Extrem states, operator spaces and ternary rings of operators

In this survey article on extreme states of operator spaces in C¡-algebras and related ternary ring of operators an extension result for rectangular operator extreme states on operator spaces in ternary rings of operators is discussed. We also observe that in the spacial case of operator spaces in rectangular matrix spaces, rectangular extreme states are conjugates of inclusion or identity maps implemented by isometries or unitaries. A characterization result for operator spaces of matrices for which the inclusion map is an extreme state is deduced using the above mentioned results.