A Simple C * -Algebra Generated by Two Finite-Order Unitaries

Source abstract

We present an example which illustrates several peculiar phenomena that may occur in the theory of C *-algebras. In particular, we show that a C *-subalgebra of a nuclear (amenable) C *-algebra need not be nuclear (amenable). The central object of this paper is a pair of abstract unitary matrices, acting on a common Hilbert space. For an explicit construction, we may decompose an infinite-dimensional Hilbert space H into H = H 0 ⴲ H 1 , H 1 = H α ⴲ H β with dim H 0 = dim H 1 = dim H α = dim H β , letting u, v Є B(H) be any two unitary operators such that and u 2 = 1, v 3 = 1. Whereas many choices of u, v are possible, it might be surprising to see that C * (u, v), the C *-algebra generated by u and v, is algebraically unique; namely, if (u 1 ,V 1 ) is another pair of such unitaries, then C*(u, v) is canonically *-isomorphic with C *( u 1 , v 1 ) (Theorem 2.6).