Full Record

Author | Gao, Li |

Title | On quantum Euclidean spaces: Continuous deformation and pseudo-differential operators |

URL | http://hdl.handle.net/2142/101546 |

Publication Date | 2018 |

Date Accessioned | 2018-09-27 16:17:44 |

Degree | PhD |

Discipline/Department | Mathematics |

Degree Level | doctoral |

University/Publisher | University of Illinois – Urbana-Champaign |

Abstract | Quantum Euclidean spaces are noncommutative deformations of Euclidean spaces. They are prototypes of locally compact noncommutative manifolds in Noncommutative Geometry. In this thesis, we study the continuous deformation and Pseudo-differential calculus of quantum Euclidean spaces. After reviewing the basic definitions and representation theory of quantum Euclidean spaces in Chapter 1, we prove in Chapter 2 a Lip^(1/2) continuous embedding of the family of quantum Euclidean spaces. This result is the locally compact analog of U. Haagerup and M. R\o rdom's work on Lip^(1/2) continuous embedding for quantum 2-torus. As a corollary, we also obtained Lip^(1/2) embedding for quantum tori of all dimensions. In Chapter 3, we developed a Pseudo-differential calculus for noncommuting covariant derivatives satisfying the Canonical Commutation Relations. Based on some basic analysis on quantum Euclidean spaces, we introduce abstract symbol classs following the idea of abstract pseudo-differential operators introduced by A. Connes and H. Moscovici. We proved the two main ingredients pseudo-differential calculus – the L2-boundedness of 0-order operators and the composition identity. We also identify the principal symbol map in our setting. Chapter 4 is devoted to application in the local index formula in noncommutative Geometry. We show that our setting with noncommuting covariant derivatives is an example of locally compact noncommutative manifold. After developed the Getzler super-symmetric symbol calculus, we calculate the local index formula for the a noncommutative analog of Bott projection. |

Subjects/Keywords | Noncommutative Euclidean spaces; Moyal Deformation; Pseudo-differential operators; |

Contributors | Junge, Marius (advisor); Ruan, Zhong-Jin (Committee Chair); Boca, Florin P. (committee member); Oikhberg, Timur (committee member) |

Language | en |

Rights | Copyright 2018 by Li Gao |

Country of Publication | us |

Record ID | handle:2142/101546 |

Repository | uiuc |

Date Indexed | 2020-03-09 |

Grantor | University of Illinois at Urbana-Champaign |

Issued Date | 2018-07-10 00:00:00 |

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…It facilitates the recognition of quantum mechanics as a deformation of classical mechanics,
with the Planck constant
on the parameter
h,
h
being a deformation parameter. The *Moyal* product, depending
gives a continuous family of deformations from…

…x5B;43]) and deformation of
C ∗ -algebras
by actions of
Rd
[39].
In the rst part of this thesis, we study the continuity of *Moyal* deformation, or more
precisely, the continuity of the quantization map
λh
depending on
h…

…algebraically and then allows the commutative function algebras to be
generalized to possibly non-commutative algebras. From this point of view, the Heisenberg
relation and the CCR algebra are noncommutative deformation of Euclidean space, which are
called *Moyal*…

…x28;ξ)dξ , f ∈ S(Rd ),
Rd
is the Fourier transform of
f.
the quantized Schwartz class. The matrix multiplication of
product
?θ
associated to the parameter
We denote by
Sθ
is equivalent to the *Moyal*
θ,
−d
Z
Z
λθ (f…

…fˆ(ξ − η)ĝ(η)e 2 ξ·θη λθ (ξ)dξdη ,
2d
(2π)
d
d
Z RZ R
i
1
f\
?θ g(ξ) =
fˆ(ξ − η)ĝ(η)e 2 ξ·θη dξdη
d
(2π) Rd Rd
7
The *Moyal* product is bilinear, associative and…

…reversed under complex conjugation
f ?θ g = g ?θ f ,
which makes
(S(Rd ), ?θ )
a
∗-algebra.
We refer to [17, 47] for more information about *Moyal*
analysis.
The
algebra
C ∗ -quantum
(S(Rd ), ?θ )…

…equivalent
denitions:
i) the C ∗ -enveloping of the *Moyal* product algebra (S(Rd ), ?θ );
ii) the full twisted group C ∗ -algebra C ∗ (Rd , σθ )
iii) the reduced twisted group C ∗ -algebra Cr∗ (Rd , σθ )…

…coordinate
(xj g)(x) = xj g(x)
2. the *Moyal* product
3.
E0 = C0 (Rd )
Rθ
is the usual point-wise multiplication of functions;
is the space of continuous functions on
{R0 = L∞ (Rd )}
Thus
?0…