On Injective Envelopes of G-AF-algebras

Subject Areas : Statistics

ALi Mahmoodi ¹ (Department of Mathematics, Mako International Unit, Islamic Azad University, Mako, Iran.)
Mohammadreza Mardanbeigi ² (Department of Mathematics / Faculty of Science / Science and Research Branch / Islamic Azad University)

Received: 2021-05-20 Accepted : 2022-03-07 Published : 2023-03-21

Keywords: AF -G-جبر, G-پوش انژکتیو, C^* -G-لیمینال و پست‌لیمینال جبر, G-ایده‌آل اساسی ساده, C^* -G- جبر, W^* -G- جبر,

Abstract :

Let G be a discrete group that acts on C^*-algebra A. We prove that in the category of G-AF-algebras and completely positive linear G-equivariant the finite-dimensional G-AF-algebra must be injective. One consequence of this is that no infinite-dimensional G-AF-algebra could be injective in the category of G-C^∗-algebras.Also, we show that the following statements are equivalent for a separable essentially simple G-C^*-algebra A: (i) A̅_G, regular completion of A, is a G-W^*-algebra (von-Neumann algebra).(ii) I_G(A), injective envelope of A, is a G-W*-algebra (von-Neumann algebra).(iii) A is G-isomorphic to a direct sum of elementary G-C^*-algebras K(H_n), which H_n is a Hilbert space. Since the K(H_n) is G-AF-algebra, and we know that the category of G-AF-algebras is closed under taking the countable direct sum. Therefore, we prove this G-C^*-algebra is a G-AF-algebra.Further, we show that if separable essentially simple G-C^∗-algebra A is post liminal, then A is liminal G-C^∗-algebra.

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