• Volume 113, Issue 1

February 2003,   pages  1-90

• Preface

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

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

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 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

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

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)?

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

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.

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 3
June 2019