The main theorem of the thesis asserts that if B is a JB*-subtriple of a JB*triple A such that B separates oe(An U {O}, then if A or B is postliminal, A=B. The main theorem and many of the other key results of the thesis are generalisations of the results of Kaplansky (1951) and Glimm (1960) on the Stone-Weierstrass conjecture for C* -algebras. We first prove a Stone-Weierstrass theorem for postliminal JB-algebras. This plays an essential role in the proof of the main theorem and is also important in the proof of our second main result, the Glimm-Stone-Weierstrass theorem for JB-algebras. Vital to the Glimm-Stone-Weierstrass proof, we show that if A is a universally reversible prime and antiliminal JB-algebra, then S(A) C P(A). Conversely, if A is universally reversible and of dimension greater than one, S(A) C P(A) implies A is prime and antiliminal. The C* -algebra version of this theorem is due to Tomiyama and Takesaki (1961). By means of the universal enveloping C*-algebra functor, we show that if , the Stone-Weierstrass conjecture is true for C* -algebras then it is true for JB-algebras. Employing a similar technique we prove Stone-Weierstrass theorems for semi-finite JW-algebras and type I JW-algebras, building on results of Akemann (1969- 70). The crucial result of the thesis reduces the Stone-Weierstrass separation condition for JB*-triples locally to that of JB*-algebras. Using this in conjunction with the Stone-Weierstrass theorem for postliminal JB-algebras is an essential part of the proof of the main theorem