چكيده فارسي :
هدف اين پايان نامه، مطالعه‌ي ويژگي‌هاي جبري و توپولوژيك حاصل‌ضرب نيم‌مستقيم جبرهاي باناخ به ويژه اولين گروه كوهومولوژيك آن‌ها است. ابتدا حاصل‌ضرب‌هاي نيم‌مستقيم جبرهاي باناخ را معرفي و ويژگي‌هاي مقدماتي آن را بررسي مي‌كنيم. سپس، ساختار مشتق‌ها روي حاصل‌ضرب‌هاي نيم‌مستقيم جبرهاي باناخ را بررسي خواهيم نمود و از دستاورد‌هاي پيوستگي خودكار مشتق‌ها بر روي جبرهاي باناخ و همچنين تجزيه‌ي مشتق‌ها نسبت به جمع استفاده خواهيم كرد. در بخش بعد، گروه‌هاي كوهومولوژي از حاصل‌ضرب‌هاي نيم‌مستقيم جبرهاي باناخ را مطرح مي‌كنيم، انواع مختلف آن را محاسبه و نتايج مختلف آن در اين زمينه را اثبات مي‌كنيم. در پايان از نتايج به دست آمده در قسمت‌هاي قبلي مرتبط با ضرب‌هاي نيم‌مستقيم جبرهاي باناخ استفاده كرده و پيوستگي خودكار از مشتق‌ها تحت رده‌هاي مختلفي از جبرهاي باناخ را بررسي مي‌كنيم.
چكيده انگليسي :
In this M.Sc. thesis, we give an expanded account of the paper: H. Farhadi. and H. Ghahramani., The first cohomology group of semidirect products of Banach algebras, Iran J. Sci. Technol. Trans. Sci. 45 (2021), 695–706. Let A and u be Banach algebras such that u is also a Banach Abimodule with compatible algebra operations, module actions and norm. By defining an approprite action, we turn l-direct product Au into a Banach algebra such that A is closed subalgebra and u is a closed ideal of it. This algebra, is in fact semidirect product of A and u which we denote it by A⋉u and every semidirect products of Banach algebras can be represented as this form. Here, we consider the Banach algebra A ⋉ u as mentioned and study the derivations on it. In fact we consider the automatic continuity of the derivations on A⋉u and obtain some results in this context and study its relation with the automatic continuity of the derivations on A and u. Also we calculate the first cohomology group of A ⋉ u in some different cases and establish relations among the first cohomology group of A ⋉ u and those of A and u. As applications of these contents, we present various results about the automatic continuity of derivations and the first cohomology group of direct products of Banach algebras, module extension Banach algebras and -Lau products of Banach algebras. The study of the first cohomology group of Banach algebras is also a considerable topic which may be used to study the structure of Banach algebras. Johnson, using the first cohomology group, has defined amenable Banach Algebras and then various types of amenability defined by using the first cohomology group. Also among the interesting problems in the theory of derivations is either characterizing algebras on which every continuous derivation is inner, that is, the first cohomology group is trivial or characterizing the first cohomology group of Banach algebras up to vector spaces isomorphism. Some results have also been obtained in the case of non-self-adjoint operator algebras. Christian showed that every continuous derivation on a nest algebra onH to itself and to B(H) is inner and then this result generalized to some other forms among which we may refer the references therein. Gilfeather and Smith have calculated the first cohomology group of some operator algebras called joins. The cohomology group of operator algebras called seminest algebras has been calculated. All of these operator algebras and nest algebras have a structure like triangluar Banach algebras, so motivated by these studies, Forrest and Marcoux verified the first cohomology group of triangular Banach algebras. Also, some results concerning the automatic continuity of derivations on triangluar Banach algebras are presented. A generalization of triangluar Banach algebras is module extensions of Banach algebras which Zhang has studied the weak amenability of them and then Medghalchi and Pourmahmood computed the first cohomology group of module extensions of Banach algebras and using those results, they gave various examples of Banach algebras with non-trivial cohomology group and in fact they calculated the first cohomology group of the given examples. Another class of Banach algebras which considered during the last thirty years, is the class of algebras obtained by a special product called -Lau product. This product is firstly introduced by Lau for a special class of Banach algebras which are pre-dual of von Neumann algebras where the dual unit element (the unit element of the dual) is a multiplicative linear functional. Afterwards, various studies have been performed to it.
