Projectionless real $C^\ast$-algebras

The report is dedicated to the projectionless real $C^\ast$-algebras. Following construction of B.Blackadar (A simple $C^\ast$-algebra with no nontrivial projections. Proc. of AMS, 78, 4, 1980, 504-508) a real $C^\ast$-algebra is constructed, which is separable, simple, nuclear, nonunital, and contains no nonzero projections. It is proved that a real $C^\ast$-algebra is projectionless if and only if the enveloping $C^\ast$-algebra is projectionless. An example of a projectionless real Banach $\ast$-algebra with the $C^\ast$-property is constructed, the complexification of which contains a non-trivial projection.