Projectivity and injectivity of $\mathsf{G}$-Hilbert $\Im$-modules

Subject Areas : Functional analysis

A. Yousefi ¹ , M. R. Mardanbeigi ²

1 - Department of Mathematics, Faculty of Science, Science and Research Branch, Islamic Azad University (IAU), Tehran, Iran
2 - Department of Mathematics, Faculty of Science, Science and Research Branch, Islamic Azad University (IAU), Tehran, Iran

Received: 2023-07-25 Accepted : 2023-09-04 Published : 2023-10-01

Keywords: $G$-projective‎, ‎$G$-projective cover‎, ‎extremally $G$-disconnected‎, ‎$G$-$C^*$-algebra‎, ‎$G$-self dual‎, ‎$G$-monotone complete‎, ‎$G$-$*$-representation, ‎$G$-Hilbert $\Im$-module‎, ‎$G$-injective Hilbert $\Im$-module‎, ‎$G$-projective Hilbert $\Im$-module‎,

Abstract :

‎Let $\mathsf{G}$ be a discrete group acting on $C^*$-algebra $\Im$‎. ‎In this paper‎, ‎we investigate projectivity and injectivity of $G$-Hilbert $\Im$-modules and study the equivalent conditions characterizing $\mathsf{G}$-$C^*$-subalgebras of the algebra of compact operators on $\mathsf{G}$-Hilbert spaces in terms of general properties of $\mathsf{G}$-Hilbert $\Im$-modules‎. ‎In particular‎, ‎we show that $\mathsf{G}$-Hilbert $\Im$-(bi)modules on $\mathsf{G}$-$C^*$-algebra of compact operators are both projective and injective‎.

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