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For compact orientable manifolds, this is accomplished by the notion of a spectral triple.

See the question Commutative spectral triples for additional information, including a precise statement of the theorem.

See also the answer to Noncommutative smooth manifolds for additional remarks about orientability.

As a side remark, if one is willing to use smooth functions instead of continuous functions, then there is a very satisfactory answer: the resulting contravariant functor from second countable Hausdorff smooth manifolds to commutative real algebras is a fully faithful functor, i.e., manifolds are (contravariantly) identified with a full subcategory of real algebras. This is known as Milnor's exercise.

For compact orientable manifolds, this is accomplished by the notion of a spectral triple.

See the question Commutative spectral triples for additional information, including a precise statement of the theorem.

See also the answer to Noncommutative smooth manifolds for additional remarks about orientability.

As a side remark, if one is willing to use smooth functions instead of continuous functions, then there is a very satisfactory answer: the resulting contravariant functor from second countable Hausdorff smooth manifolds to commutative real algebras is a fully faithful functor, i.e., manifolds are (contravariantly) identified with a full subcategory of real algebras. This is known as Milnor's exercise. See also the answer How much of differential geometry can be developed entirely without atlases? for a discussion on how to characterize the essential image of the embedding functor.

For compact orientable manifolds, this is accomplished by the notion of a spectral triple.

See the question Commutative spectral triples for additional information, including a precise statement of the theorem.

See also the answer to Noncommutative smooth manifolds for additional remark about orientability.

For compact orientable manifolds, this is accomplished by the notion of a spectral triple.

See the question Commutative spectral triples for additional information, including a precise statement of the theorem.

See also the answer to Noncommutative smooth manifolds for additional remarks about orientability.

As a side remark, if one is willing to use smooth functions instead of continuous functions, then there is a very satisfactory answer: the resulting contravariant functor from second countable Hausdorff smooth manifolds to commutative real algebras is a fully faithful functor, i.e., manifolds are (contravariantly) identified with a full subcategory of real algebras. This is known as Milnor's exercise.

For compact manifolds, this is accomplished by the notion of a spectral triple.

See the question Commutative spectral triples for additional information, including a precise statement of the theorem.

For compact orientable manifolds, this is accomplished by the notion of a spectral triple.

See the question Commutative spectral triples for additional information, including a precise statement of the theorem.

See also the answer to Noncommutative smooth manifolds for additional remark about orientability.