the topology of uniform convergence. This is a C*-algebra with the *-operation given by pointwise complex conjugation. The complex unital *-algebra generated by consists of all those functions that can be obtained from the elements of by throwing in the constant function and adding them, multiplying them, conjugating them, or multiplying them with complex scalars, and repeating finitely many times. This theorem implies the real version, because if a net of complex-valued functions uniformly approximates a given function, , then the real parts of those functions uniformly approximate the real part of that function, , and because for real subsets, taking the real parts of the generated complex unital (selfadjoint) algebra agrees with the generated real unital algebra generated. As in the real case, an analog of this theorem is true for locally compact Hausdorff spaces. Stone–Weierstrass theorem, quaternion version Following , consider the algebra of quaternion-valued continuous functions on the compact space , again with the topology of uniform convergence. If a quaternion is written in the form its scalar part is the real number . Likewise the scalar part of is which is the real number . the scalar part of is which is the real number . the scalar part of is which is the real number . Then we may state: Stone–Weierstrass theorem, C*-algebra version The space of complex-valued continuous functions on a compact Hausdorff space i.e. is the canonical example of a unital commutative C*-algebra . The space X may be viewed as the space of pure states on , with the weak-* topology. Following the above cue, a non-commutative extension of the Stone–Weierstrass theorem, which remains unsolved, is as follows: In 1960, Jim Glimm proved a weaker version of the above conjecture. Lattice versions Let be a compact Hausdorff space. Stone's original proof of the theorem used the idea of lattices in . A subset of is called a lattice if for any two elements , the functions also belong to . The lattice version of the Stone–Weierstrass theorem states: The above versions of Stone–Weierstrass can be proven from this version once one realizes that the lattice property can also be formulated using the absolute value which in turn can be approximated by polynomials in . A variant of the theorem applies to linear subspaces of closed under max : More precise information is available: Suppose is a compact Hausdorff space with at least