• Volume 113, Issue 1

February 2003,   pages  1-90

• Preface

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

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

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

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

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

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

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

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.

• # Proceedings – Mathematical Sciences

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