C* - algebre i njihove reprezentacije

Abstract

Banachove algebre s involucijom koje zadovoljavaju $\| a ^\ast a \| = \|a\|^2$ za svaki njihov element $\alpha$ nazivaju se $C^\ast$-algebrama. U ovom su radu pokazana osnovni rezultati teorije $C^\ast$-algebri, a razradi problema pristupljeno je najprije predstavljajući neke od rezultata teorije Banachovih algebri. Nakon što je pokazano postojanje Gelfandove transformacije na komutativnim Banachovim algebrama, taj se rezultat proširuje na komutativne $C^\ast$-algebre gdje je navedeno preslikavanje izometrički $\ast$-izomorfizam. Osim toga, pokazana je uska veza zatvorenih lijevih ideala i hereditarnih $C^\ast$-podalgebri koje se, dodatno, u slučaju separabilnosti početnog prostora, usko povezuju s pozitivnim elementima originalne $C^\ast$-algebre. Konačno, u poznatom teoremu Gelfanda i Naimarka pokazano je postojanje vjerne reprezentacije svake $C^\ast$-algebre.A Banach algebra supplied with an involution and a norm satisfying the condition $\| a ^\ast a \| = \|a\|^2$, for every element $\alpha$, is called a $C^\ast$-algebra. The approach used in this thesis was to first introduce the main ideas and theorems of the theory of Banach algebras and then to move on to the theory $C^\ast$-algebra and presented its important results. After proving the existence of Gelfand transformation of abelian Banach algebras, this result has been generalized to abelian $C^\ast$-algebras, where the said mapping has been proved to be an isometric $\ast$-isomophism. Additionally, we have shown a close relationship between closed left ideals and hereditary $C^\ast$-subalgebras, and the latter are, in the case of separability of the original space, closely related to positive elements of the observed $C^\ast$-algebra. Finally, we have presented a famous resut of Gelfand and Naimark stating the existence of a faithful representation of an arbitrary $C^\ast$-algebra