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A C*-algebra is a complex Banach algebra together with an isometric antilinear involution satisfying (a b)* = b*a* and the C*-identity ‖ a*a ‖ = ‖ a ‖2.

For bounded operators on a given Hilbert space, C*-algebras characterize topologically closed subalgebras of ${\mathcal B}({\mathcal H})$ (in operator norm), also closed under taking the adjoint operator. C*-algebras are at the heart of and are extensively used in .

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