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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:

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