1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

490 F. Bimodules over von Neumann Algebras

Exercises

Exercise F.l. Let 1i be an M-N-bimodule and e E 1i be a unit vector

such that (ex' e) = TN ( x) for every x E N. Prove that the formula

\ip(a)x,YJL2(N) = \aex,eYhi

defines a normal u.c.p. map ip: M---+ N.

Exercise F.2. Let A C JIB(H) be a von Neumann algebra and e E A be a

projection. Prove that (eAe)' = A'e in lIB(eH).

Exercise F.3. Let N be a semifinite von Neumann algebra with a faithful

normal semifinite trace Tr and let

0 = {x EN: [/x//2 = Tr(x*x)^2 :S 1} c N.

Prove that the formal inclusion 0 '-----7 L^2 (N, Tr) is ultraweak-weak continu-

ous. As a corollary, deduce that if C is an ultraweakly closed subset of N

which is bounded in both the operator norm and the £^2 -norm, then C is

closed in L^2 (N, Tr).

Exercise F .4 .. Let e E £^2 ( M) and u, v E M be a partial isometry such that

u*uevv* = e. Prove that Lu~v = uL~v.

Exercise F.5. Let ACM be finite von Neumann algebras and (M, A) be

the basic construction. Prove that the conditional expectation EA : M ---+ A

extends to \M,A) by the relation eAXeA = EA(x)eA.

Exercise F .6. Let A C M be finite von Neumann algebras and take a family

{ vkh of partial isometries in (M, A) such that v'kvk :S eA and ~k vkvk = 1.

(Such a family exists since the central support of eA is 1.) Define a normal

weight Tr on \M,A) by

Tr(z) = 2:)zvS,vS).

k

Prove that Tr is a faithful semifinite trace such that Tr(xeAy*) = r(xy*) for

every x, y E M.

Exercise F.7. Let ACM be finite von Neumann algebras and Tr be the

canonical faithful normal semifinite trace on \M, A). Prove that Tr(P) =

dimA P L^2 ( M) for every projection P E (M, A).

Exercise F:8. Let Ai, A2 c r be groups and suppose that for every s Er

one has sAis-i nA2 = {e} (e.g., r =Ai *A2). Let Ao c L(Ai) be a diffuse

von Neumann subalgebra. Prove that there is no unitary element u E L(r)

such that uAou* C L(A2).