Algebraic Rieffel induction, formal Morita equivalence, and applications to deformation quantization
Algebraic Rieffel induction, formal Morita equivalence, and applications to deformation quantization
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代数 Rieffel 归纳、形式 Morita 等价以及在变形量化中的应用
DOI：
10.1016/s0393-0440(00)00035-8
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发表时间：
1999
影响因子：
1.5
通讯作者：
S. Waldmann
中科院分区：
文献类型：
--
作者：
H. Bursztyn;S. Waldmann
本文考虑了环C上的对合代数是由序环R的二次扩张i给出的。我们讨论了C上的pre-Hilbert空间上的这类代数的表示理论，并发展了这类代数的Rieffel归纳和形式Morita等价的概念，类似于C上的代数的情形.在本文中，正泛函和正代数元的概念对于所有的构造都是至关重要的。在C-代数的情况下，我们表明，GNS建设的表示可以理解为Rieffel归纳，而且，形式的Morita等价的两个的代数，这是定义的双模的存在与某些额外的结构，意味着等价的范畴的强非退化的表示的两个的代数。我们讨论了各种例子，如有限秩算子的准希尔伯特空间和矩阵代数上的矩阵代数代数。形式的森田等价暗示森田等价环理论框架。最后，我们将我们的考虑变形理论，特别是变形量子化和讨论的经典极限和变形的等价双模。
In this paper, we consider algebras with involution over a ring C which is given by the quadratic extension by i of an ordered ring R . We discuss the∗-representation theory of such∗-algebras on pre-Hilbert spaces over C and develop the notions of Rieffel induction and formal Morita equivalence for this category analogously to the situation for C∗-algebras. Throughout this paper, the notion of positive functionals and positive algebra elements will be crucial for all constructions. As in the case of C∗-algebras, we show that the GNS construction of∗-representations can be understood as Rieffel induction and, moreover, that formal Morita equivalence of two∗-algebras, which is defined by the existence of a bimodule with certain additional structures, implies the equivalence of the categories of strongly non-degenerate∗-representations of the two∗-algebras. We discuss various examples like finite rank operators on pre-Hilbert spaces and matrix algebras over∗-algebras. Formal Morita equivalence is shown to imply Morita equivalence in the ring-theoretic framework. Finally, we apply our considerations to deformation theory and in particular to deformation quantization and discuss the classical limit and the deformation of equivalence bimodules.