• Volume 113, Issue 1

      February 2003,   pages  1-90

    • Preface

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    • A Remark on the Unitary Group of a Tensor Product of 𝑛 Finite-Dimensional Hilbert Spaces

      K R Parthasarathy

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      Let $H_i, 1 ≤ i ≤ n$ be complex finite-dimensional Hilbert spaces of dimension $d_i, 1 ≤ i ≤ n$ respectively with $d_i ≥ 2$ for every 𝑖. 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 product $H = H_1 \otimes H_2 \otimes\ldots \otimes H_n$ can be expressed as a composition of a finite number of unitary operators living on pair products $H_i \otimes H_j, 1 ≤ i, j ≤ n$. 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 an 𝑀-ideal in the space of bounded operators, a very smooth operator 𝑇 attains its norm at a unique vector 𝑥 (up to a constant multiple) and 𝑇(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 a 𝐶*-Algebra

      Anil K Karn

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

    • Questions Concerning Matrix Algebras and Invariance of Spectrum

      Bruce A Barnes

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      Let 𝐴 and 𝐵 be unital Banach algebras with 𝐴 a subalgebra of 𝐵. Denote the algebra of all 𝑛 × 𝑛 matrices with entries from 𝐴 by $M_n(A)$. In this paper we prove some results concerning the open question: If 𝐴 is inverse closed in 𝐵, then is $M_n(A)$ inverse closed in $M_n(B)$? We also study related questions in the setting where 𝐴 is a symmetric Banach -algebra.

    • When is $f(x_1, x_2, \ldots , x_n) = u_1 (x_1) + u_2(x_2) + \cdots + u_n (x_n)$?

      A Kłopotowski M G Nadkarni K P S Bhaskara Rao

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      We discuss subsets 𝑆 of $\mathbb{R}^n$ such that every real valued function 𝑓 on 𝑆 is of the form

      $$f(x_1, x_2, \ldots , x_n) = u_1(x_1) + u_2(x_2) + \cdots + u_n (x_n),$$

      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 0 belongs to the boundary 𝜕 𝐾 . Let $\mathcal{A}(K)$ denote the space of all functions 𝑓 on 𝐾 such that 𝑓 is holomorphic in a neighborhood of 𝐾 and 𝑓(0) = 0. Also for any given positive integer 𝑚, let $\mathcal{A}(m, K)$ denote the space of all 𝑓 such that 𝑓 is holomorphic in a neighborhood of 𝐾 and $f(0) = f'(0) = \cdots = f^{(m)}(0) = 0$. Then $\mathcal{A}(m, K)$ is dense in $\mathcal{A}(K)$ under the supremum norm on 𝐾 provided that there exists a sector $W = \{re^{i𝜃}; 0 ≤ r ≤ 𝛿, 𝛼 ≤ 𝜃 ≤ 𝛽\}$ such that $W \cap K = \{0\}$. (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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