In some disciplines such properties are axiomatized and algebras with certain topological structure become the subject of the research.
In this context, the best studied examples are self-adjoint operator algebras, meaning that they are closed under taking adjoints.
C*-algebras can be easily characterized abstractly by a condition relating the norm, involution and multiplication.
Such abstractly defined C*-algebras can be identified to a certain closed subalgebra of the algebra of the continuous linear operators on a suitable Hilbert space.
Commutative self-adjoint operator algebras can be regarded as the algebra of complex-valued continuous functions on a locally compact space, or that of measurable functions on a standard measurable space.