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NORTH BRITISH FUNCTIONAL ANALYSIS SEMINAR
Report on Meeting at Sheffield.
NBFAS first meeting at Sheffield
took place on 4th March 1996
with Dr Christain Le Merdy (Franche-Comte) as speaker.
Operator algebras structures on commutative and non commutative $ l^p$ spaces.
The lectures considered
which Banach spaces could be viewed as algebras of operators on Hilbert space.
This topic attracted the attention of Varopoulos, Davie and others in the late 1960's
and the speaker began with a review of the classical results. Since then, the theory of operator
spaces has developed, and the first lecture included a detailed description of the new formalism of
matricially normed spaces.
In the second lecture the operator space theory was applied to describe operator algebra
structures on the Schatten ideals $c^p ,$ and on row/column spaces of matrices.
Two of the most significant results were:
- $c^p ,$ is a $Q$-algebra for $2 p 4 ,$, and
- $ row / column / ,$ space is a matricial $Q$-algebra.
The proof of the second result depends upon an operator algebra version
of Khintchine's inequality, as
interpreted in the non-commutative setting by Lust-Piquard and Pisier.
The attendance of 16 was relatively low, presumably due to the travel time to Sheffield.
Report on Meeting at Edinburgh
The 1996 Summer Meeting at Edinburgh was held at George Square on the Late Spring Bank Holiday with
Ken Dykema (Odense) and V. Vasyunin (St. Petersburg) as speakers.
Free probability theory and von Neumann algebras related to free groups
Dykema's first lecture was an introduction to non commutative probability theory. The fundamental notion of freeness
was introduced, motivated by Voiculescu's matrix model. Although this idea arises in statistical mechanics,
the speaker's main concern was with $L( F _n) ,$ the von Neumann algebras
generated by the left regular representation of
the free group $ F _n ,$. A problem of Kadison asks whether different $n \geq 2 ,$ give rise to distinct
algebras. The speaker discussed several results relating to the dimension of
$L( F _n) ,$ and interpolated groups $ F _t ,$.
The second lecture described recent work of Dykema, Haagerup and Rordam
on free entropy. This concept was introduced by Voiculescu to analyse maximal abelian subalgebras of
$L( F _n) ,$. Some of the continuity properties of entropy depend upon deep probabilitic inequalities
related to the isoperimetric inequality.
This lecture was an attractive introduction to a subject which
will develop in importance.
After de Branges' proof of the Bieberbach conjecture
At the Leeds meeting of 1992, Nikolskii described an opertor-theoretic
approach to the de Branges' Theorem on coefficients $\hat\varphi (n)\,$ of univalent functions
$\var
phi (z)\,$ on ${\Bbb D}\,$.
Vasyunin, the co-author of this epic work, showed how the Robertson, Milin and Bieberbach
conjecture
s could be viewed
as operator inequalities for the composition operator $f\mapsto f\circ \varphi\,$ on Dirichlet
space
$\{f\, {\rm holomorphic\/}\,: \int\!\!\int_{\Bbb D}\vert f'\vert^2<\infty \}\,$.
\indent The lecture included as clear discussion of the Lowner equation. This was originally
introduced to
settle particular cases of the Bieberbach conjecture, but more generally could be viewed
as a differential equation satisfied by extremal vectors for an evolution
family of composition operators.
Beurling's description of the shift invariant subspaces and the Riemann
Hypothesis
The second lecture of Vasyunin began with a striking statement of how the Riemann
Hypothesis
is implied by
a density condition for subspaces of Hilbert space
$L^2((1,\infty ) ,dx/x^2)\,$. Several other Hilbert space
approahes to the Riemann Hypothesis were described, some of which made sense of Mellin
transform calculations involving
$\zeta (s)\,$ which had previously baffled this reviewer.
Fair weather, good talks and an attendance of over thirty contributed to a successful
meeting.
Dr. G. Blower (Secretary),
e-mail:
G.Blower@lancaster.ac.uk
Tel: 01524 593962