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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 47 (6 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)
Poisson Transforms On Some ...Algebras Generated By Isometries
 J. Func. Anal
, 1999
"... . A noncommutative Poisson transform associated to a certain class of sequences of operators on Hilbert spaces, with property (P), is defined on some universal C algebras (resp. nonselfadjoint algebras) generated by isometries. Its properties are described and used to study these universal alge ..."
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Cited by 38 (8 self)
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. A noncommutative Poisson transform associated to a certain class of sequences of operators on Hilbert spaces, with property (P), is defined on some universal C algebras (resp. nonselfadjoint algebras) generated by isometries. Its properties are described and used to study these universal algebras and their representations. As consequences, we obtain a functional calculus, isometric (resp. unitary) dilations, and commutant lifting theorem for the class of sequences of operators with property (P). Our "geometrical" approach leads also to new and elementary proofs as well as extensions of some classical results. 1. Introduction and preliminaries Let H be a Hilbert space and B(H) be the algebra of all bounded linear operators on H. Let T 2 B(H) be a contraction, i.e., kTk 1 and denote \Delta(T ) := I H \Gamma TT . It is easy to see that for each 0 ! r ! 1 (1.1) 1 X n=0 (rT ) n \Delta(rT )(rT ) n = I H : Let S be the unilateral shift on l 2 (C) and fe i g 1 i=0 be...
private communication
"... Stability problem; Cauchy–Jensen mappings; Euler–Lagrange mappings; Fixed point alternative. In 1940 S.M. Ulam proposed the famous Ulam stability problem. In 1941 D.H. Hyers solved the wellknown Ulam stability problem for additive mappings subject to the Hyers condition on approximately additive ma ..."
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Cited by 27 (4 self)
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Stability problem; Cauchy–Jensen mappings; Euler–Lagrange mappings; Fixed point alternative. In 1940 S.M. Ulam proposed the famous Ulam stability problem. In 1941 D.H. Hyers solved the wellknown Ulam stability problem for additive mappings subject to the Hyers condition on approximately additive mappings. In this paper we introduce generalized additive mappings of Jensen type mappings and establish new theorems about the Ulam stability of additive and alternative additive mappings. This study was financially supported by research fund of Chungnam National University in 2007. Euler–Lagrange Additive Mappings
Onesided Mideals and multipliers in operator spaces
 I, Pacific J. Math
"... The theory of Mideals and multiplier mappings of Banach spaces naturally generalizes to left (or right) Mideals and multiplier mappings of operator spaces. These subspaces and mappings are intrinsically characterized in terms of the matrix norms. In turn this is used to prove that the algebra of l ..."
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Cited by 24 (11 self)
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The theory of Mideals and multiplier mappings of Banach spaces naturally generalizes to left (or right) Mideals and multiplier mappings of operator spaces. These subspaces and mappings are intrinsically characterized in terms of the matrix norms. In turn this is used to prove that the algebra of left adjointable mappings of a dual operator space X is a von Neumann algebra. If in addition X is an operator A–Bbimodule for C ∗algebras A and B, then the module operations on X are automatically weak ∗ continuous. One sided Lprojections are introduced, and analogues of various results from the classical theory are proved. An assortment of examples is considered. 1. Introduction. It has long been recognized that the algebraic structure of a C∗algebra A is closely linked to its geometry as a Banach space (see [25]). This principle was illustrated in [5],and [2],p. 237,where it was shown that the closed
Inner quasidiagonality and strong NF algebras
 Pacific J. Math
"... Continuing the study of generalized inductive limits of finitedimensional C ∗algebras, we define a refined notion of quasidiagonality for C ∗algebras, called inner quasidiagonality, and show that a separable C∗algebra is a strong NF algebra if and only if it is nuclear and inner quasidiagonal. Ma ..."
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Cited by 22 (2 self)
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Continuing the study of generalized inductive limits of finitedimensional C ∗algebras, we define a refined notion of quasidiagonality for C ∗algebras, called inner quasidiagonality, and show that a separable C∗algebra is a strong NF algebra if and only if it is nuclear and inner quasidiagonal. Many natural classes of NF algebras are strong NF, including all simple NF algebras, all residually finitedimensional nuclear C∗algebras, and all approximately subhomogeneous C∗algebras. Examples are given of NF algebras which are not strong NF. 1. Introduction. This paper is a sequel to Blackadar & Kirchberg [BKb], to which we will frequently refer. In Blackadar & Kirchberg, we studied a generalized inductive limit construction for C ∗algebras and gave various characterizations of C ∗algebras which can be written as generalized inductive limits of finitedimensional
Onesided ideals and approximate identities in operator algebras
 J. Australian Math. Soc
"... Abstract. A left ideal of any C ∗algebra is an example of an operator algebra with a right contractive approximate identity (r.c.a.i.). Indeed left ideals in C ∗algebras may be characterized as the class of such operator algebras, which happen also to be triple systems. Conversely, we show here an ..."
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Cited by 10 (3 self)
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Abstract. A left ideal of any C ∗algebra is an example of an operator algebra with a right contractive approximate identity (r.c.a.i.). Indeed left ideals in C ∗algebras may be characterized as the class of such operator algebras, which happen also to be triple systems. Conversely, we show here and in a sequel to this paper [9], that operator algebras with r.c.a.i. should be studied in terms of a certain left ideal of a C ∗algebra. We study left ideals from the perspective of ‘Hamana theory ’ and using the multiplier algebras introduced by the author. More generally, we develop some general theory for operator algebras which have a 1sided identity or approximate identity, including a BanachStone theorem for these algebras, and an analysis of the ‘multiplier operator algebra’.
Modules over operator algebras, and the maximal C ∗ dilation
, 1999
"... Abstract. We continue our study of the general theory of possibly nonselfadjoint algebras of operators on a Hilbert space, and modules over such algebras, developing a little more technology to connect ‘nonselfadjoint operator algebra ’ with the C ∗ −algebraic framework. More particularly, we make u ..."
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Cited by 8 (6 self)
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Abstract. We continue our study of the general theory of possibly nonselfadjoint algebras of operators on a Hilbert space, and modules over such algebras, developing a little more technology to connect ‘nonselfadjoint operator algebra ’ with the C ∗ −algebraic framework. More particularly, we make use of the universal, or maximal, C ∗ −algebra generated by an operator algebra, and C ∗ −dilations. This technology is quite general, however it was developed to solve some problems arising in the theory of Morita equivalence of operator algebras, and as a result most of the applications given here (and in a companion paper) are to that subject. Other applications given here are to extension problems for module maps, and characterizations of C ∗ −algebras. * Supported by a grant from the NSF. The contents of this paper were announced at the January 1999 meeting of the American Mathematical Socety. 1 2 DAVID P. BLECHER 1. Introduction Modules
Gheondea: Representations of hermitian kernels by means of Kreĭn spaces
 Publ. RIMS. Kyoto Univ
"... Abstract. In this paper we study hermitian kernels invariant under the action of a semigroup with involution. We characterize those hermitian kernels that realize the given action by bounded operators on a Kreĭn space. Applications to the GNS representation of ∗algebras associated to hermitian func ..."
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Cited by 7 (3 self)
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Abstract. In this paper we study hermitian kernels invariant under the action of a semigroup with involution. We characterize those hermitian kernels that realize the given action by bounded operators on a Kreĭn space. Applications to the GNS representation of ∗algebras associated to hermitian functionals are given. We explain the key role played by the Kolmogorov decomposition in the construction of Weyl exponentials associated to an indefinite inner product and in the dilation theory of hermitian maps on C ∗algebras. 1.