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

الموضوعات :

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

تاريخ الإرسال : 07 الثلاثاء , محرم, 1445 تاريخ التأكيد : 19 الإثنين , صفر, 1445 تاريخ الإصدار : 16 الأحد , ربيع الأول, 1445

الکلمات المفتاحية: $G$-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,

ملخص المقالة :

‎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.

المصادر:

[1] D. Bakic, B. Guljas, Extensions of Hilbert C∗-modules I, Houston Math. J. 30 (2004), 537-558.

[2] M. D. Choi, E. G. Effros, Injectivity and operator spaces, J. Func. Anal. 24 (1977), 156-209.

[3] E. Feizi, J. Soleymani, Injectivity and projectivity of some classes of Frechet algebras, UPB. Sci. Ser. A. 82 (3) (2020), 53-60.

[4] M. Frank, Hahn-Banach type theorems for Hilbert C^∗-modules, Inter. J. Math. 13 (2002), 675-693.

[5] M. Frank, Geometrical aspects of Hilbert C^∗-module, Positivity. 3 (1999), 215-243.

[6] M. Frank, Self-duality and C^∗-reflexivity of Hilbert C^∗-modules, Zeitschr. Anal. Anwendungen. 9 (1990), 165-176.

[7] M. Frank, D. R. Larson, Frames in Hilbert C^∗-modules and C^∗-algebras, J. Operator Theory. 48 (2002), 273-314.

[8] M. Frank, V. I. Paulsen, Injective and Projective Hilbert C^∗-module and C^∗-algebras of Compact Operators, Preprint, 2008.

[9] H. Gonshor, Injective hulls of C^∗-algebras, Trans. Amer. Math. Soc. 131 (1968), 315-322.

[10] H. Gonshor, Injective hulls of C^∗-algebras II, Proc. Amer. Math. Soc. 24 (1970), 468-491.

[11] D. Hadwin, V. I. Paulsen, Injectivity and projectivity in analysis and topology, Sci. China. Math. 59 (2011), 2347-2359.

[12] F. Kasch, Module and Rings, B. G. Teubner, Stuttgart, 1977.

[13] E. C. Lance, Hilbert C^∗-modules a Toolkit for Operator Algebraists, Lecture Notes Series, Vol. 210, Cambridge University Press, England, 1995.

[14] H. Lin, Bounded module maps and pure completely positive maps, J. Operator Theory. 26 (1991), 121-138.

[15] H. Lin, Extensions of multipliers and injective Hilbert modules, Chinese Ann. Math. Ser. B. 14 (1993), 387-396.

[16] H. Lin, Injective Hilbert C*-modules, Pacific J. Math. 154 (1992), 131-164.

[17] A. Mahmoodi, M. R. Mardanbeigi, On injective envelopes of AF-algebras, Thai. J. Math. 19 (2021), 1661-1669.

[18] V. M. Manuilov, E. V. Troisky, Hilbert C^∗-modules, American Mathematical Society, 2005.

[19] T. Oikhberg, Injectivity and projectivity in p-multinormed spaces, Positivity. 22 (4) (2018), 1023-1037.

[20] W. L. Paschke, Inner product module over B^∗-algebras, Trans. Amer. Math. Soc. 182 (1973), 443-468.

[21] W. L. Paschke, The double B-dual of inner product module over a C^∗-algebra, Canad. J. Math. 26 (1971), 1272-1280.

[22] N. E. Wegge-Olsen, K-theory and C^∗-algebras, a Frtendly Approach, Oxford University Press, Oxfored, 1993.

[23] Z. T. Xu, Hilbert C^∗-modules and C^∗-algebras, I (Engl./Chin.), Nanjing University Journal, Mathematical Biquarterly. 13 (1) (1996), 101-108.