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191
The algebraic structure of noncommutative analytic Toeplitz algebras
 Math.Ann
, 1998
"... Abstract. 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 homomorphi ..."
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Cited by 76 (12 self)
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Abstract. 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. In [6, 17, 18, 20], a good case is made that the appropriate analogue for the analytic Toeplitz algebra in n noncommuting 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
A Unifying Construction of Orthonormal Bases for System Identification
 IEEE TRANSACTIONS ON AUTOMATIC CONTROL
, 1994
"... In this paper we develop a general and very simple construction for complete orthonormal bases for system identification. This construction provides a unifying formulation of all known orthonormal bases since the common FIR and recently popular Laguerre and Kautz model structures are restrictive spe ..."
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Cited by 54 (20 self)
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In this paper we develop a general and very simple construction for complete orthonormal bases for system identification. This construction provides a unifying formulation of all known orthonormal bases since the common FIR and recently popular Laguerre and Kautz model structures are restrictive special cases of our construction as is another construction method based on balanced realisations of all pass functions. However, in contrast to these special cases, the basis vectors in our unifying construction can have nearly arbitrary magnitude frequency response according to the prior information the user wishes to inject into the problem. We also provide results characterising the completeness properties of our bases.
Ideal structure in free semigroupoid algebras from directed graphs
 J. Operator Theory
"... Abstract. 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 fo ..."
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Cited by 49 (5 self)
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Abstract. 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. In [18] and [19], the second author and Stephen Power began studying a class of operator algebras called free semigroupoid algebras. These are the wotclosed (nonselfadjoint) algebras LG generated by the left regular representations of directed graphs G. Earlier work of Muhly and Solel [24, 25] considered the norm closed algebras generated by these representations in the finite graph case; they called them quiver algebras. In the case of single vertex graphs, the LG obtained include the classical analytic Toeplitz algebra H ∞ [13, 14, 29] and the noncommutative analytic Toeplitz algebras Ln studied by Arias, Popescu,
Free semigroupoid algebras
, 2003
"... Abstract. 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 a ..."
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Cited by 36 (14 self)
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Abstract. 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. 1.
Recurrence and asymptotics for orthogonal rational functions on an interval
 IMA Journal of Numerical Analysis
"... Let {α1, α2,...} be a sequence of real numbers outside the interval [−1, 1] and µ a positive bounded Borel measure on this interval. We introduce rational functions ϕn(x) with poles {α1,..., αn} orthogonal on [−1, 1] and establish some ratio asymptotics for these orthogonal rational functions, i.e. ..."
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Cited by 30 (22 self)
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Let {α1, α2,...} be a sequence of real numbers outside the interval [−1, 1] and µ a positive bounded Borel measure on this interval. We introduce rational functions ϕn(x) with poles {α1,..., αn} orthogonal on [−1, 1] and establish some ratio asymptotics for these orthogonal rational functions, i.e. we discuss the convergence of ϕn+1(x)/ϕn(x) as n tends to infinity under certain assumptions on the measure and the location of the poles. From this we derive asymptotic formulas for the recurrence coefficients in the three term recurrence relation satisfied by the orthonormal functions. 1
KullbackLeibler approximation of spectral density functions
 IEEE Trans. Inform. Theory
, 2003
"... Abstract—We introduce a Kullback–Leiblertype distance between spectral density functions of stationary stochastic processes and solve the problem of optimal approximation of a given spectral density 9 by one that is consistent with prescribed secondorder statistics. In general, such statistics are ..."
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Cited by 26 (15 self)
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Abstract—We introduce a Kullback–Leiblertype distance between spectral density functions of stationary stochastic processes and solve the problem of optimal approximation of a given spectral density 9 by one that is consistent with prescribed secondorder statistics. In general, such statistics are expressed as the state covariance of a linear filter driven by a stochastic process whose spectral density is sought. In this context, we show i) that there is a unique spectral density 8 which minimizes this Kullback–Leibler distance, ii) that this optimal approximate is of the form 9 where the “correction term ” is a rational spectral density function, and iii) that the coefficients of can be obtained numerically by solving a suitable convex optimization problem. In the special case where 9=1, the convex functional becomes quadratic and the solution is then specified by linear equations. Index Terms—Approximation of power spectra, crossentropy minimization, Kullback–Leibler distance, mutual information, optimization, spectral estimation. I.
Analytic algebras of higher rank graphs
 Math. Proc. Royal Irish Acad
"... Abstract. We begin the study of a new class of operator algebras that arise from higher rank graphs. Every higher rank graph generates a Fock space Hilbert space and creation operators which are partial isometries acting on the space. We call the weak operator topology closed algebra generated by th ..."
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Cited by 24 (9 self)
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Abstract. We begin the study of a new class of operator algebras that arise from higher rank graphs. Every higher rank graph generates a Fock space Hilbert space and creation operators which are partial isometries acting on the space. We call the weak operator topology closed algebra generated by these operators a higher rank semigroupoid algebra. A number of examples are discussed in detail, including the single vertex case and higher rank cycle graphs. In particular the cycle graph algebras are identified as matricial multivariable function algebras. We obtain reflexivity for a wide class of graphs and characterize semisimplicity in terms of the underlying graph. In [22] Kumjian and Pask introduced kgraphs as an abstraction of the combinatorial structure underlying the higher rank graph C ∗algebras of Robertson and Steger [31, 32]. A kgraph generalizes the set of finite paths of a countable directed graph when viewed as a partly defined multiplicative semigroup with vertices considered as degenerate paths. The C ∗algebras associated with kgraphs include kfold tensor products of graph C ∗algebras, and much more [2, 21, 26, 27, 30]. On the other hand, as a generalization of the nonselfadjoint free semigroup
Isomorphisms of algebras associated with directed graphs
, 2003
"... Abstract. 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 invari ..."
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Cited by 24 (11 self)
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Abstract. 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. 1. introduction and Preliminaries Let G be a countable directed graph with vertex set V(G), edge set E(G) and range and source maps r and s respectively. The Toeplitz algebra of G, denoted as T (G), is the universal C ∗algebra generated by a set of partial isometries {Se}e∈E(G) and projections {Px}x∈V(G) satisfying the relations (1) PxPy = 0 for all x, y ∈ V(G), x ̸ = y (2) S ∗ eSf = 0 for all e, f ∈ E(G), e ̸ = f (3) S ∗ eSe = Ps(e) for all e ∈ E(G)
Generalized interpolation in H∞ with a complexity constraint
 TRANS. AMER. MATH. SOC
, 2006
"... In a seminal paper, Sarason generalized some classical interpolation problems for H∞ functions on the unit disc to problems concerning lifting onto H² of an operator T that is defined on K = H² ⊖φH² (φ is an inner function) and commutes with the (compressed) shift S. In particular, he showed that in ..."
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Cited by 24 (12 self)
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In a seminal paper, Sarason generalized some classical interpolation problems for H∞ functions on the unit disc to problems concerning lifting onto H² of an operator T that is defined on K = H² ⊖φH² (φ is an inner function) and commutes with the (compressed) shift S. In particular, he showed that interpolants (i.e., f ∈ H ∞ such that f(S)=T) having norm equal to �T � exist, and that in certain cases such an f is unique and can be expressed as a fraction f = b/a with a, b ∈ K. In this paper, we study interpolants that are such fractions of K functions and are bounded in norm by 1 (assuming that �T � < 1, in which case they always exist). We parameterize the collection of all such pairs (a, b) ∈ K × K and show that each interpolant of this type can be determined as the unique minimum of a convex functional. Our motivation stems from the relevance of classical interpolation to circuit theory, systems theory, and signal processing, where φ is typically a finite Blaschke product, and where the quotient representation is a physically meaningful complexity constraint.