The algebraic structure of non-commutative analytic Toeplitz algebras

Abstract

Abstract. The non-commutative analytic Toeplitz algebra is the wotclosed algebra generated by the left regular representation of the free semigroup on n generators. We develop a detailed picture of the algebraic structure of this algebra. In particular, we show that there is a canonical homomorphism of the automorphism group onto the group of conformal automorphisms of the complex n-ball. The k-dimensional representations form a generalized maximal ideal space with a canonical surjection onto the ball of k × kn matrices which is a homeomorphism over the open ball analogous to the fibration of the maximal ideal space of H ∞ over the unit disk.In [6,17,18,20], a good case is made that the appropriate analogue for the analytic Toeplitz algebra in n non-commuting variables is the wotclosed algebra generated by the left regular representation of the free semigroup on n generators. The papers cited obtain a compelling analogue of Beurling's theorem and inner-outer factorization. In this paper, we add further evidence. The main result is a short exact sequence determined by a canonical homomorphism of the automorphism group onto this algebra onto the group of conformal automorphisms of the unit ball of C n . The kernel is the subgroup of quasi-inner automorphisms, which are trivial modulo the wot-closed commutator ideal. Additional evidence of analytic properties comes from the structure of k-dimensional (completely contractive) representations, which have a structure very similar to the fibration of the maximal ideal space of H ∞ over the unit disk. An important tool in our analysis is a detailed structure theory for wot-closed right ideals. Curiously, left ideals remain more obscure.The non-commutative analytic Toeplitz algebra L n is determined by the left regular representation of the free semigroup F n on n generators z 1 , . . . , z n which acts on 2 (F n ) by λ(w)ξ v = ξ wv for v, w in F n . In particular, the algebra L n is the unital, wot-closed algebra generated by the isometries L i = λ(z i ) for 1 ≤ i ≤ n. This algebra and its norm-closed version (the noncommutative disk algebra) were introduced by Popescu [19] in an abstract sense in connection with a non-commutative von Neumann inequality and

Isomorphisms of algebras associated with directed graphs

“…Therefore, x∈V(G) τ (P x ) = I, as desired. The proof of the following theorem is modelled on the proof from [3] for the special case of L n . Theorem 3.16.…”

Erratum to: Multipliers of Embedded Discs

“…Most of the results in Section 8 also hold, with the exception of Lemma 8. 4 and statements (1) and (3) in Theorem 8.5. Statements (1) and (3) are not known to hold in full generality, but do hold if one assumes that one of the spaces is H s , s < −1, so Example 8.6 still exhibits an uncountable family of non-isomorphic multiplier algebras associated to compact varieties.…”

Polynomial Endomorphisms of the Cuntz Algebras Arising from Permutations: I ? General Theory

“…The permutative representation was introduced by [3,5,6]. We generalize and give another characterization of them in [11,12,13,14].…”

“…This fact disturbs an intention to study an ordinary representation theory of operator algebras like that of semisimple Lie algebras and quantum groups. In spite of this, permutative representations of the Cuntz algebra O N ( [3,5,6]) are completely reducible and their irreducible decompositions are unique up to unitary equivalences. Roughly speaking, there are two kinds of (cyclic)permutative representations, "cycle" and "chain".…”