Projections from a von Neumann algebra onto a subalgebra
نویسندگان
چکیده
— This paper is mainly devoted to the following question : let M, N be Von Neumann algebras with M C N. If there is a completely bounded projection P : N -^ M, is there automatically a contractive projection P : N -^ M? We give an affirmative answer with the only restriction that M is assumed semi-finite. The main point is the isometric identification of the complex interpolation space (AO, Ai )@ associated to the couple (Ao,Ai) defined as follows : AQ (resp. Ai) is the Banach space of all n-tuples x = ( a i , . . . , x-n) of elements in M equipped with the norm ii 1 1 1 1 \~^ * nl/2 / 1 1 1 1 1 1 V^ *n l /2\ \\X\\AQ = II ^L^Z^HM O^PII^HAI = \\l^^i \\M )• Introduction This paper is mainly devoted to the following question. Let M, N be von Neumann algebras with M C N ; if there is a completely bounded (c.b. in short) projection P : N -^ M, is there automatically a contractive projection P : N -^ M7 We give an affirmative answer with the only restriction that M is assumed semi-finite. At the time of this writing, the case when the (*) Texte recu Ie 19 juillet 1993, revise Ie 26 fevrier 1994. G. PISIER, Texas A & M University, College Station, TX 77843, U. S. A. and Universite Paris VI, Equipe d'Analyse, Boite 186, 75252 Paris CEDEX 05 (France). Email : [email protected]. Partially supported by the N.S.F. AMS classification : 46L10, 46L50, 46B70, 47 A 68. BULLETIN DE LA SOCIETE MATHEMATIQUE DE FRANCE 0037-9484/1995/139/$ 5.00 (c) Societe mathematique de France
منابع مشابه
Conditional Expectations onto Maximal Abelian *-subalgebras
We determine when there is a unique conditional expectation from a semifinite von Neumann algebra onto a singly-generated maximal abelian *-subalgebra. Our work extends the results of Kadison and Singer via new methods, notably the observation that a unique conditional expectation onto a singly-generated maximal abelian *-subalgebra must be normal.
متن کاملClassically Normal Pure States
A pure state f of a von Neumann algebra M is called classically normal if f is normal on any von Neumann subalgebra of M on which f is multiplicative. Assuming the continuum hypothesis, a separably represented von Neumann algebra M has classically normal, singular pure states iff there is a central projection p ∈ M such that pMp is a factor of type I∞, II, or III. DEFINITION: A pure state f of ...
متن کاملSolid Von Neumann Algebras
We prove that the relative commutant of a diffuse von Neumann subalgebra in a hyperbolic group von Neumann algebra is always injective. It follows that any non-injective subfactor in a hyperbolic group von Neumann algebra is non-Γ and prime. The proof is based on C∗-algebra theory.
متن کاملRemarks on Complemented Subspaces of Von-neumann Algebras*
In this note we include two remarks about bounded (not necessarily contractive) linear projections on a von Neumann-algebra. We show that if M is a von Neumann-subalgebra of B(H) which is complemented in B(H) and isomorphic to M ⊗ M then M is injective (or equivalently M is contractively complemented). We do not know how to get rid of the second assumption on M. In the second part,we show that ...
متن کاملVarious topological forms of Von Neumann regularity in Banach algebras
We study topological von Neumann regularity and principal von Neumann regularity of Banach algebras. Our main objective is comparing these two types of Banach algebras and some other known Banach algebras with one another. In particular, we show that the class of topologically von Neumann regular Banach algebras contains all $C^*$-algebras, group algebras of compact abelian groups and ...
متن کاملA double commutant theorem for Murray-von Neumann algebras.
Murray-von Neumann algebras are algebras of operators affiliated with finite von Neumann algebras. In this article, we study commutativity and affiliation of self-adjoint operators (possibly unbounded). We show that a maximal abelian self-adjoint subalgebra A of the Murray-von Neumann algebra A(f)(R) associated with a finite von Neumann algebra R is the Murray-von Neumann algebra A(f)(A(0)), wh...
متن کامل