# C*-algebre i njihove reprezentacije

Malenica, Ante (2015) C*-algebre i njihove reprezentacije. Diploma thesis, Faculty of Science > Department of Mathematics.

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Language: Croatian

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.