B(h) Is a Free Semigroup Algebra
نویسنده
چکیده
We provide a simplified version of a construction of Charles Read. For any n ≥ 2, there are n isometries with orthogonal ranges with the property that the nonself-adjoint weak-∗-closed algebra that they generate is all of B(H). A free semigroup algebra is the weak operator topology closed algebra generated by an n-tuple of isometries with pairwise orthogonal ranges. The C*-algebra generated by these isometries is called the Cuntz algebra On, introduced in [4]. Cuntz algebras arise frequently in the theory of C*-algebras as fundamental building blocks. They also play a crucial role in the analysis of endomorphisms of B(H) beginning with the seminal work of Powers [20]. Free semigroup algebras have been studied extensively in recent years [1, 5, 6, 7, 8, 9, 10, 11, 14, 16, 17, 18, 19]. In part, this is because of their intrinsically interesting properties as nonself-adjoint algebras. However it is also the case that these algebras reveal invariants of the corresponding C*-algebra representations. As such, they have allowed the classification of certain families of representations of the Cuntz algebra [9, 7] which are relevant to recent work of Bratteli and Jorgensen, who use the Cuntz algebras to generate wavelets [2, 3, 12, 13]. In [9], we first raised the question of whether a free semigroup algebra could be a von Neumann algebra. In [6, 8], the special structure theory of these algebras has been revealed to a significant degree, and this question was seen as a central unresolved issue. Recently, Charles Read [21] constructed a representation of O2 for which the free semigroup is all of B(H). The purpose of this note is to provide a more transparent view of his construction.
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