A Note on Absolute Continuity in Free Semigroup Algebras

An absolutely continuous free semigroup algebra is weak-∗ type L. A free semigroup algebra is the wot-closed (nonself-adjoint, unital) algebra S generated by n isometries S1, . . . , Sn with pairwise orthogonal ranges. See [4] for an introduction. There is now a significant literature on these algebras [1, 2, 9, 10, 11, 7, 6, 8, 5, 16, 15, 18, 17, 20]. The prototype is the non-commutative Toeplitz algebra Ln, the algebra generated by the left regular representation λ of the free semigroup on n letters, Fn . A free semigroup algebra is type L if it is isomorphic to Ln. Algebraic isomorphism implies the much stronger property that they are completely isometrically isomorphic and weak-∗ homeomorphic. An open problem of central importance is whether every type L representation has a wandering vector; i.e. a vector ξ such that the set {Swξ : w ∈ Fn } is orthonormal. Here we write Sw = Sik . . . Si1 for a word w = ik . . . i1 in Fn . The C*-algebra generated by n isometries is ∗-isomorphic to either the Cuntz algebra On if ∑n i=1 SiS ∗ i = I or the Cuntz-Toeplitz algebra En if ∑n i=1 SiS ∗ i < I. The norm closed nonselfadjoint subalgebra generated by S1, . . . , Sn is denoted by An, the non-commutative analytic disk algebra introduced by Popescu [16]. The quotient map of En onto On is completely isometric on An. So it may also be considered as a subalgebra of On, which is its C*-envelope (because On is simple). So An sits isometrically inside σ(En) for every ∗-representation σ. Let the abstract generators of An and En be denoted by s1, . . . , sn. We consider ∗-representations of En and On as a natural way of describing n-tuples of isometries with orthogonal ranges. If σ is a representation of En, then let Sσ = σ(An) wot denote the corresponding free semigroup algebra. Note that σ splits as σ = λ ⊕ τ , where λ is the identity representation of En and τ factors through the quotient onto On. This is the C*-algebra equivalent of the Wold decomposition. In [6], a structure theorem was established which shows that there is a canonical projection P in S which is coinvariant so that SP = WP , where W is the von Neumann algebra generated by S, and S|P⊥H is type L. In [8], a notion of absolute continuity was introduced in order to further refine the analysis of free semigroup algebras, and of weaker type L representations in particular. A linear functional φ on An is absolutely continuous if it extends to a weak-∗-continuous functional on Ln. In this case, there are vectors ζ, η in the Fock space ` (Fn ) so that φ(A) = 〈λ(A)ζ, η〉. In particular, the wot topology and the weak-∗ topology coincide on Ln [9]. A vector in Hσ is called absolutely continuous if the functional φ(A) = 〈σ(A)x, x〉 is 2000 Mathematics Subject Classification. 47L55,46L35.