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Vol.31 (1) 2009 — 5 results found.

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Cordes characterization for pseudodifferential operators with symbols valued in a noncommutative C$^*$-algebra
C. R. Math. Rep. Acad. Sci. Canada Vol. 31 (1) 2009, pp. 24–32
Severino T. Melo; Marcela I. Merklen (Received: 2009/01/23)

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Given a separable unital C\(^*\)-algebra \({\mathcal A}\) with norm \(\|\cdot\|\), let \(E\) denote the Banach-space completion of the \({\mathcal A}\)-valued Schwartz space on \(\mathbb{R}^n\) with norm \(\|f\|_2=\|\langle f,f\rangle\|^{1/2}\), \(\langle f,g\rangle=\int f(x)^*g(x)\,dx\). The assignment of the pseudodifferential operator \(B=b(x,D)\) with \({\mathcal A}\)-valued symbol \(b(x,\xi)\) to each smooth function with bounded derivatives \(b\in\mathcal{B}^{\mathcal{A}(\mathbb{R}^{2n})}\) defines an injective mapping \(O\) from \(\mathcal{B}^{\mathcal{A}(\mathbb{R}^{2n})}\) to the set \(\mathcal{H}\) of all operators with smooth orbit under the canonical action of the Heisenberg group on the algebra of all adjointable operators on the Hilbert C\(^*\)-module \(E\). It is known that \(O\) is surjective if \({\mathcal A}\) is commutative. In this paper, we show that if \(O\) is surjective for \({\mathcal A}\), then it is also surjective for \(M_k({\mathcal A})\).

Étant donné une C\(^*\)-algèbre \(A\), séparable et avec unité, soit \(E\) l’espace de Banach obtenu par complétation de l’espace de Schwartz sur \(\mathbb{R}^n\) avec valeurs dans \(A\) par la norme induite par le produit interne à valeurs dans A: \(\langle f,g\rangle=\int f(x)^*g(x)\,dx\). L’association de l’opérateur pseudo-différentiel \(B=b(x,D)\), ayant symbol \(b(x,\xi)\) à valeurs dans A, à chaque fonction smooth \(b\), à derivées bornées, define une application injective \(O\) de l’ensemble de tous ces symbols dans l’ensemble de tous les opérateurs ayant orbite lisse par l’action du group de Heisenberg sur l’algèbre de tous les opérateurs adjointables sur le C\(^*\)-module de Hilbert \(E\). Il est bien connu que, si \(A\) est commutatif, alors \(O\) est surjective. Dans cet article nous montrons que, si \(O\) est surjective pour une algèbre quelconque \(A\), alors elle est surjective aussi pour l’algèbre des matrices \(k\) par \(k\) à coefficients dans \(A\).

Classification of compact homogeneous manifolds with pseudo-Kählerian structures
C. R. Math. Rep. Acad. Sci. Canada Vol. 31 (1) 2009, pp. 20–23
Daniel Guan (Received: 2008/11/11)

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In this note we apply a modification theorem for compact homogeneous solvmanifolds to compact complex homogeneous manifolds with pseudo-Kählerian structures. We are then finally able to classify these compact pseudo-Kählerian manifolds as certain products of projective rational homogeneous spaces, tori, and simple and double reduced primitive pseudo-Kähler spaces.

Dans cette note, nous appliquons un théorème de modification pour des “solv-variétés” compactes et homogènes aux variétés compactes complexes equipées d’une structure pseudo-kählérienne. Nous obtenons une classification de ces variétés compactes pseudo-kählériennes sous la forme de certains produits d’espaces projectifs rationnels et homogènes, de tores, et d’espaces pseudo-kählériens réduits et primitifs simples ou doubles.

On AF embeddability of continuous fields
C. R. Math. Rep. Acad. Sci. Canada Vol. 31 (1) 2009, pp. 16–19
Marius Dadarlat (Received: 2008/12/18)

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Let \(A\) be a separable and exact \(C^*\)-algebra which is a continuous field of \(C^*\)-algebras over a connected, locally connected, compact metrizable space. If at least one of the fibers of \(A\) is AF embeddable, then so is \(A\). As an application we show that if \(G\) is a central extension of an amenable and residually finite discrete group by \(\mathbb{Z}^n\), then the \(C^*\)-algebra of \(G\) is AF embeddable.

Soit A une \(C^*\)-algèbre séparable et exacte qui est un champ continu de \(C^*\)-algèbres sur un espace connexe, localement connexe, compact et metrizable. Si au moins l’une des fibres de \(A\) est embeddable dans une AF algèbre donc la \(C^*\)-algèbre \(A\) est aussi. Comme application, nous montrons que si \(G\) est une extension centrale d’un groupe discret amenable et résiduellement fini par le groupe \(\mathbb{Z}^n\), alors la \(C^*\)-algèbre de \(G\) est embeddable dans une AF algèbre.

On controllability of partially prescribed pairs of matrices
C. R. Math. Rep. Acad. Sci. Canada Vol. 31 (1) 2009, pp. 7–15
Gloria Cravo (Received: 2008/07/18, Revised: 2008/10/25)

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Let \(F\) be an infinite field and let \(n,p_{1},p_{2},p_{3}\) be positive integers such that \(n=p_{1}+p_{2}+p_{3}.\) Let \[(C_{1},C_{2})=\left( \begin{bmatrix} C_{1,1} & C_{1,2} \\ C_{2,1} & C_{2,2} \end{bmatrix} , \begin{bmatrix} C_{1,3} \\ C_{2,3} \end{bmatrix} \right) ,\] where the blocks \(C_{i,j}\) are of type \(p_{i}\times p_{j},i\in \{1,2\},j\in \{1,2,3\}.\) We analyse the possibility of the pair \((C_{1},C_{2})\) being completely controllable, when

(i) \(C_{1,2},\) \(C_{1,3}\), and \(C_{2,1}\) are fixed and the other blocks vary;

(ii) \(C_{1,1},\) \(C_{1,2}\), and \(C_{2,1}\) are fixed and the other blocks vary.

We still describe the possible characteristic polynomials of a partitioned matrix of the form \(C=[ C_{i,j}] \in F^{n\times n},\) where the blocks \(C_{i,j}\) are of type \(p_{i}\times p_{j},i,j\in \{1,2,3\}\), when one of the conditions (i) or (ii) occurs.

Soit \(F\) un corps infini et soient \( n,p_{1},p_{2},p_{3}\) des entiers positifs tels que \(n=p_{1}+p_{2}+p_{3}.\) Soit \[(C_{1},C_{2})=\left( \begin{bmatrix} C_{1,1} & C_{1,2} \\ C_{2,1} & C_{2,2} \end{bmatrix} , \begin{bmatrix} C_{1,3} \\ C_{2,3} \end{bmatrix} \right) ,\] où les blocs \(C_{i,j}\) sont de type \(p_{i}\times p_{j},i\in \{1,2\},j\in \{1,2,3\}.\) Nous établions conditions pour lesquelles \((C_{1},C_{2})\) est controllable, quand

(i) \(C_{1,2},C_{1,3}\), et \(C_{2,1}\) sont connus et les autres blocs varient;

(ii) \(C_{1,1},C_{1,2}\), et \(C_{2,1}\) sont connus et les autres blocs varient.

Soit \(C=[ C_{i,j}] \in F^{n\times n},\) où les blocs \(C_{i,j}\) sont de type \(p_{i}\times p_{j},i,j\in \{1,2,3\}.\) Nous étudions le polynôme caractéristique de la matrice \(C,\) quand une des conditions (i) ou (ii) est satisfait.

A note on projection equivalence in von Neumann algebras
C. R. Math. Rep. Acad. Sci. Canada Vol. 31 (1) 2009, pp. 1–6
Richard V. Kadison (Received: 2008/03/09)

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A new structural result in the comparison theory of projections for von Neumann algebras is proved: two monotone-increasing nets of projections indexed by the same directed set have unions that are equivalent when pairs of projections with the same index are equivalent. The same is not true, in general, for intersections of monotone-decreasing nets of projections. Counterexamples are given indicating limitations on extensions, variants, and methods for proving that result.

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Most used Keywords

algebraic number theory approximation property automorphisms Bessel functions Boson-fermion correspondence C*-algebra Carmichael number center problem Chebyshev transform classification Classification of simple C*-algebras composition operators continued fractions Cuntz Semigroup elliptic curves fixed point Fourier transform function fields. functoriality general relativity generic property ideals indefinite inner product inductive limits of sub-homogeneous C*- algebras Irrational rotation algebra J-Hermitian matrix K-theory Kahler manifolds L-functions maximal ideal space nonexpansive mapping numerical range orthogonal polynomials Predual space prime number property SP Renormalization rotation algebras Salem number semi-reciprocal polynomials tracially approximate splitting interval algebras unbounded traces uniqueness Weak Markov set Whitney problems

Most used AMS

05C05 11A07 11A55 11B37 11B68 11D09 11D25 11D41 11E04 11F11 11F66 11F67 11G05 11R09 11R11 13B25 14J26 14M25 14P10 17B37 17B67 19K14 19K56 26A51 30C15 30H05 35B 37E10 37E20 37F25 39B72 42C05 43A07 46B20 46L05 46L35 46L40 46L55 46L80 47H10 53B25 53C55 54C60 60F10 83C05

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