• Volume 113, Issue 1

      February 2003,   pages  1-90

    • Preface

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    • A remark on the unitary group of a tensor product ofn finite-dimensional Hilbert spaces

      K R Parthasarathy

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      LetHi, 1 ≤ i ≤n be complex finite-dimensional Hilbert spaces of dimension di,1 ≤ i ≤n respectively withdi ≥ 2 for everyi. By using the method of quantum circuits in the theory of quantum computing as outlined in Nielsen and Chuang [2] and using a key lemma of Jaikumar [1] we show that every unitary operator on the tensor productH =H1H2 ⊗... ⊗Hn can be expressed as a composition of a finite number of unitary operators living on pair productsHiHj,1 ≤i,jn. An estimate of the number of operators appearing in such a composition is obtained.

    • The planar algebra associated to a Kac algebra

      Vijay Kodiyalam Zeph Landau V S Sunder

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      We obtain (two equivalent) presentations — in terms of generators and relations — of the planar algebra associated with the subfactor corresponding to (an outer action on a factor by) a finite-dimensional Kac algebra. One of the relations shows that the antipode of the Kac algebra agrees with the ‘rotation on 2-boxes’.

    • Very smooth points of spaces of operators

      T S S R K Rao

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      In this paper we study very smooth points of Banach spaces with special emphasis on spaces of operators. We show that when the space of compact operators is anM-ideal in the space of bounded operators, a very smooth operatorT attains its norm at a unique vectorx (up to a constant multiple) andT(x) is a very smooth point of the range space. We show that if for every equivalent norm on a Banach space, the dual unit ball has a very smooth point then the space has the Radon-Nikodým property. We give an example of a smooth Banach space without any very smooth points.

    • Order units in aC* -algebra

      Anil K Karn

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      Order unit property of a positive element in aC* -algebra is defined. It is proved that precisely projections satisfy this order theoretic property. This way, unital hereditary C*-subalgebras of aC* -algebra are characterized.

    • Questions concerning matrix algebras and invariance of spectrum

      Bruce A Barnes

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      LetA andB be unital Banach algebras withA a subalgebra ofB. Denote the algebra of alln xn matrices with entries fromA byMn (A). In this paper we prove some results concerning the open question: IfA is inverse closed inB, then isMn (A) inverse closed inMn (B)? We also study related questions in the setting where A is a symmetric Banach *-algebra.

    • When isf(x1, x2, ..., xn) =u1(x1) +u2(x2) + ... +un(xn)?

      A Kłopotowski M G Nadkarni K P S Bhaskara Rao

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      We discuss subsetsS of ℝn such that every real valued functionf onS is of the formf(x1, x2, ..., xn) =u1(x1) +u2(x2) +...+un(xn), and the related concepts and situations in analysis.

    • Some approximation theorems

      N V Rao

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      The general theme of this note is illustrated by the following theorem:Theorem 1.Suppose K is a compact set in the complex plane and 0belongs to the boundary ∂K. Let A(K) denote the space of all functions f on K such that f is holo morphic in a neighborhood of K and f(0) = 0.Also for any givenpositive integer m, let A(m, K) denote the space of all f such that f is holomorphic in a neighborhood of K and f(0) =f′(0) = ... =f(m)(0) = 0.Then A(m, K) is dense in A(K) under the supre mum norm on K provided that there exists a sector W = re; 0≤r≤ δ,α≤ θ≤ βsuch that W ∩ K = 0. (This is the well- known Poincare’s external cone condition). We present various generalizations of this result in the context of higher dimensions replacing holomorphic with harmonic.

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