整数群约化交叉积C*-代数上半范数的下半连续性
The Lower Semicontinuity of Seminorms on Crossed Product C*-Algebras of Integer Group
DOI: 10.12677/aam.2024.136281, PDF,   
作者: 沈文涛：南京航空航天大学数学学院，江苏 南京
关键词: 长度函数；*-半范数；下半连续性；Length Function； *-Seminorms； Lower Semi-Continuity
摘要: 紧量子度量空间结构是算子代数领域非常重要的研究内容，既有重要的理论意义，又有广泛的应用前景.本文利用一般的长度函数构造出一类*-半范数。同时，利用约化交叉积C*-代数的共变表示可以构造出另一类*-半范数。通过讨论它们下半连续性，发现其中一类*-半范数是下半连续的，另一类*-半范数与C*-代数的半范数L的下半连续性是等价的。进一步构造出一类与对应的紧量子度量空间紧密相关的*-半范数。
Abstract: Compact quantum metric space structure is a very important research topic in the field of operator algebras. It has great theoretical significance and a wide range of application prospects. In this paper, a class of *-seminorms is constructed by using the general length function. At the same time, another kind of *-seminorms can be constructed by using the covariant representation of reduced cross product algebras. By discussing their lower semicontinuity, we can find that one type of *-seminorms is lower semicontinuous, while the other type of *-seminorms is equivalent to the lower semicontinuity of the seminorm of C*-algebra. Furthermore, we can a class of *-seminorms that is closely related to the corresponding compact quantum metric space.
文章引用：沈文涛. 整数群约化交叉积C*-代数上半范数的下半连续性[J]. 应用数学进展, 2024, 13(6): 2943-2951. https://doi.org/10.12677/aam.2024.136281

参考文献

[1] Connes, A. (1985) Non-Commutative Differential Geometry. Publications Mathématiques de l’Institut des Hautes Études Scientifique, 62, 257-360.
[2] Connes, A. (1989) Compact Metric Spaces, Fredholm Modules, and Hyperfiniteness. Ergodic Theory and Dynamical Systems, 9, 207-220. [Google Scholar] [CrossRef] 
[3] Connes, A. (1994) Noncommutative Geometry. Academic Press.
[4] Connes, A. (1996) Gravity Coupled with Matter and the Foundation of Non-Commutative Geometry. Communications in Mathematical Physics, 182, 155-176. [Google Scholar] [CrossRef] 
[5] Kantorovich, L.V. (1945) On an Effective Method of Solving Extremal Problems for Quadratic Funcional. Soviet Mathematics, 7, 455-460.
[6] Rieffel, M.A. (1999) Metrics on State Spaces. Documenta Mathematica, 4, 559-600. [Google Scholar] [CrossRef] 
[7] Rieffel, M.A. (2004) Gromov-Hausdorff Distance for Quantum Metric Spaces. Memoirs of the American Mathematical Society, 168, Article No. 796. [Google Scholar] [CrossRef] 
[8] Antonescu, C. and Christensen, E. (2004) Metrics on Group C*-Algebras and a Non-Commutative Arzelà-Ascoli Theorem. Journal of Functional Analysis, 214, 247-259. [Google Scholar] [CrossRef] 
[9] Hawkins, A., Skalski, A., White, S. and Zacharias, J. (2013) On Spectral Triples on Crossed Products Arising from Equicontinuous Actions. Mathematica Scandinavica, 113, 262-291. [Google Scholar] [CrossRef] 
[10] Christ, M. and Rieòel, M.A. (2017) Nilpotent Group C*-Algebras as Compact Quantum Metric Spaces. Canadian Mathematical Bulletin, 60, 77-94. [Google Scholar] [CrossRef] 
[11] Bellissard, J., Marcolli, M. and Reihani,K. (2010) Dynamical Systems on Spectralmetric Spaces. arXiv: 1008.4617.
[12] Kaad, J. and Kyed, D. (2020) Dynamics of Compact Quantum Metric Spaces. Ergodic Theory and Dynamical Systems, 41, 2069-2109. [Google Scholar] [CrossRef] 
[13] Brown, N. and Ozawa, N. (2008) C*-Algebras and Finite-Dimensional Approximations. None American Mathematical Society, 88. [Google Scholar] [CrossRef] 
[14] Davidson, K. (1996) C*-Algebras by Example. None American Mathematical Society, 6. [Google Scholar] [CrossRef] 
[15] Oh, C.L. (2017) Geometric Group Theory: An Introduction. Springer.
[16] Rudin, W. (1987) Real and Complex Analysis. Third Edition, McGraw-Hill Book.

为你推荐