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In this paper, we investigate homomorphisms in proper CQ-algebras, proper Lie CQ-algebras and proper Jordan CQ-algebras and deriva-tions on proper CQ-algebras, proper Lie CQ-algebras and proper JordanCQ-algebras associated with the Cauchy-Jensen functional equation2f(x + y2+ z)= f(x) + f(y) + 2f(z);which was introduced and investigated in [3, 28].Furthermore, Isometries and isometric isomorphisms in proper CQ-algebrasare studied.

ISOMORPHISMS,DERIVATIONS AND ISOMETRIES IN PROPER <i>CQ</i><sup>&#8727;</sup>-ALGEBRAS

ISOMORPHISMS,DERIVATIONS AND ISOMETRIES IN PROPER <i>CQ</i><sup>∗</sup>-ALGEBRAS