**4'th Öresund Symposium on**

**Non-commutative Geometry and Non-commutative Analysis**

**Lund, Sweden, **

** **

**Thursday, November 25, 2004**

**Organisers**

**Lund:**
Sergei Silvestrov
**
Copenhagen:**
Søren Eilers, Frank Hansen,

**Location**

Centre for Mathematical Sciences,
Lund University, Lund, Sweden

Third floor, Auditorium 332A, (10.00-15.00), Auditorium 362D, (15.30-17.30)

10.10-10.15 ** Welcome Address** (Auditorium 332A)

10.15-11.05 **
Yacin Ameur**, Kalmar university, Sweden

Calderon couples, interpolation functions and monotone matrix functions.

**Abstract:** Let $X_0$ and $X_1$ be banach spaces such
that $X_0\cap X_1$

is dense in $X_0$ and in $X_1$. A basic goal of interpolation theory is

to characterize those spaces $X_*$ which fulfill

$X_0\cap X_1\subseteq X_*\subseteq X_0+X_1$ and the interpolation

inequality:

$$\|T\|_{{\cal L}(X_*)}\le \max(\|T\|_{{\cal L}(X_0)},\|T\|_{{\cal L}(X_1)})$$

whenever $T$ is an operator which is defined and bounded on $X_0$ and

on $X_1$.

In the case when $X_0$ and $X_1$ are Hilbert spaces, the corresponding

Banach spaces $X_*$ can be completely characterized, and they

generalize the monotone matrix functions of Löwner. I will speak about this and

some realted results and problems.

11.05-11.55 ** Gunnar Sparr,** Lund
University (LTH), Sweden.

Monotone matrix functions and the Foias-Lions
interpolation problem

**Abstract:** The Foias-Lions problem concerns
characterization of the exact

interpolation functions for weighted $L^p$-spaces, i.e. functions $h$

obeying

$$

||Tf||_{L^p_{w_i}} \le ||f||_{L^p_{v_i}}, \ i=0,1 \Longrightarrow ||Tf||_{L^p_{h(w_0,w_1}} \le ||f||_{L^p_{h(v_0,v_1)}}\ .

$$

For $p=2$, a characterization is provided by the Löwner theorem for

monotone matrix functions.

The presentation will reveal this connection and discuss an

unsuccessful attempt 25 years ago to solve the Foias-Lions problem, starting by a

non-Hilbertian proof of Löwner's theorem in the case $p=2$,

cf. G. Sparr, A new proof of Löwner's theorem on monotone matrix

functions, Math. Scand. 1980.

12.00-13.10 **Lunch**

13.15-14.05 **Sten Kaijser,** Uppsala University,
Sweden

Interpolation of Banach algebras and Hilbert spaces

14.10-15.00 **Frank Hansen,**
University of Copenhagen, Denmark

Monotone trace functions of several variables.

15.00-15.30
Coffee at the department (in the lunch room at the
4:th floor)

Note change of room to Auditorium 362D

15.30-16.20 **
Niels Jakob Laustsen**, University of
Copenhagen, Denmark

Involutions on Banach algebra of operators on a Banach
space

**Abstract:** I shall report on ongoing joint work with
Matt Daws (Oxford,

formerly Leeds) and Charles Read (Leeds), where we define and study

involutions on the Banach algebra $\mathcal{B}(E)$ of all bounded,

linear operators on a Banach space $E$. Our motivating example is the standard

involution on $\mathcal{B}(H)$ for a Hilbert space $H$.

16.25-17.15 **Toke Carlsen,** NTNU,
Trondheim, Norway

On C*-algebras of actions of inverse semigroup

**Abstract:** I will talk about C*-algebras associated with actions of
inverse

semigroups on sets (without topology or any other structure).

These C*-algebras can be described as universal C*-algebras

generated by partial isometries subject to conditions given

by the inverse semigroup and a Boolean algebra.

I will describe how Cuntz-Krieger algebras (both for finite

and infinite matrices), C*-algebras associated to shift spaces

and C*-algebras of higher-rank graphs in a very natural

way can be constructed as C*-algebras of actions of inverse semigroups.

This allow us to make natural generalizations of these

C*-algebras.

18.00- If weather is enough good, then interested participants join for a walk

to a dinner in one of the restaurants in the centre of Lund