480 F. Bimodules over von Neumann AlgebrasExample F.2. We recall that the formula llxll2 = T(x*x)^112 defines a
Hilbertian norm on M and the GNS Hilbert space L^2 (M) is the comple-
tion of M with respect to this norm. The vector in £^2 ( M) associated with
x E M is denoted by x. The identity bimodule (or the trivial bimodule) over
M is L^2 (M) with the action given by axb = ~ for a, b, x E M.
Example F.3. Let 1i be an M-module. A Hilbert subspace JC c 1i is called
an M-submodule if MJC c JC. Let e E lffi(H) be a projection. Observe that
eH is an M-submodule if and only if e E M'. In passing, we note that if
e EM, then eH is an eMe-module.
Example F.4. The coarse M-N-bimodule is L^2 (M)®L^2 (N) with the action
given by a(17®()b = (a17)®((b) for a EM, b EN, 17 E L^2 (M) and ( E L^2 (N).
Example F.5. Let r be a group and M = L(I'). For every unitary repre-
sentation (rr, H) of r, we associate the M-M-bimodule fl= 1i ® £^2 (r) with
left and right actions given by
I' 3 st-> (1 ® .\)(s) E lffi(H) and I' 3 t 1-> (7r ® p)(C^1 ) E lffi(H).
Notice that 7r ®pis unitarily equivalent to 1 ® p (cf. Fell's absorption prin-
ciple) and thus we have a canonical isomorphism ( 7r ® p(I') )" ~ M^0 P.
Example F.6. For a normal u.c.p. map cp: M-+ N, we define an M-N-
bimodule Hrp via the minimal Stinespring dilation of cp: Equip M 8 L^2 (N)
with the semi-inner product(Lai®%L)j®(j) = L(cp(bjai)"li,(j)
i j ij
and promote i.t to a Hilbert space Hrp by separation and completion. Abusing
notation, denote by b ® ( the vector in Hrp which it represents. The action
is given by
a(b ® ()x =(ab)® ((x).
For the unit vector erp = 1 ® l E Hrp we have
(aerpx,erpY)?-lcp = (cp(a)X,fi)P(N)
for every a E M and x, y E N. In particular, ( erpx, erp) = TN ( x) for every
x E N. For the converse direction, see Exercise F .1.Comparison of projections and finite bimodules. Comparison theory
for projections in von Neumann algebras culminates in the existence of the
center-valued trace on a finite von Neumann algebra. The center of M is
denoted by Z(M). As usual, for projections e, f E M, we write e ;:) f if
there exists a partial isometry v EM such that vv = e and vv :S f. We
refer to Theorem V.2.6 and Corollary 2.8 in [183] for the following result.