6.2. Amenable traces 215It follows that if b is a linear combination of unitaries in A, then <p(Tb) =
<p(bT) and, since unitaries span A, this shows T is a trace.
One should ask whether or not the definition of amenable trace depends
on the choice of embedding AC JE(7i). Luckily, it doesn't.
Proposition 6.2.2. Assume A C JE(7i) and T is an amenable trace. For
every faithful representation 7r: A ----+ JE(JC), there exists a state 1/JK on JE(JC)
such that 'lf;K(7r(u)T7r(u*)) = 'lj;K(T), for all TE JE(JC) and unitaries u EA,
and 'l/JK o 7r = T.Proof. By Arveson's Extension Theorem, there is a u.c.p. map : JE(JC) ----+
JE(H) which extends the inverse map 7r-l: 7r(A) ----+ A C lE(H). Note that
7r(A) is in the multiplicative domain of . Defining 'l/JK = 'lj; o , where 'lj; is
any state on JE(H) as in the definition of amenable trace, we get a state on
JE(JC). Evidently we have 'l/JK o 7r = T and the rest of the proof is a simple
multiplicative domain argument. Indeed, for T E JE(JC) and a unitary u E A
we have
'lf;K(7r(u)T7r(u)) = 'lj;(u(T)u) = 'lj;K(T),
as desired. D
Recall that S1, S2 C JE(7i) denote the trace class and Hilbert-Schmidt op-
erators, respectively. An easy, but important, fact is that both the L^1 -norm
llTll1 = Tr(ITI) and L^2 -norm llTll2 = JTr(ITl^2 ) are unitarily invariant,
meaning that lluTll1 = llTull1 = llTll1 for all unitary operators u E JE(7i)
(same for the L^2 -norm).
Lemma 6.2.3. For any finite-rank h, k E JE(7i), arbitrary x E JE(H) and
p E {1,2},^5 one has llh*llP = llhllP, llxhlb :S llxllllhllp, llhxllP :S llxllllhllp·
Moreover, llhklli :S llhll2llkll2 and
llhll1 =sup{! Tr(hy)I: YE lE(7i), llYll :S 1}.
Proof. The last equation is well known (JE(7i) is the dual of S1 (1-l)) and
easily implies llh*ll1 = llhll1; that llh*ll2 = llhll2 is trivial. The inequality
llxhll~ = Tr(h*x*xh)::; llxll^2 Tr(h*h) = llxll^2 llhll~
also implies llhxll2 ::; llxll llhll2, after taking adjoints. Operator monotonicity
of the square root function (applied to lxhl^2 ::; llxll^2 lhl^2 ) implies lxhl :S
llxll lhl; thus llxhll1 :S llxll llhll1·
The Cauchy-Schwarz inequality says that5rn fact, this holds for any 1 ::::; p < oo.