroup. On locally compact quantum groups we
obtain a bijective correspondence between such convolution semigroups and a
class of C_0-semigroups of maps which we characterise.

We study properties of C*-algebras obtained from the nonstandard hull
construction (a generalization of the ultraproduct of C*-algebras). Among
others, we prove that the properties of being an infinite and a properly
infinite C*-algebra are both preserved and reflected by the nonstandard hull
construction. We also show that the property of being generated by mutually
orthogonal projections is preserved by the nonstandard hull.

Kadison and Kastler introduced a natural metric on the collection of all
C*-subalgebras of the bounded operators on a separable Hilbert space. They
conjectured that sufficiently close algebras are unitarily conjugate. We
establish this conjecture when one algebra is separable and nuclear. We also
consider one-sided versions of these notions, and we obtain embeddings from
certain near inclusions involving separable nuclear C*-algebras.

We present a general setting to investigate U_fin-cocycle superrigidity for
Gaussian actions in terms of closable derivations on von Neumann algebras. In
this setting we give new proofs to some U_fin-cocycle superrigidity results of
S. Popa and we produce new examples of this phenomenon. We also use a result of
K. Schmidt to give a necessary cohomological condition on a group
representation in order for the resulting Gaussian action to be U_fin-cocycle
superrigid.

A Markov operator $P$ acting on $C(X)$, where $X$ is compact, gives rise to a
natural topological quiver. We use the theory of such quivers to attach a
$C^{*}$-algebra to $P$ in a fashion that reflects some of the probabilistic
properties of $P$.

Let $X$ be a Cantor set, and let $A$ be a unital separable simple amenable
$C$*-algebra with tracial rank zero which satisfies the Universal Coefficient
Theorem, we use $C(X,A)$ to denote the set of all continuous functions from $X$
to $A$, let $\alpha$ be an automorphism of $C(X,A)$. Suppose that $C(X,A)$ is
$\alpha$-simple and $\alpha_{*i}|_{(1\otimes K_{i}(A))}={id}|_{(1\otimes
K_{i}(A))}$ for $i=0,1$, we show that $C(X,A)\rtimes_{\alpha}\mathbb{Z}$ has
tracial rank zero.

This paper studies the problems of embedding and isomorphism for countably
generated Hilbert C*-modules over commutative C*-algebras. When the fibre
dimensions differ sufficiently, relative to the dimension of the spectrum, we
show that there is an embedding between the modules. This result continues to
hold over recursive subhomogeneous C*-algebras. For certain modules, including
all modules over $C_0(X)$ when $dim X \leq 3$, isomorphism and embedding are
determined by the restrictions to the sets where the fibre dimensions are
constant.

Let A be a unital AF-algebra (simple or non-simple) and let \alpha be an
automorphism of A. Suppose that \alpha has certain Rokhlin property and A is
\alpha-simple. Suppose also that there is an integer J\geq1 such that
\alpha^{J}_{*0}=id_{K_{0}(A)}, we show that A\rtimes_{\alpha}\mathbb{Z} has
tracial rank zero.

Recently, M. Daws introduced a notion of co-representation of abelian
Hopf--von Neumann algebras on general reflexive Banach spaces. In this note, we
show that this notion cannot be extended beyond subhomogeneous Hopf--von
Neumann algebras. The key is our observation that, for a von Neumann algebra
$\M$ and a reflexive operator space $E$, the normal spatial tensor product $\M
\bar{\tensor} \CB(E)$ is a Banach algebra if and only if $\M$ is subhomogeneous
or $E$ is completely isomorphic to column Hilbert space.

Let $A$ be a $C^*$-algebra. Let $E$ and $F$ be Hilbert $A$-modules with $E$
being full. Suppose that $\theta : E\to F$ is a linear map preserving
orthogonality, i.e., $<\theta(x), \theta(y) > = 0$ whenever $<x, y > = 0$. We
show in this article that if, in addition, $A$ has real rank zero, and $\theta$
is an $A$-module map (not assumed to be bounded), then there exists a central
positive multiplier $u\in M(A)$ such that $<\theta(x), \theta(y) > = u < x, y>$
($x,y\in E$).

We consider II$_1$ factors $M$ which can be realized as inductive limits of
subfactors, $N_n \nearrow M$, having spectral gap in $M$ and satisfying the
bi-commutant condition $(N_n'\cap M)'\cap M=N_n$. Examples are the enveloping
algebras associated to non-Gamma subfactors of finite depth, as well as certain
crossed products of McDuff factors by amenable groups. We use
deformation/rigidity techniques to obtain classification results for such
factors.

Let A be an approximately subhomogeneous (ASH) C*-algebra with slow dimension
growth. We prove that if A is unital and simple, then the Cuntz semigroup of A
agrees with that of its tensor product with the Jiang-Su algebra Z. In tandem
with a result of W. Winter, this yields the equivalence of Z-stability and slow
dimension growth for unital simple ASH algebras.

The purpose of the present paper is to study tensor products of operator
systems. After giving an axiomatic definition of tensor products in this
category, we examine in detail several particular examples of tensor products,
including a minimal, maximal, maximal commuting, maximal injective and some
asymmetric tensor products. We characterize these tensor products in terms of
their universal properties and give descriptions of their positive cones.

For any sequence of matrix algebras that converge to a coadjoint orbit we
give explicit formulas that show that the distances between the matrix algebras
(viewed as quantum metric spaces) converges to 0. In the process we develop a
general point of view that is likely to be useful in other related settings.

We introduce a non-symmetric tensor product of representations of UHF
algebras by using Kronecker products of matrices. We prove tensor product
formulae of GNS representations by product states and show examples.

We prove an index theorem for Toeplitz operators on irreducible tube--type
domains and we extend our results to Toeplitz operators with matrix symbols. In
order to prove our index theorem, we proved a result asserting that a
non--vanishing function on the Shilov boundary of a tube--type bounded
symmetric domain, not necessarily irreducible, is equal to a unimodular
function defined as the product of powers of generic norms times an exponential
function.

We show that every infinite-dimensional commutative unital C*-algebra has a
Hilbert C*-module admitting no frames. In particular, this shows that
Kasparov's stabilization theorem for countably generated Hilbert C*-modules can
not be extended to arbitrary Hilbert C*-modules.

We provide a simple algorithm for finding the optimal upper bound for sums of
products of matrix entries of the form

S_pi(N) := sum_{j_1, ..., j_2m = 1}^N t^1_{j_1 j_2} t^2_{j_3 j_4} ...
t^m_{j_2m-1 j_2m} where some of the summation indices are constrained to be
equal. The upper bound is easily obtained from a graph G associated to the
constraints in the sum.

We construct a subfactor planar algebra, and as a corollary a subfactor, with
the `extended Haagerup' principal graph pair. This is the last open case from
Haagerup's 1993 list of potential principal graphs of subfactors with index in
the range (4,3+\sqrt{3}). We prove that the subfactor planar algebra with these
principal graphs is unique. We give a skein theoretic description, and a
description as a subalgebra generated by a certain element in the graph planar
algebra of its principal graph.

The stabilization theorem for $A$-Hilbert modules was established by G. G.
Kasparov. The equivariant version, in which a locally compact group $H$ acts
properly on a locally compact space $Y$, was proved by N. C. Phillips. This
equivariant theorem involves the Hilbert $(H,C_{0}(Y))$-module
$C_{0}(Y,L^{2}(H)^{\infty})$. It can naturally be interpreted in terms of a
stabilization theorem for proper groupoids, and the paper establishes this
theorem within the general proper groupoid context. The theorem has
applications in equivariant KK-theory and groupoid index theory.

Let $A$ be a unital $C^*$-algebra. Its unitary group, $UA$, contains a wealth
of topological information about $A$. However, the homotopy type of $UA$ is out
of reach even for $A = M_2(\CC)$. There are two simplifications which have been
considered. The first, well-traveled road, is to pass to $\pi_*(U(A\otimes \KK
))$ which is isomorphic (with a degree shift) to $K_*(A)$. This approach has
led to spectacular success in many arenas, as is well-known.

Let S_k be the set of separable states on B(C^m \otimes C^n) admitting a
representation as a convex combination of k pure product states, or fewer. If
m>1, n> 1, and k \le max(m,n), we show that S_k admits a subset V_k such that
V_k is dense and open in S_k, and such that each state in V_k has a unique
decomposition as a convex combination of pure product states, and we describe
all possible convex decompositions for a set of separable states that properly
contains V_k.

We show that if $\AA$ is a Fell bundle over a locally compact group $G$, then
there is a natural coaction $\delta$ of $G$ on the Fell-bundle $C^*$-algebra
$C^*(G,\AA)$ such that if $\hat{\delta}$ is the dual action of $G$ on the
crossed product $C^*(G,\AA) \rtimes_{\delta} G$, then the full crossed product
$(C^*(G,\AA) \rtimes_{\delta}G)\rtimes_{\hat{\delta}}G$ is canonically
isomorphic to $C^*(G,\AA) \otimes\KK(L^2(G))$. Hence the coaction $\delta$ is
maximal.

We consider two categories of C*-algebras; in the first, the isomorphisms are
ordinary isomorphisms, and in the second, the isomorphisms are Morita
equivalences. We show how these two categories, and categories of dynamical
systems based on them, crop up in a variety of C*-algebraic contexts. We show
that Rieffel's construction of a fixed-point algebra for a proper action can be
made into functors defined on these categories, and that his Morita equivalence
then gives a natural isomorphism between these functors and crossed-product
functors.

We find a class of algebras A satisfying the following property: for every
nontrivial noncommutative polynomial, the linear span of all its values in A
equals A. This class includes the algebras of all bounded and all compact
operators on an infinite dimensional Hilbert space.

Let $X$ be a compact metric space which is locally absolutely retract and let
$\phi: C(X)\to C(Y, M_n)$ be a unital homomorphism, where $Y$ is a compact
metric space with ${\rm dim}Y\le 2.$ It is proved that there exists a sequence
of $n$ continuous maps $\alfa_{i,m}: Y\to X$ ($i=1,2,...,n$) and a sequence of
sets of mutually orthogonal rank one projections $\{p_{1, m},
p_{2,m},...,p_{n,m}\}\subset C(Y, M_n)$ such that $$ \lim_{m\to\infty}
\sum_{i=1}^n f(\alfa_{i,m})p_{i,m}=\phi(f) for all f\in C(X). $$

We construct inner amenable groups G with infinite conjugacy classes and such
that the associated II_1 factor does not have property Gamma of Murray and von
Neumann. This solves a problem posed by Effros in 1975.

We establish axiomatic characterizations of $K$-theory and $KK$-theory for
real C*-algebras. In particular, let $F$ be an abelian group-valued functor on
separable real C*-algebras. We prove that if $F$ is homotopy invariant, stable,
and split exact, then $F$ factors through the category $KK$. Also, if $F$ is
homotopy invariant, stable, half exact, continuous, and satisfies an
appropriate dimension axiom, then there is a natural isomorphism $K(A) \to
F(A)$ for a large class of separable real C*-algebras $A$.

Our main result is a theorem saying that a bounded operator $A$ on a Hilbert
space belongs to a certain set associated with its self-commutator $[A^*,A]$,
provided that $A-zI$ can be approximated by invertible operators for all
complex numbers $z$. The theorem remains valid in a general $C^*$-algebra of
real rank zero under the assumption that $A-zI$ belong to the closure of the
connected component of unity in the set of invertible elements.

We complete the realisation by braided subfactors, announced by Ocneanu, of
all SU(3)-modular invariant partition functions previously classified by
Gannon.

We determine the cells, whose existence has been announced by Ocneanu, on all
the candidate nimrep graphs except $\mathcal{E}_4^{(12)}$ proposed by di
Francesco and Zuber for the SU(3) modular invariants classified by Gannon. This
enables the Boltzmann weights to be computed for the corresponding integrable
statistical mechanical models and provide the framework for studying
corresponding braided subfactors to realise all the SU(3) modular invariants as
well as a framework for a new SU(3) planar algebra theory.

We consider several orthogonal quantum groups satisfying the easiness
assumption axiomatized in our previous paper. For each of them we discuss the
computation of the asymptotic law of Tr(u^k) with respect to the Haar measure,
u being the fundamental representation. For the classical groups O_n, S_n we
recover in this way some well-known results of Diaconis and Shahshahani.

We demonstrate how virtually all common cardinal invariants associated to a
von Neumann algebra M can be computed from the decomposability number, dec(M),
and the minimal cardinality of a generating set, gen(M).

We give a new Banach module characterization of $W^*$-modules, also known as
selfdual Hilbert $C^*$-modules over a von Neumann algebra. This leads to a
generalization of the notion, and the theory, of W*-modules, to the setting
where the operator algebras are $\sigma$-weakly closed algebras of operators on
a Hilbert space. That is, we find the appropriate weak* topology variant of our
earlier notion of {\em rigged modules}, and their theory, which in turn
generalizes the notions of C*-module, and Hilbert space, successively.

In an earlier paper of mine relating vector bundles and Gromov-Hausdorff
distance for ordinary compact metric spaces, it was crucial that the Lipschitz
seminorms from the metrics satisfy a strong Leibniz property. In the present
paper, for the now non-commutative situation of matrix algebras converging to
the sphere (or to other spaces) for quantum Gromov-Hausdorff distance, we show
how to construct suitable seminorms that also satisfy the strong Leibniz
property.

We define the notion of a principal S-bundle where S is a groupoid group
bundle and show that there is a one-to-one correspondence between principal
S-bundles and elements of a sheaf cohomology group associated to S. We also
define the notion of a locally unitary action and show that the spectrum of the
crossed product is a principal bundle. Furthermore, we prove that the
isomorphism class of the spectrum determines the exterior equivalence class of
the action and that every principal bundle can be realized as the spectrum of
some locally unitary crossed product.

Let $\cX$ be the unital *-algebra generated by the unilateral shift operator.
It is shown that for any nonnegative operator $X\in \cX$ there is an element
$Y\in \cX$ such that $X=Y^*Y$.

We give a spectral theorem for unital representions of Hermitian commutative
unital *-algebras by possibly unbounded operators in a pre-Hilbert space. A
more general result is known for the case in which the *-algebra is countably
generated.