Stability of certain relations in C*-algebras

Jiajie Hua »¨¼Ò½Ü

*(Jiaxing University)*

9:00-10:00,April 14,2023 A503

__Abstract:__

In 1976, Halmos raised the following question, that is, can two unitary elements satisfying almost commutative relation in a matrix be approximated by two unitary elements with exact commutative relation. The difficulty of this problem is that this approximation is consistent and independent of the order of the matrix. Voiclescu first realized that this problem is generally incorrect, and he cited a counterexample. From the counterexample of Voiclescu, Exel and Loring found that in the example of Voiclescu, the Bott element generated by two unitary elements (in K theory) is an obstacle that the problem can not be realized. Later, it was proved by Lin Huaxin and other scholars that once the obstacle of Bott element disappears (i.e. Bott element is zero), the Halmos problem is established, and Bott element is zero is still a necessary condition for the establishment of this problem. We will consider some more general cases, that is, to study the stability of some important relations of unitary elements in C*-algebra. Specifically, we will study under what conditions a group of unitary elements satisfying the approximate relations can be approximated by a group of unitary elements satisfying the exact relations.

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