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\title{{\small
Special Session for the first AMS-UMI\\ Joint Meeting to 
be held in Pisa June 12-16 2002:}\\[1eM]
OPERATOR ALGEBRAS\\ Saturday June 15, 2002}
\author{(S.Doplicher, E.G.Effros)}
\date{}
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\maketitle
\begin{center}
{\large\bf SCHEDULE:}\\[2eM]
\begin{tabular}{cl}
\phantom{0}9:00 -- 9:45&  E.G.Effros       \\
\phantom{0}9:50 -- 10:20&  C.Pinzari            \\
\phantom{k}& \\
10:50 -- 11:35&  D.V.Voiculescu   \\
11:40 -- 12:10&  D.Guido            \\  
12:20 -- 12:50&  N.P.Brown         \\
\phantom{k}& \\
15:00 -- 15:45&  M.Roerdam      \\
15:50 -- 16:20&  J.E.Roberts      \\      
\phantom{k}&\\
16:40 -- 17:25&  R.Longo        \\
17:30 -- 18:15&  S.Popa         \\
18:20 -- 18:35&  T.Isola          \\         
18:35 -- 18:50&  S.Carpi            \\       
\end{tabular}\\[2eM]

\newpage
{\large\bf TITLES and ABSTRACTS:}
\end{center}


\paragraph*{Nathaniel P. Brown: 
Aspects of Representation Theory of C*-algebras}

\begin{abstract} 

We discuss some general questions in the  representation theory of
C*-algebras and how these  questions are related to other
(non-representation theoretic) questions in operator algebras. 

\end{abstract}


\paragraph*{Sebastiano Carpi: The Virasoro algebra and sectors with
infinite statistical dimension}

\begin{abstract}

We describe recent results on the existence of sectors with
infinite statistical dimension for the local net of von Neumann algebras
on the circle associated to the Virasoro algebra with central charge c=1.
\end{abstract}


\paragraph*{Edward G.Effros: Aspects of smoothness and rotundity for operator spaces}


\begin{abstract}
We show that the classical geometrical notions of smoothness and
rotundity indeed have operator space analogues.
\end{abstract}


\paragraph*{Daniele Guido: Modular localization and Wigner particles}

\begin{abstract}
A framework for the free field construction of algebras of local observables
is proposed, which uses as an input the Bisognano-Wichmann relations and a
representation of the Poincare' group on the one-particle Hilbert space.
The abstract real Hilbert subspace version of the Tomita-Takesaki theory
enables one to bypass some limitations of the Wigner formalism by
introducing an intrinsic spacetime localization. 
This approach works also for continuous spin representations to which a
net of von Neumann algebras on spacelike cones with the Reeh-Schlieder
property is associated. The positivity of the energy in the representation
turns out to be equivalent to the isotony of the net, in the spirit of
Borchers theorem. 
This procedure extends to other spacetimes homogeneous under a group of
geometric transformations as in the case of conformal symmetries and de
Sitter spacetime.
(Joint work with Romeo Brunetti and Roberto Longo)
\end{abstract}


\paragraph*{Tommaso Isola: Fractals in Noncommutative Geometry }

\begin{abstract}

     To any spectral triple $(A,D,H)$ of the Alain Connes'
     noncommutative geometry a dimension $d$ is associated,
     in analogy with the Hausdorff dimension for metric spaces. Indeed
     $d$ is the unique number, if any, such that $|D|^{-d}$ has non
     trivial logarithmic Dixmier trace. Moreover, when $d$ is finite
     nonzero, there always exists a singular trace which is nontrivial on
     $|D|^{-d}$, giving rise to a noncommutative integration on $A$.
     However $d$ is not characterized by this property in general:
     such numbers form an interval, the traceability interval.
     Such results are applied to fractals in $R$, using Connes'
     spectral triple, and to limit fractals in $R^{n}$, a class
     which generalises self-similar fractals, using a new spectral
     triple.  The noncommutative dimension or measure can be
     computed in some cases.  They are shown to coincide with the
     (classical) Hausdorff dimension and measure in the case of
     self-similar fractals. For a class of fractals, the endpoints of the
     traceability interval coincide with some (upper and lower) dimensions.
     (Joint work with D. Guido)
\end{abstract}


\paragraph*{Roberto Longo: Classification of Local Conformal Nets. The Discrete Series}

\begin{abstract}

We explain the structure of diffeomorphism invariant local
conformal nets of von Neumann algebras on the circle (chiral components of
2-dimensional conformal quantum field theories). Each net corresponds to an
extension of an infinite dimensional Lie algebra, the Virasoro algebra. The
case the central charge of the Virasoro algebra is less than 1 gives the
discrete series. We give a complete classification of the discrete series
(joint work with Y. Kawahigashi).
\end{abstract}


\paragraph*{Claudia Pinzari: Jones index for countably generated Hilbert C*-bimodules.}

\begin{abstract}

We introduce a notion of Jones index in the C*--category
of (countably generated) Hilbert C*-bimodules over nonunital
$C^*$--algebras with centres. Our definition relies on two axioms:
 an anlytic one and an algebraic one. The analytic axiom is essentially a
Pimsner--Popa inequality, while the algebraic axiom requires
that the left actions are contained in the compact operators.
We then show the existence of countable unconditionally convergent bases,
and we construct the index value as a central element in the coefficient
algebra of the bimodule.
Frank and Kirchberg have recently developped an index theory for
conditional expectations requiring only a Pimsner--Popa inequality.
However, the Jones basic construction does not exist in general. We show
that under our additional algebraic axiom, a Jones basic construction
can always be performed.
We characterize bimodules of finite index algebraically as those
with finite intrinsic dimension in the sense of Longo and Roberts.
We discuss several examples. In particular we show that 
a typical inclusion of commutative unital C*-algebras for which a finite 
Watatani basis does not exist is actually determined by a nonunital 
subinclusion with finite Jones index in our sense. 
(Joint work with T. Kajiwara and Y. Watatani)
\end{abstract}

\newpage

\paragraph*{Sorin Popa: Betti numbers and rigidity for group-measure space factors.} 

\begin{abstract}

I will explain a new strategy  for studying cross product algebras 
$M=B \rtimes_\sigma \Gamma_0$,  which works whenever a combination of rigidity 
and weak-amenability properties for $\Gamma_0$ and $\sigma$ are met.
This method allows to obtain very fine invariants and rigidity results 
for large classes of factors, and solve a multitude of problems, 
including partial answers to some conjectures of Connes. 
A key application is the consideration of a notion of Betti numbers 
for factors $M = L^\infty(X, \mu) \rtimes_\sigma \Gamma_0$, 
by proving that the L$^2$-Betti numbers for the groups 
$\Gamma_0$ considered by Cheeger-Gromov, and Gaboriau's 
L$^2$-Betti numbers for the measurable equivalence relations 
implemented by $\Gamma_0$ on the  probability space $(X, \mu)$, 
pass to the factors $M$. 
\end{abstract}


\paragraph*{John E.Roberts:  Sector Structure in Curved Spacetime}

\begin{abstract}

We discuss the extent to which results on sector structure
can be generalized to curved spacetime pointing out the problems that
arise and the new techniques designed to overcome them.
\end{abstract}


\paragraph*{Mikael Roerdam : A nuclear simple C*-algebra with a finite
and an infinite projection}

\begin{abstract}

It was asked around 1980 if every simple finite C*-algebra must
be stably finite and if every infinite simple C*-algebra must be purely
infinite. These questions are also relevant in the current classification
program of Elliott.

In this talk I will present a nuclear, separable counterexample to both
questions. More specifically, there is a C*-algebra $A$, which is an
inductive limit of type I C*-algebras (hence nuclear and in the UCT class)
and an automorphism on A such that the crossed product, $B$, of $A$ by the
$Z$-action induced on $A$ by that automorphism, is simple, separable, and
contains an infinite and a non-zero finite projection.

The C*-algebra $B$ provides a counterexample to Elliott's classification
conjecture (as it is formulated today - but if one replaces $K_0$ with the
dimension range, then the conjecture is still open). The C*-algebra $B$ is
also an example of a nuclear, simple C*-algebra that is tensorially prime
(i.e, is not a tensor product of two non-type I C*-algebras).

The example is a refinement of an earlier, non-nuclear example of the same
type of phenomenon, and its construction uses ideas of Villadsen.
\end{abstract}


\paragraph*{Dan V.Voiculescu: Topics in free probability theory.}

\begin{abstract}

Free probability theory is a highly noncommutative probability theory ,
with independence modelled on free products instead of tensor products.
There are important  applications to operator algebras and to
random matrices . The talk will deal with recent advances in the theory .

\end{abstract}


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