## Taiwanese Journal of Mathematics

### Invariant Subsets and Homological Properties of Orlicz Modules over Group Algebras

#### Abstract

Let $G$ be a locally compact group with left Haar measure. We study the closed convex left invariant subsets of $L^{\Phi}(G)$ and characterize affine mappings from the space of nonnegative functions in $L^{1}(G)$ of norm $1$ into $L^{\Phi}(G)$ spaces. We apply the results to the study of the multipliers of $L^{\Phi}(G)$. We also investigate the homological properties of $L^{\Phi}(G)$ as a Banach left $L^{1}(G)$-module such as projectivity, injectivity and flatness.

#### Article information

Source
Taiwanese J. Math., Advance publication (2019), 15 pages.

Dates
First available in Project Euclid: 19 September 2019

https://projecteuclid.org/euclid.twjm/1568858572

Digital Object Identifier
doi:10.11650/tjm/190903

#### Citation

Üster, Rüya; Öztop, Serap. Invariant Subsets and Homological Properties of Orlicz Modules over Group Algebras. Taiwanese J. Math., advance publication, 19 September 2019. doi:10.11650/tjm/190903. https://projecteuclid.org/euclid.twjm/1568858572

#### References

• I. Akbarbaglu and S. Maghsoudi, Banach-Orlicz algebras on a locally compact group, Mediterr. J. Math. 10 (2013), no. 4, 1937–1947.
• C. A. Akemann, Some mapping properties of the group algebras of a compact group, Pacific J. Math. 22 (1967), 1–8.
• C. Bennett and R. Sharpley, Interpolation of Operators, Pure and Applied Mathematics 129, Academic Press, London, 1988.
• B. Brainerd and R. E. Edwards, Linear operators which commute with translations I: Representation theorems, J. Austral. Math. Soc. 6 (1966), 289–327.
• H. G. Dales, Banach Algebras and Automatic Continuity, London Mathematical Society Monographs, New Series, 24, Oxford University Press, New York, 2000.
• H. G. Dales and M. E. Polyakov, Homological properties of modules over group algebras, Proc. London Math. Soc. (3) 89 (2004), no. 2, 390–426.
• R. S. Doran and J. Wichmann, Approximate Identities and Factorization in Banach Modules, Lecture Notes in Mathematics 768, Springer-Verlag, Berlin, 1979.
• J. Faraut, Analysis on Lie Groups: An introduction, Cambridge Studies in Advanced Mathematics 110, Cambridge University Press, Cambridge, 2008.
• A. Ya. Helemskii, The Homology of Banach and Topological Algebras, Mathematics and its Applications (Soviet Series) 41, Kluwer Academic Publishers, Dordrecht, 1989.
• E. Hewitt and K. A. Ross, Abstract Harmonic Analysis II: Structure and analysis for compact groups analysis on locally compact abelian groups, Springer-Verlag, Berlin, 1997.
• B. E. Johnson, Cohomology in Banach Algebras, Memoirs of the American Mathematical Society 127, American Mathematical Society, Providence, R.I., 1972.
• R. Larsen, An Introduction to the Theory of Multipliers, Die Grundlehren der mathematischen Wissenschaften, Band 175, Springer-Verlag, Heidelberg, 1971.
• A. T. M. Lau, Closed convex invariant subsets of $L_{p}(G)$, Trans. Amer. Math. Soc. 232 (1977), 131–142.
• A. T. M. Lau and V. Losert, Complementation of certain subspaces of $L_{\infty}(G)$ of a locally compact group, Pacific J. Math. 141 (1990), no. 2, 295–310.
• W. A. Majewski and L. E. Labuschagne, On applications of Orlicz spaces to statistical physics, Ann. Henri Poincaré 15 (2014), no. 6, 1197–1221.
• A. Osançl\iol and S. Öztop, Weighted Orlicz algebras on locally compact groups, J. Aust. Math. Soc. 99 (2015), no. 3, 399–414.
• S. Öztop and E. Samei, Twisted Orlicz algebras I, Studia Math. 236 (2017), no. 3, 271–296.
• ––––, Twisted Orlicz algebras II, Math. Nachr. 292 (2019), no. 5, 1122–1136.
• S. Öztop, E. Samei and V. Shepelska, Weak amenability of weighted Orlicz algebras, Arch. Math. (Basel) 110 (2018), no. 4, 363–376.
• G. Racher, On the projectivity and flatness of some group modules, in: Banach Algebras 2009, 315–325, Banach Center Publications 91, Polish Acad. Sci. Inst. Math., Warsaw, 2010.
• M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker, New York, 1991.
• R. A. Ryan, Introduction to Tensor Products of Banach Spaces, Springer Monographs in Mathematics, Springer-Verlag London, London, 2002.
• S. Sakai, Weakly compact operators on operator algebras, Pacific J. Math. 14 (1964), 659–664.
• J. G. Wendel, Left centralizers and isomorphisms of group algebras, Pacific J. Math. 2 (1952), 251–261.