matrices. as above. The Jordan algebra freely generated by Rn with the relations where the right-hand side is defined using the usual inner product on Rn. This is sometimes called a spin factor or a Jordan algebra of Clifford type. The Jordan algebra of 3×3 self-adjoint octonionic matrices, as above (an exceptional Jordan algebra called the Albert algebra). Of these possibilities, so far it appears that nature makes use only of the n×n complex matrices as algebras of observables. However, the spin factors play a role in special relativity, and all the formally real Jordan algebras are related to projective geometry. Peirce decomposition If e is an idempotent in a Jordan algebra A (e2 = e) and R is the operation of multiplication by e, then R(2R − 1)(R − 1) = 0 so the only eigenvalues of R are 0, 1/2, 1. If the Jordan algebra A is finite-dimensional over a field of characteristic not 2, this implies that it is a direct sum of subspaces A = A0(e) ⊕ A1/2(e) ⊕ A1(e) of the three eigenspaces. This decomposition was first considered by for totally real Jordan algebras. It was later studied in full generality by and called the Peirce decomposition of A relative to the idempotent e. Special kinds and generalizations Infinite-dimensional Jordan algebras In 1979, Efim Zelmanov classified infinite-dimensional simple (and prime non-degenerate) Jordan algebras. They are either of Hermitian or Clifford type. In particular, the only exceptional simple Jordan algebras are finite-dimensional Albert algebras, which have dimension 27. Jordan operator algebras The theory of operator algebras has been extended to cover Jordan operator algebras. The counterparts of C*-algebras are JB algebras, which in finite dimensions are called Euclidean Jordan algebras. The norm on the real Jordan algebra must be complete and satisfy the axioms: These axioms guarantee that the Jordan algebra is formally real, so that, if a sum of squares of terms is zero, those terms must be zero. The complexifications of JB algebras are called Jordan C*-algebras or JB*-algebras. They have been used extensively in complex geometry to extend Koecher's Jordan algebraic treatment of bounded symmetric domains to infinite dimensions. Not all JB algebras can be realized as Jordan algebras of self-adjoint operators on a Hilbert space, exactly as in finite dimensions. The exceptional Albert algebra is the common obstruction. The Jordan algebra analogue of von Neumann algebras is played by JBW algebras. These