Some homological properties of Orlicz space

The subject of this thesis is in the scope of harmonic analysis in general and abstract harmonic analysis in particular. The main purpose of this study is to examine some homological properties of Orlicz spaces which are generalizations of L^p (G) Lebesgue spaces, including 1≤p<∞ and locally compact group G. Homological properties of Banach algebras are examined in the book of A. YA. Helemskii for the first time. Also in 2003, G. Dales and M.E. Polyakov examined the homological properties of L^p Banach modules, including the locally compact group G and 1≤p<∞, by using the fact that L^p (G) is a Banach module over the Banach algebra L^1 (G) with respect to the convolution product. On the other hand, it is important to investigate the multipliers of Banach algebras and Banach modules. It is especially important in the theory of representation of compact groups. Multipliers space for the Lebesgue spaces has been studied by many mathematicians (C.A Akemann, E. Brainerd, R.E. Edwards, A. Figa-Talamanca, G.I. Gaudry, L. Hörmander, J. Wendel). In this thesis, homological properties of Orlicz spaces, including the locally compact group G and Young function Ф, such as projectivity, injectivity and flatness are examined and multipliers space, in particular compact multipliers space is determined by using the fact Orlicz spaces are Banach modules over L^1 (G) with respect to the convolution. Thus, the studies in Lebesgue space have been generalized. Again, based on Lau's study, closed convex invariant subsets of Orlicz spaces are determined and compact multipliers space is examined by affine transformations by using the fact that Orlicz spaces are L^1 (G) Banach modules. Thus, a criterion is obtained to determine the compactness of the group by the compactness of the multiplier space.