We prove a quantized version of a theorem by M. V. Shĕınberg: A uniform algebra equipped with its canonical, i.e. minimal, operator space structure is operator amenable if and only if it is a commutative C∗-algebra.

Let A be a unital Banach algebra, and denote the spectral radius of f ∈ A by ρ(f). If A is a uniform algebra and ρ(fh + 1) = ρ(gh + 1) for all h ∈ A, then it can be shown that f = g, a result that also carries in algebras of bounded linear operators on Banach spaces. On the other hand ρ(fh) = ρ(gh) does not imply f = g in any unital algebra, marking a distinction between the polynomials p(z, w)...

We give a characterization of the ”uniform closure” of the dual of a C-algebra. Some applications in harmonic analysis are given. 1. Four topologies on the unit ball of a C-algebra Let A be a C-algebra and P (A),S(A),A1, and A denote the pure state space, state space, closed dual unit ball, and the dual of A, respectively, all equipt with the w-topology. Let A1denote the closed unit ball of A. ...

Generalizing the case of an infinite discrete metric space finite diameter, we say that a (X,d) is Kuiper space, if group invertible elements its uniform Roe algebra norm-contractible. Various sufficient conditions on to be or not are obtained.

These lectures given in Montreal in Summer 1997 are mainly based on, and form a condensed survey of the book by N. Chriss and V. Ginzburg, Representation Theory and Complex Geometry , Birkhäuser 1997. Various algebras arising naturally in Representation Theory such as the group algebra of a Weyl group, the universal enveloping algebra of a complex semisimple Lie algebra, a quantum group or the ...