\(C^*\) -Algebras

\(C^*\) -algebras are to Banach algebras what Hilbert spaces are to Banach spaces. The presence of an ‘involution’ leads to the concepts of normal, self-adjoint, and unitary elements. For any normal element, the associated state space is the closed convex hull of its spectrum. Operators in \(B(H)\) also have an associated set, called the numerical range, which is related to the spectrum by the Hausdorff-Toeplitz theorem. The culmination is the Spectral theorem for compact normal operators, the associated Singular Value Decomposition, and the various ideals of compact operators, with emphasis on the Hilbert-Schmidt operators. Additional results are the continuous functional calculus for normal elements, the characterization of self-adjoint and unitary elements in terms of their spectra, positive self-adjoint elements, the polar decomposition, and von Neumann’s Spectral theorem for normal operators. Finally, the Gelfand-Naimark shows how every \(C^*\) -algebra is embedded in some \(B(H)\) .