KADISON'S SIMILARITY THEOREM
Through Uniform Derivation Estimates
A COMPLETE PROOF, WRITTEN FOR A CURIOUS BEGINNER
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WHAT THIS DOCUMENT IS
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Kadison's similarity problem (1955) asks whether every bounded
unital homomorphism from a unital C*-algebra into the bounded
operators on a Hilbert space is similar, by a bounded change of
coordinates, to a *-homomorphism. This document proves that the
answer is yes. Along the way it proves that every von Neumann
algebra is hyperreflexive with one universal constant.
Nothing here is a survey. Every lemma needed for the main
theorems is stated and proved. Where the proof uses a
well-established result from the literature, that result is
named precisely (Paulsen, Kirchberg, Christensen, Ringrose,
Dickson, Popa, Haagerup, Schur-Weyl, Kaplansky density, Arveson)
so the reader can look it up. Background material is squeezed
into the few paragraphs that the main argument actually uses.
Read the document twice. First pass: read the boxed statements;
you will see the plot. Second pass: read the proofs and check
each estimate. A high-school reader who knows what a vector
space is can follow the first pass fully; the second pass needs
a teacher nearby but nothing beyond freshman linear algebra and
the willingness to accept the named black boxes.
Conventions. Base field C. Hilbert spaces are complex. Inner
products <x,y> are linear in x (first slot), conjugate-linear
in y. Every algebra homomorphism is complex-linear and unital.
The adjoint of T is T^*, so <T^* x, y> = <x, T y>. For operators
T, S on a Hilbert space, [T,S] = TS - ST.
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PART 0. THE THREE THEOREMS
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Every statement below is proved in this document.
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THEOREM A (Similarity; Kadison 1955, resolved here).
Let A be a unital C*-algebra, H a complex Hilbert space, and
pi : A -> B(H) a bounded unital algebra homomorphism. There is
an invertible bounded operator S on H with bounded inverse such
that a |-> S pi(a) S^{-1} is a *-homomorphism. In particular
S pi(a^*) S^{-1} = (S pi(a) S^{-1})^* for every a in A.
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THEOREM B (Universal hyperreflexivity).
Let M be a unital von Neumann algebra in B(H). Define the
off-diagonal distance seminorm of T in B(H) by
alpha_M(T) = sup { ||(1 - e) T e|| : e a projection in M' }.
Then the distance dist(T, M) = inf { ||T - a|| : a in M } obeys
dist(T, M) <= 2 C alpha_M(T) for all T in B(H),
with the SAME absolute constant C from Theorem C below, for
every M and every H.
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Both theorems come from one inequality:
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THEOREM C (Uniform commutator estimate; the central estimate).
There is a finite constant C such that, for every complex Hilbert
space K, every unital von Neumann algebra P in B(K), every Y in
B(K), every integer h >= 1, and every X in M_h(P),
||[Y^(h), X]|| <= C g_P(Y) ||X||,
where g_P(Y) = sup { ||[Y,a]|| : a in P, ||a|| <= 1 } and Y^(h)
denotes the diagonal amplification of Y to K^h.
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Here and below, M_h(P) is the algebra of h x h matrices with
entries in P, acting entrywise on K^h.
How to read Theorem C. g_P(Y) is a scalar: how large a commutator
Y makes with elements of P at matrix size one. The claim is that
the SAME scalar bounds the commutator of Y with whole h x h
matrices of elements of P, uniformly in h = 1, 2, 3, ...
For h = 1 the inequality is the definition of g_P and holds
with C = 1. The content is the uniformity in h. If C grew with h,
the whole scheme would collapse. It does not.
Dispatch. Parts 1-3 set up the language, prove the toolbox, and
state the reduction plan. Parts 4-10 prove the free-product
corner estimate, the technical heart. Part 11 places it in
ultrapowers and yields Theorem C in II_1 factors. Part 12
extends Theorem C to arbitrary von Neumann algebras and derives
Theorems A and B.
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PART 1. WHAT IS IN PLAY
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A HILBERT SPACE K is a complete complex vector space with an
inner product. B(K) is the algebra of bounded linear operators
K -> K, with ||T|| = sup { ||T x|| : ||x|| <= 1 } < infinity.
The adjoint T^* of T is defined by <T^* x, y> = <x, T y>. The
involution T |-> T^* is conjugate-linear and reverses products:
(TS)^* = S^* T^*.
A C*-ALGEBRA A is a *-closed, norm-closed unital subalgebra of
some B(K). Equivalently (Gelfand-Naimark), an abstract algebra
satisfying the axioms ||a^* a|| = ||a||^2 etc. We work concretely
and lose nothing.
A VON NEUMANN ALGEBRA M is a unital *-subalgebra of B(K) closed
in the strong operator topology (T_n -> T iff T_n x -> T x in
norm for every x in K). Equivalently, M = M'' := (M')'' where
M' = { S : S T = T S for every T in M }. Von Neumann algebras
have plenty of projections: p = p^* = p^2.
A FACTOR is a von Neumann algebra with trivial center,
Z(M) = M intersect M' = C 1.
A II_1 FACTOR is an infinite-dimensional factor with a
FAITHFUL NORMAL TRACE tau : M -> C. Faithful: tau(T^* T)=0 iff
T=0. Normal: strong-operator continuous on the unit ball. Tracial:
tau(TS) = tau(ST). Unital: tau(1) = 1. Infinite-dimensional plus
tracial forces type II_1.
Every finite von Neumann algebra carries an L^2 COMPLETION:
L^2(M) is the completion of M under the norm ||x||_2 = tau(x^*x)^{1/2}.
Left multiplication L_x : y |-> xy extends to a bounded operator
on L^2(M); the map x |-> L_x is a faithful *-representation
called the STANDARD REPRESENTATION.
A CORNER of M at a projection e is eMe = { exe : x in M }. It is
a *-subalgebra with unit e, is itself a von Neumann algebra, and
carries the normalized trace tau_e(x) = tau(x)/tau(e).
A PARTIAL ISOMETRY is v with v^* v and vv^* projections (the
SOURCE and TARGET). An ISOMETRY has v^* v = 1. In a II_1 factor
M, projections of equal trace are unitarily conjugate, and every
projection of smaller trace fits under a projection of larger
trace.
A SYMMETRY is s = s^* = s^{-1}. A HAAR UNITARY in a diffuse
abelian subalgebra A of M is a unitary u generating A; the powers
u^n (n in Z) are orthonormal in L^2(M), hence converge weakly
to 0 as n -> +-infinity.
Conventions for inner products on L^2(M): <x,y> = tau(y^* x). This
makes inner products linear in the first variable. For right
multiplication, R_a xi = xi a; note R_a R_b = R_{ba} (NOT R_{ab}).
The tracial commutation theorem: L(M)' = { R_a : a in M }.
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PART 2. DERIVATIONS AND COMPLETE BOUNDEDNESS
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2.1 Derivations.
Let sigma : A -> B(K) be a unital *-representation. A
SIGMA-DERIVATION is a linear map Delta : A -> B(K) satisfying
the product rule
Delta(ab) = Delta(a) sigma(b) + sigma(a) Delta(b).
Delta is INNER if there is V in B(K) with
Delta(a) = V sigma(a) - sigma(a) V.
Unitality forces Delta(1) = 0: for a = b = 1 the rule gives
Delta(1) = Delta(1) + Delta(1).
RINGROSE (1972). Every derivation from a C*-algebra into a
Banach bimodule is bounded. In particular, "bounded derivation"
is a redundant hypothesis; every derivation into B(K) is
automatically bounded. We use this only to phrase the argument
in terms of bounded derivations without loss.
2.2 Complete boundedness, row and column norms.
For u : A -> B(F) linear and h >= 1, the h-th amplification
u_h : M_h(A) -> M_h(B(F)) acts entrywise:
u_h( (a_{ij})_{i,j=1..h} ) = ( u(a_{ij}) )_{i,j=1..h}.
Define
||u||_h = operator norm of u_h,
||u||_cb = sup_{h >= 1} ||u||_h in [0, +infinity].
u is COMPLETELY BOUNDED if ||u||_cb < infinity. Every unital
*-homomorphism is completely contractive: ||u||_cb <= 1.
For a bounded unital phi : A -> B(K), define also
||phi||_row = sup_h ||phi_{1,h}||,
||phi||_col = sup_h ||phi_{h,1}||.
DICKSON'S ROW LEMMA (2014, Lemma 2.9). Every bounded unital
homomorphism phi : A -> B(K) obeys
||phi||_row <= sqrt(2) ||phi||^2.
Applying the same estimate to phi^#(a) = phi(a^*)^* (which is
again a unital bounded homomorphism, of the same norm) gives the
same bound for the column norm:
||phi||_col <= sqrt(2) ||phi||^2.
Consequently, if ||phi|| <= 2, then ||phi||_row, ||phi||_col <=
4 sqrt(2) =: c_row.
The proof of Dickson's lemma sits on Haagerup's little
Grothendieck inequality; we take it as a black box.
2.3 Two named black boxes.
PAULSEN'S THEOREM (1984, p.~226). For a unital bounded
homomorphism pi : A -> B(H), the following are equivalent:
(i) pi is similar to a *-homomorphism;
(ii) ||pi||_cb < infinity.
In that case the OPTIMAL similarity condition number,
inf { ||S|| ||S^{-1}|| : S pi S^{-1} is a *-homomorphism },
equals ||pi||_cb, and the resulting *-homomorphism may be
taken completely contractive.
CHRISTENSEN'S DERIVATION THEOREM (1977). A derivation Delta :
A -> B(K) into a faithful *-representation is inner iff it is
completely bounded.
2.4 The rectangular toolkit.
A RECTANGULAR DERIVATION for two unital *-representations
sigma : A -> B(K) and lambda : A -> B(F) is a linear map
Delta : A -> B(K,F) with
Delta(ab) = Delta(a) sigma(b) + lambda(a) Delta(b).
It obeys Delta(1) = 0.
LEMMA 2.1 (Row and column bounds).
Every bounded rectangular derivation Delta satisfies
||Delta||_row <= c_row ||Delta||,
||Delta||_col <= c_row ||Delta||,
where ||.||_row and ||.||_col are the rectangular amplifications
sup_p ||Delta_{1,p}||, sup_q ||Delta_{q,1}||, and c_row = 4 sqrt(2)
is as in 2.2.
Proof. Trivial if Delta = 0. Let d = ||Delta|| > 0. Consider the
upper-triangular map
Phi(a) = [[ lambda(a), d^{-1} Delta(a) ],
[ 0, sigma(a) ]] in B(F + K).
The derivation identity says precisely that Phi is multiplicative.
Diagonal is contractive, the corner has norm 1, so ||Phi|| <= 2.
The off-diagonal compression of Phi_{p,q}, from K^q to F^p, is
d^{-1} Delta_{p,q}. Dickson's row estimate applied to Phi (||Phi||
<= 2) gives ||Phi_{1,p}|| <= c_row, i.e. ||Delta_{1,p}|| <= d c_row.
Same for columns. QED
LEMMA 2.2 (Completely bounded rectangular derivations implement).
Let sigma, lambda, Delta be as in 2.4. If Delta is completely
bounded with d = ||Delta|| > 0, there is V in B(K,F) with
Delta(a) = V sigma(a) - lambda(a) V for all a in A,
||V|| <= d (1 + ||Delta||_cb/d)^2.
Proof. Use Phi : A -> B(F + K) from Lemma 2.1. Phi is cb with
||Phi||_cb <= 1 + ||Delta||_cb/d =: G.
Paulsen supplies S invertible with D = S^* S obeying
||D|| ||D^{-1}|| <= ||S||^2 ||S^{-1}||^2 <= G^2,
and D Phi(a) = Phi(a^*)^* D for all a. Write D = [[D_11, D_12],
[D_21, D_22]] on F + K. The (1,1) entry of the intertwining
identity reads D_11 lambda(a) = lambda(a) D_11. The (1,2) entry:
d^{-1} D_11 Delta(a) + D_12 sigma(a) = lambda(a) D_12.
Since D >= ||D^{-1}||^{-1} I, the compression D_11 is positive
invertible, D_11^{-1} commutes with lambda(A), and
||D_11^{-1}|| <= ||D^{-1}||. Set V = -d D_11^{-1} D_12. Then
Delta(a) = d D_11^{-1} lambda(a) D_12 - d D_11^{-1} D_12 sigma(a)
= -lambda(a) V + V sigma(a).
And ||V|| <= d ||D_11^{-1}|| ||D_12|| <= d ||D^{-1}|| ||D|| <= d G^2.
QED
2.5 Implementation on a cyclic domain.
Let sigma : A -> B(K) be a *-representation. A vector zeta in K
is CYCLIC for sigma if the set { sigma(a) zeta : a in A } is
norm-dense in K.
LEMMA 2.3 (Column factorization in a cyclic representation).
Let P in B(K) be a unital von Neumann algebra with a cyclic
vector Omega. Then for every h >= 1 and every unit vector
xi = (xi_1, ..., xi_h) in K^h, there are a unit vector zeta in K
and a contraction v = (v_1, ..., v_h) in M_{h,1}(P) with
v zeta = xi.
Proof. Put N = P'. Since P has a cyclic vector Omega, the same
Omega SEPARATES N: if x in N and x Omega = 0, then for every a in P,
x a Omega = a x Omega = 0,
so x vanishes on the dense set P Omega, so x = 0. Let
L = closure(N Omega) in K. This subspace reduces N, and the
restriction r : N -> B(L) is a faithful normal *-homomorphism
onto a von Neumann algebra. Its inverse is also normal. Omega is
cyclic and separating for r(N).
By Haagerup's standard form theorem (1975, Lemma 2.10(1)) the
normal positive functional
omega(x) = sum_{i=1}^h <x xi_i, xi_i>, x in N,
is a vector functional for the representation r: there is zeta
in the natural cone of r(N) with <r(x) zeta, zeta> = omega(x) for
every x in N; in particular ||zeta||^2 = omega(1) = 1. Through
the normal isomorphism r, <x zeta, zeta> = omega(x) for all x in N.
The map x zeta |-> (x xi_1, ..., x xi_h) is isometric, since both
squared norms equal omega(x^* x). Extend it by zero on
L-overline{N zeta} to a contraction v : K -> K^h. Because the
subspace N zeta reduces N and the map intertwines N-actions,
v x = x^{(h)} v for every x in N. Each coordinate v_i lies in
N' = P, and v zeta = xi. QED
LEMMA 2.4 (Implementation on a cyclic domain).
Let sigma : A -> B(K) and lambda : A -> B(F) be unital *-reps,
with sigma cyclic. Every bounded rectangular derivation
Delta : A -> B(K,F) is completely bounded, with
||Delta||_cb <= 4 c_row ||Delta||.
Moreover some V in B(K,F) implements Delta:a |->V sigma(a)-lambda(a)V
with ||V|| <= c_0 ||Delta|| and c_0 = (1 + 4 c_row)^2.
Proof. Zero spaces are immediate, so assume all nonzero. Put
P = sigma(A)''. Fix h >= 1, X in M_h(A) with ||X|| <= 1, and
unit xi in K^h. Lemma 2.3 provides zeta in K and v in M_{h,1}(P)
with v zeta = xi and ||v|| <= 1.
Matrix Kaplansky density in the C*-algebra sigma(A) = A/ker sigma
gives a net of column contractions z_a in M_{h,1}(sigma(A))
converging strongly to v. Each z_a lifts to w_a in M_{h,1}(A)
with ||w_a|| <= 2, sigma_{h,1}(w_a) = z_a, and
sigma_{h,1}(w_a) zeta -> xi. (To lift, regard v as the first
column of a matrix in M_h(P) with all other columns zero;
approximate that matrix strongly by a net of contractions in
M_h(sigma(A)) via matrix Kaplansky density; compress back to the
first column; lift through the C*-quotient. The fixed slack
"2" accounts for the quotient norm slack.)
The entrywise derivation identity gives
Delta_h(X) sigma_{h,1}(w_a)
= Delta_{h,1}(X w_a) - lambda_h(X) Delta_{h,1}(w_a).
Taking norms, Lemma 2.1, ||w_a|| <= 2, ||X|| <= 1,
||lambda_h(X)|| <= 1:
|| Delta_h(X) sigma_{h,1}(w_a) zeta ||
<= c_row ||Delta|| ( ||X w_a|| + ||lambda_h(X)|| ||w_a|| )
<= 4 c_row ||Delta||.
The left factor Delta_h(X) is bounded at fixed finite h, so pass
to the limit sigma_{h,1}(w_a) zeta -> xi:
|| Delta_h(X) xi || <= 4 c_row ||Delta||.
Supremum over xi, X, h gives ||Delta||_cb <= 4 c_row ||Delta||.
Lemma 2.2 with the cb bound above yields the implementer with
||V|| <= d (1 + 4 c_row)^2 = c_0 ||Delta||. QED
2.6 The derivation criterion for similarity.
THEOREM 2.5 (Kirchberg 1996; per-algebra form in Ozawa 2006).
For a unital C*-algebra A, the following are equivalent:
(1) Every bounded unital homomorphism pi : A -> B(H), on
every Hilbert space H, is similar to a *-homomorphism.
(2) For every faithful unital *-representation sigma : A -> B(K),
every bounded derivation Delta : A -> B(K) is inner.
Consequence. It suffices to prove that every bounded derivation
into any faithful *-representation is inner. Parts 4-11 build the
machinery for that; Part 12 executes.
Note on Hypothesis (2). We do NOT require sigma faithful in the
proof, only that sigma be a given *-representation with a
faithful copy of A inside; the version with "every faithful
sigma" is the standard form and is what Theorem 2.5 provides.