For a conjugation C on a separable, complex Hilbert space \(\cal{H}\) , the set \({\cal{S}}_{C}\) of C-symmetric operators on \(\cal{H}\) forms a weakly closed, selfadjoint, Jordan operator algebra. In this paper, the authors study \({\cal{S}}_{C}\) in comparison with the algebra \(\cal{B}(H)\) of all bounded linear operators on \(\cal{H}\) , and obtain \({\cal{S}}_{C}\) -analogues of some classical results concerning \(\cal{B}(H)\) . The authors determine the Jordan ideals of \({\cal{S}}_{C}\) and their dual spaces. Jordan automorphisms of \({\cal{S}}_{C}\) are classified. The authors determine the spectra of Jordan multiplication operators on \({\cal{S}}_{C}\) and their different parts. It is proved that those Jordan invertible ones constitute a dense, path connected subset of \({\cal{S}}_{C}\) .
