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

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 = H0 ⴲ H1 , H1 = Hα ⴲ Hβ with dim H0 = dim H1 = dim Hα = dim Hβ, letting u, v Є B(H) be any two unitary operators such thatand u2 = 1, v3 = 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 (u1,V1) is another pair of such unitaries, then C*(u, v) is canonically *-isomorphic with C*(u1, v1) (Theorem 2.6).