On Jones' index for real $\mathrm W^*$-algebras

The notions of the real coupling constant and the index for real $\mathrm W^*$-algebras are introduced and investigated. The main tool in our approach is the reduction of real $\mathrm W^*$-algebras to involutive $^*$-anti-automorphisms of their complex enveloping von Neumann algebras. Similar to the complex case the values of the index for type II$_1$ real factors are calculated.