REAL STRUCTURE IN NONCOMMUTATIVE Lp-SPACES

Abstract

Our work is devoted to the construction of real valued noncommutative $L_p$ spaces associated with real $W^{\ast }$-algebras of different types. We construct such real noncommutative $L_p$-spaces, , and we prove the theorem of isomorphism for real noncommutative $L_p$-spaces. Finally, we build an approximation of $L_p$-spaces associated with real $W^{\ast }$-algebras of type $III$ via $L_p$-spaces associated with finite real $W^{\ast }$-algebras, analogously to the work of U. Haagerup, M. Junge and Q. Xu (Trans. Amer. Math. Soc. 362:4 (2010), 2125–2165).