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Vol.37 (3) 2015 — 5 results found.

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Periodic Integral Transforms and Associated Noncommutative Orbifold Projections
C. R. Math. Rep. Acad. Sci. Canada Vol. 37 (3) 2015, pp. 114-120
Sam Walters (Received: 2014/11/02, Revised: 2015/02/04)

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We report on recent results on the existence of Cubic and Hexic integral transforms on self-dual locally compact groups (orders 3 and 6 analogues of the classical Fourier transform) and their application in constructing a canonical continuous section of smooth projections \(\mathcal E(t)\) of the continuous field of rotation C*-algebras \(\{A_t\}_{0 \le t \le 1}\) that is invariant under the noncommutative Hexic transform automorphism. This leads to invariant matrix (point) projections of the irrational noncommutative tori \(A_\theta\). We also present a quick method for computing the (quantized) topological invariants of such projections using techniques from classical Theta function theory.

On décrit des résultats récents sur l’existence d’une transformation intégrale d’ordre trois (ou d’ordre six) sur un groupe localement compact abélien self-dual. On étudie l’application possible à la construction d’un champs continu de projecteurs invariants sous l’automorphisme associé du champs de C*-algèbres de rotation. On calcule certains invariants topologiques de ces projecteurs.

Constructive Geometrization of Thurston Maps
C. R. Math. Rep. Acad. Sci. Canada Vol. 37 (3) 2015, pp. 100-113
Nikita Selinger; Michael Yampolsky (Received: 2014/10/06, Revised: 2014/11/19)

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We prove that every Thurston map can be constructively geometrized in a canonical fashion. According to Thurston’s theorem, a map with hyperbolic orbifold has a canonical geometrization – a combinatorially equivalent postcritically finite rational map of the Riemann sphere – if and only if there is no Thurston obstruction. We follow Pilgrim’s idea of a canonical decomposition of a Thurston map to handle the obstructed case. A key ingredient of our proof is a geometrization result for marked Thurston maps with parabolic orbifolds – an analogue of Thurston’s theorem for the exceptional case not covered by it.

On montre que toute application de Thurston peut être géométrisée de façon constructive et canonique. Selon le théoreme de Thurston, une telle application ayant un orbifold hyperbolique possède une géométrisation canonique, c’est-à-dire une fonction rationnelle combinatoriellement équivalente dont les orbites critiques sont finies, si et seulement s’il n’existe pas d’obstruction de Thurston. On traite le cas où il existe une obstruction en utilisant l’idée de Pilgrim d’une décomposition canonique d’une application de Thurston. L’ingrédient principal de la preuve est un résultat de géométrisation pour les applications de Thurston marquées ayant un orbifold parabolique – un analogue du théorème de Thurston pour le cas exceptionnel.

Topological Obstruction to Approximating the Irrational Rotation C*-algebra by Certain Fourier Invariant C*-subalgebras
C. R. Math. Rep. Acad. Sci. Canada Vol. 37 (3) 2015, pp. 94-99
Sam Walters (Received: 2014/07/10, Revised: 2014/07/10)

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We demonstrate, in a rather quantitative manner, the existence of topological obstructions to approximating the irrational rotation C*-algebra \(A_\theta\) by Fourier invariant unital C*-subalgebras of either of the forms \[M \oplus B \oplus \sigma(B), \qquad M \oplus N \oplus D \oplus \sigma(D) \oplus \sigma^2(D) \oplus \sigma^3(D),\] where \(M, N\) are Fourier invariant matrix algebras (over \(\mathbb C\)), \(B\) is a C*-subalgebra whose unit projection is flip invariant and orthogonal to its Fourier transform, and \(D\) is a C*-subalgebra whose unit projection is orthogonal to its orbit under the Fourier transform. Here, \(\sigma\) is the noncommutative Fourier transform automorphism of \(A_\theta\) defined by \(\sigma(U) = V^{-1},\ \sigma(V)=U\) on the canonical unitary generators \(U,V\) obeying the unitary Heisenberg commutation relation \(VU = e^{2\pi i\theta}UV\).

On montre l’existence d’obstructions topologiques à l’approximation du tore non-commutatif par sous-algèbres de certains types qui sont invariantes sous l’automorphisme de Fourier.

Commutativity Criteria in Banach Algebras
C. R. Math. Rep. Acad. Sci. Canada Vol. 37 (3) 2015, pp. 89-93
Cheikh O. Hamoud (Received: 2014/05/30, Revised: 2014/11/10)

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We consider complex Banach algebras satisfying the condition \(\displaystyle (xy)^k=x^ky^k\) for all \(x\,,\,y\,\) in the algebra where \(k\) is an integer \((k\geq 2)\).

We show that for \(k=2\) and \(k=3\), this condition yields commutativity in unital Banach algebras. For higher values of \(k\), commutativity is obtained for semi-simple algebras and the conclusions are quite similar to the ones in Cheikh 1995.

The extension of the results to wider classes of algebras is also considered.

Nous considérons des algèbres de Banach complexes vérifiant la condition \(\displaystyle (xy)^k=x^ky^k\) pour tout \(x\,,\,y\,\) dans l’algèbre, \(k\) étant un entier \((k\geq 2)\).

Nous montrons que pour \(k=2\) et \(k=3\), cette condition entraine la commutativité dans les algèbres de Banach unitaires. Pour les valeurs plus elevées de \(k\), la commutativité est établie dans les algèbres semi-simples avec des résultats similaires à ceux obtenus dans Cheikh 1995.

L’extension des résultats à d’autres classes d’algèbres topologiques est également considérée.

The Parallelism of a Certain Tensor of Real Hypersurfaces in a Nonflat Complex Space Form
C. R. Math. Rep. Acad. Sci. Canada Vol. 37 (3) 2015, pp. 81-88
Tatsuyoshi Hamada; Katsufumi Yamashita (Received: 2014/10/15, Revised: 2014/11/04)

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In Theorem 1, we show a new condition for a real hypersurface \(M\) isometrically immersed into a nonflat complex space form to be a hypersurface of type (A). This condition is expressed by the parallelism of a certain tensor of type (1, 1) on \(M\) . Furthermore, using the discussion in the proof of Theorem 1, we can give a condition for a Kähler manifold to be a complex space form (see Theorem 2).

Dans le théorème 1, nous donnons une nouvelle condition pour qu’une hypersurface réelle \(M\) immergée dans une “space form” complexe non plate soit une hypersurface de type (A). Cette condition est exprimée par le parallélisme d’un certain tenseur de type (1, 1) sur \(M\) . De plus, en utilisant la discussion dans la démonstration du théorème 1, nous donnons une condition pour qu’une variété de Kähler soit une “space form” complexe (voir le théorème 2).

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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 noninterlacing 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 Weak Markov set Whitney problems

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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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