Algebre z ločljivim spektrom in Banachovimi moduli nad njimi - Bračič, Janko (visokošolska dela, 2001)
Let ?$\mathfrak{A}$? be a unital commutative Banach algebra. Under what conditions on a left Banach ?$\mathfrak{A}$?-module ?$\mathcal{X}$? is it true that each elament in ?$\mathfrak{A}$? induces a decomposable multiplication operator on ?$\mathcal{X}$?? In order to give an ansver to this question we introduce a representation theory for modules and this theory is a natural extension of the representation theory of algebras. There are many notions from the theory of algebras which are extended in a natural way to modules. For instance, we introduce a notion of spectrally separable module and answer the above question in the fillowing way. If ?$\mathcal{X}$? is a left Banach ?$\mathfrak{A}$?-module such that its dual module ?$\mathcal{X}^*$? is spectrally separable, then each element in ?$\mathfrak{A}$? induces a decomposable multiplication operator on ?$\mathcal{X}$?. By the help of the theory of module representations we define simple multipliers on a given Banach module. For example, all multipliers on a semisimple commutative Banach algebra are simple. We show that simple multipliers have similar properties as multipliers on algebras. Under some additional conditions on a module we can prove that simple multipliers on this module form a semisimple unital commutative Banach algebra. Then the assertions which are similar to the known results about multipliers on algebras can be proven.
Keywords: funkcionalna analiza; teorija operatorjev; algebra z ločljivim spektrom; Arvesonov spekter; dekomponibilen operator množenja; komutativna Banachova algebra; krepko harmoničen operator; modul z ločljivim spektrom; multiplikatorji; točkasti multiplikatorji; upodobitev modula