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104
The algebraic structure of noncommutative analytic Toeplitz algebras
 MATH.ANN
, 1998
"... The noncommutative analytic Toeplitz algebra is the wot–closed 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 a ..."
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Cited by 117 (17 self)
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The noncommutative analytic Toeplitz algebra is the wot–closed 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 nball. The kdimensional 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.
Subalgebras of C*algebras III: Multivariable operator theory
 ACTA MATH
, 1997
"... A dcontraction is a dtuple (T1,..., Td) of mutually commuting operators acting on a common Hilbert space H such that ‖T1ξ1 + T2ξ2 + · · · + Tdξd ‖ 2 ≤ ‖ξ1 ‖ 2 + ‖ξ2 ‖ 2 + · · · + ‖ξd ‖ 2 for all ξ1, ξ2,..., ξd ∈ H. These are the higher dimensional counterparts of contractions. We show that man ..."
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Cited by 103 (4 self)
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A dcontraction is a dtuple (T1,..., Td) of mutually commuting operators acting on a common Hilbert space H such that ‖T1ξ1 + T2ξ2 + · · · + Tdξd ‖ 2 ≤ ‖ξ1 ‖ 2 + ‖ξ2 ‖ 2 + · · · + ‖ξd ‖ 2 for all ξ1, ξ2,..., ξd ∈ H. These are the higher dimensional counterparts of contractions. We show that many of the operatortheoretic aspects of function theory in the unit disk generalize to the unit ball Bd in complex dspace, including von Neumann’s inequality and the model theory of contractions. These results depend on properties of the dshift, a distinguished dcontraction which acts on a new H 2 space associated with Bd, and which is the higher dimensional counterpart of the unilateral shift. H 2 and the dshift are highly unique. Indeed, by exploiting the noncommutative Choquet boundary of the dshift relative to its generated C ∗algebra we find that there is more uniqueness in dimension d ≥ 2 than there is in dimension one.
NevanlinnaPick interpolation for noncommutative analytic Toeplitz algebras
 OPERATOR THY
, 1998
"... The noncommutative analytic Toeplitz algebra is the wot–closed algebra generated by the left regular representation of the free semigroup on n generators. We obtain a distance formula to an arbitrary wotclosed right ideal and thereby show that the quotient is completely isometrically isomorphic to ..."
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Cited by 73 (15 self)
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The noncommutative analytic Toeplitz algebra is the wot–closed algebra generated by the left regular representation of the free semigroup on n generators. We obtain a distance formula to an arbitrary wotclosed right ideal and thereby show that the quotient is completely isometrically isomorphic to the compression of the algebra to the orthogonal complement of the range of the ideal. This is used to obtain Nevanlinna–Pick type interpolation theorems.
Noncommutative interpolation and Poisson transforms
 Israel J. Math
"... Abstract. General results of interpolation (eg. NevanlinnaPick) by elements in the noncommutative analytic Toeplitz algebra F ∞ (resp. noncommutative disc algebra An) with consequences to the interpolation by bounded operatorvalued analytic functions in the unit ball of C n are obtained. Noncommu ..."
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Cited by 70 (14 self)
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Abstract. General results of interpolation (eg. NevanlinnaPick) by elements in the noncommutative analytic Toeplitz algebra F ∞ (resp. noncommutative disc algebra An) with consequences to the interpolation by bounded operatorvalued analytic functions in the unit ball of C n are obtained. Noncommutative Poisson transforms are used to provide new von Neumann type inequalities. Completely isometric representations of the quotient algebra F ∞ /J on Hilbert spaces, where J is any w ∗closed, 2sided ideal of F ∞ , are obtained and used to construct a w ∗continuous, F ∞ /J–functional calculus associated to row contractions T = [T1,..., Tn] when f(T1,..., Tn) = 0 for any f ∈ J. Other properties of the dual algebra F ∞ /J are considered. In [Po5], the second author proved the following version of von Neumann’s inequality for row contractions: if T1,..., Tn ∈ B(H) (the algebra of all bounded linear operators on the Hilbert space H) and T = [T1,...,Tn] is a contraction, i.e., ∑n ∗ i=1 TiTi ≤ IH, then for every polynomial p(X1,..., Xn) on n noncommuting indeterminates, (1)
The structure of free semigroup algebras
 J. REINE ANGEW. MATH
, 1996
"... A free semigroup algebra is the wotclosed algebra generated by an ntuple of isometries with pairwise orthogonal ranges. The interest in these algebras arises primarily from two of their interesting features. The first is that they provide useful information about unitary invariants of representati ..."
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Cited by 56 (15 self)
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A free semigroup algebra is the wotclosed algebra generated by an ntuple of isometries with pairwise orthogonal ranges. The interest in these algebras arises primarily from two of their interesting features. The first is that they provide useful information about unitary invariants of representations of the Cuntz–Toeplitz algebras. The second is that they form a class of nonselfadjoint operator algebras which are of interest in their own right. This class contains a distinguished representative, the “noncommutative Toeplitz algebra, ” which is generated by the left regular representation of the free semigroup on n letters and denoted Ln. This paper provides a general structure theorem for all free semigroup algebras, Theorem 2.6, which extends results for important special cases in the literature. The structure theorem highlights the importance of the type L representations, which are the representations which provide a free semigroup algebra isomorphic to Ln. Indeed, every free semigroup algebra has a 2 × 2 lower triangular form where
Free semigroupoid algebras
, 2004
"... Every countable directed graph generates a Fock space Hilbert space and a family of partial isometries. These operators also arise from the left regular representations of free semigroupoids derived from directed graphs. We develop a structure theory for the weak operator topology closed algebras g ..."
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Cited by 45 (17 self)
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Every countable directed graph generates a Fock space Hilbert space and a family of partial isometries. These operators also arise from the left regular representations of free semigroupoids derived from directed graphs. We develop a structure theory for the weak operator topology closed algebras generated by these representations, which we call free semigroupoid algebras. We characterize semisimplicity in terms of the graph and show explicitly in the case of finite graphs how the Jacobson radical is determined. We provide a diverse collection of examples including; algebras with free behaviour, and examples which can be represented as matrix function algebras. We show how these algebras can be presented and decomposed in terms of amalgamated free products. We determine the commutant, consider invariant subspaces, obtain a Beurling theorem for them, conduct an eigenvalue analysis, give an elementary proof of reflexivity, and discuss hyperreflexivity. Our main theorem shows the graph to be a complete unitary invariant for the algebra. This classification theorem makes use of an analysis of unitarily implemented automorphisms. We give a graphtheoretic description of when these algebras are partly free, in the sense that they contain a copy of a free semigroup algebra.
Isometric Dilations of noncommuting finite rank ntuples
 CANAD. J. MATH
, 1999
"... A contractive ntuple A = (A1,..., An) has a minimal joint isometric dilation S = (S1,..., Sn) where the Si’s are isometries with pairwise orthogonal ranges. This determines a representation of the CuntzToeplitz algebra. When A acts on a finite dimensional space, the wotclosed nonselfadjoint al ..."
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Cited by 42 (16 self)
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A contractive ntuple A = (A1,..., An) has a minimal joint isometric dilation S = (S1,..., Sn) where the Si’s are isometries with pairwise orthogonal ranges. This determines a representation of the CuntzToeplitz algebra. When A acts on a finite dimensional space, the wotclosed nonselfadjoint algebra S generated by S is completely described in terms of the properties of A. This provides complete unitary invariants for the corresponding representations. In addition, we show that the algebra S is always hyperreflexive. In the last section, we describe similarity invariants. In particular, an ntuple B of d × d matrices is similar to an irreducible ntuple A if and only if a certain finite set of polynomials vanish on B.
Ideal structure in free semigroupoid algebras from directed graphs
 J. OPERATOR THEORY
, 2003
"... A free semigroupoid algebra is the weak operator topology closed algebra generated by the left regular representation of a directed graph. We establish lattice isomorphisms between ideals and invariant subspaces, and this leads to a complete description of the wotclosed ideal structure for these ..."
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Cited by 33 (6 self)
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A free semigroupoid algebra is the weak operator topology closed algebra generated by the left regular representation of a directed graph. We establish lattice isomorphisms between ideals and invariant subspaces, and this leads to a complete description of the wotclosed ideal structure for these algebras. We prove a distance formula to ideals, and this gives an appropriate version of the Carathéodory interpolation theorem. Our analysis rests on an investigation of predual properties, specifically the An properties for linear functionals, together with a general Wold Decomposition for ntuples of partial isometries. A number of our proofs unify proofs for subclasses appearing in the literature.
Isomorphisms of algebras associated with directed graphs
, 2004
"... Given countable directed graphs G and G′, we show that the associated tensor algebras T+(G) and T+(G ′) are isomorphic as Banach algebras if and only if the graphs G are G′ are isomorphic. For tensor algebras associated with graphs having no sinks or no sources, the graph forms an invariant for alge ..."
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Cited by 31 (12 self)
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Given countable directed graphs G and G′, we show that the associated tensor algebras T+(G) and T+(G ′) are isomorphic as Banach algebras if and only if the graphs G are G′ are isomorphic. For tensor algebras associated with graphs having no sinks or no sources, the graph forms an invariant for algebraic isomorphisms. We also show that given countable directed graphs G, G′, the free semigroupoid algebras LG and LG ′ are isomorphic as dual algebras if and only if the graphs G are G′ are isomorphic. In particular, spatially isomorphic free semigroupoid algebras are unitarily isomorphic. For free semigroupoid algebras associated with locally finite directed graphs with no sinks, the graph forms an invariant for algebraic isomorphisms as well.
Hardy algebras, W ∗ correspondences and interpolation theory
"... Given a von Neumann algebra M and a W ∗correspondence E over M, we construct an algebra H ∞ (E) that we call the Hardy algebra of E. When M = C = E, then H ∞ (E) is the classical Hardy space H ∞ (T) of bounded analytic functions on the unit disc. We show that given any faithful normal representatio ..."
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Cited by 29 (7 self)
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Given a von Neumann algebra M and a W ∗correspondence E over M, we construct an algebra H ∞ (E) that we call the Hardy algebra of E. When M = C = E, then H ∞ (E) is the classical Hardy space H ∞ (T) of bounded analytic functions on the unit disc. We show that given any faithful normal representation σ of M on a Hilbert space H there is a natural correspondence E σ over the commutant σ(M) ′, called the σdual of E, and that H ∞ (E) can be realized in terms of (B(H)valued) functions on the open unit ball D((E σ) ∗ ) in the space of adjoints of elements in E σ. We prove analogues of the NevanlinnaPick theorem in this setting and discover other aspects of the value