We characterize bilinear forms V on such that V(e, e) = ‖V‖ = 1 in terms of their matrices. For such V we prove that |V(x, y)|2≦φ(|x|2)ψ(|y|2) for all x, y, where φ(x)= V(x, e), ψ(y) = V(e, y). Some other properties of such forms are given.

A class of extremal Banach algebras has Arens irregular multiplication.

Let be a finite dimensional toric variety over an algebraically closed field of characteristic zero, k. Let be the sheaf of differential operators on . We show that the ring of global sections, is a finitely generated Noetherian k-algebra and that its generators can be explicitly found. We prove a similar result for the sheaf of differential operators with coefficients in a line bundle.

Given a finite (connected) simplicial graph with groups assigned to the vertices, the graph product of the vertex groups is the free product modulo the relation that adjacent groups commute. The graph product of finitely presented infinite groups is both semistable at infinity and quasi-simply filtrated. Explicit bounds for the isoperimetric inequality and isodiametric inequality for graph products is given, based on isoperimetric and isodiametric inequalities for the vertex groups.

For a large class of C*-algebras including all von Neumann algebras, the central Haagerup tensor product of the multiplier algebra with itself has an isometric representation as completely bounded operators.

Let E, F, G be three compact sets in ℂn. We say that (E, F, G) holds if for any choice of an interpolating array in F and of an analytic function ℂ on G, the Kergjn interpolation polynomial of ℂ exists and converges to ℂ on E. Given two of the three sets, we study how to construct the third in order that (E, F, G) holds.

We prove a technical result required by Ivanov and Shpectorov in their construction of a non-split extension of 34371 by the Baby Monster simple group.