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We give a concrete description of an isometryv from l2(ℕ×ℤ×ℤ×ℤ) to l2(ℕ×ℤ×ℕ×ℤ) whose existence has recently been discovered by Woronowicz [11]. The isometryv gives the comultiplication δ on the C*-algebraA of the quantum group SU(2) q through the formula δ(x)=v(x⊗1)v*(x∈A), where 1 is the identity operator on l2(ℤ×ℤ). The matrix entries ofv are described in terms of littleq-Jacobi polynomials. Usingv, we give a concrete description of a unitary operatorV onH η ⊗H η such that (πη⊗πη)δ(x)=V(πη(x⊗1)V*, whereHη=l2(ℕ×ℤ×ℕ) and πη:A→L(Hη) is the GNS representation associated with the Haar state η onA. The operatorV satisfies the pentagonal identity of Baaj and Skandalis [1].

There are many examples of C*-algebras that are most naturally specified in terms of generators and relations. But there has been very little work on the development of a general theory of finitely-presented C*-algebras. (The papers [1], [2], [4], [5], [7] and [8] touch on the topic, although this list is by no means complete.) In this Note, we shall pose a few problems about such algebras, indicate the progress that has been made towards solving these problems, and present a case study to illustrate the theory. Most of the problems and results below are due to P. Tapper, and are expounded in more detail in [11].

Hilbert C*-modules are objects like Hilbert spaces, except that the inner product, instead of being complex valued, takes its values in a C*-algebra. The theory of these modules, together with their bounded and unbounded operators, is not only rich and attractive in its own right but forms an infrastructure for some of the most important research topics in operator algebras. This book is based on a series of lectures given by Professor Lance at a summer school at the University of Trondheim. It provides, for the first time, a clear and unified exposition of the main techniques and results in this area, including a substantial amount of new and unpublished material. It will be welcomed as an excellent resource for all graduate students and researchers working in operator algebras.

Let N be a von Neumann subalgebra of a von Neumann algebra M. A linear mapping π: M → N is called a retraction if it is idempotent and has norm one. By a result of Tomiyama(15) a retraction is a positive mapping and is a module homo-morphism over N. A retraction is normal if it is ultraweakly continuous, and faithful if it does not annihilate any nonzero positive element of M. Suppose that (Nn)n≥1 is an increasing sequence of von Neumann subalgebras of M whose union is weakly dense in M and that, for each n, πn: M → Nn is a faithful normal retraction. The sequence (πn) is called a martingale if, whenever m ≥ n