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46L40, Automorphisms — 6 results found.

      
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Cubic and Hexic Integral Transforms for Locally Compact Abelian Groups
C. R. Math. Rep. Acad. Sci. Canada Vol. 37 (4) 2015, pp. 121-130
Sam Walters (Received: 2014/10/10, Revised: 2014/10/10)

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We prove that for locally compact, compactly generated self-dual Abelian groups \(G\), there are canonical unitary integral operators on \(L^2(G)\) analogous to the Fourier transform but which have orders 3 and 6. To do this, we establish the existence of a certain projective character on \(G\) whose phase multiplication with the FT gives rise to the Cubic transform (of order 3). (Thus, although the Fourier transform has order 4, one can “make it” have order 3 (or 6) by means of a phase factor!)

Soit \(G\) un groupe localement compact, engendré par un sousensemble compact, et isomorphe à son groupe dual. On construit des operateurs intégrals unitaires canoniques qui sont analogues à la transformée de Fourier, mais qui sont d’ordres trois et six.

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.

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.

Periodic Integral Transforms and C*-Algebras
C. R. Math. Rep. Acad. Sci. Canada Vol. 26 (2) 2004, pp. 55–61
S. Walters (Received: 2003/06/22)

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No abstract available but the full text is online at the title link above.

On the irrational quartic algebra
C. R. Math. Rep. Acad. Sci. Canada Vol. 21 (3) 1999, pp. 91–96
S.G. Walters (Received: 1998/10/22)

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Index theory and quantization of boundary value problems
C. R. Math. Rep. Acad. Sci. Canada Vol. 11 (6) 1989, pp. 237–242
P.E.T. Jorgensen / G.L. Price (Received: 1989/09/08)

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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 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

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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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