arXiv:1607.04470v2 [math.OA] 22 Jun 2017

TRACIAL STABILITY FOR C ∗ -ALGEBRAS
DON HADWIN AND TATIANA SHULMAN
Abstract. We consider tracial stability, which requires that tuples of elements of a C*-algebra with a trace that nearly satisfy a relation are close to
tuples that actually satisfy the relation. Here both ”near” and ”close” are
in terms of the associated 2-norm from the trace, e.g. the Hilbert-Schmidt
norm for matrices. Precise definitions are stated in terms of liftings from tracial ultraproducts of C*-algebras. We completely characterize matricial tracial
stability for nuclear C*-algebras in terms of certain approximation properties
for traces. For non-nuclear C ∗ -algebras we find new obstructions for stability
by relating it to Voiculescu’s free entropy dimension. We show that the class
of C*-algebras that are stable with respect to tracial norms on real-rank-zero
C*-algebras is closed under tensoring with commutative C*-algebras. We show
that C(X) is tracially stable with respect to tracial norms on all C ∗ -algebras
if and only if X is approximately path-connected.

Contents
Introduction
1. Preliminaries
2. Tracial stability
3. Matricial tracial stability
3.1. Embeddable traces
3.2. Matricial tracial stability – necessary conditions
3.3. Matricial tracial stability for tracially nuclear C*-algebras
4. Matricial tracial stability and free entropy dimension
5. C ∗ -tracial stability
References

1
5
6
14
14
16
19
21
25
32

Introduction
The notion of stability is an old one. For a given equation p(x1 , . . . , xn ) = 0 of
noncommutative variables x1 , . . . , xn one can ask if it is ”stable”, meaning that for
any ǫ > 0 there is a δ > 0 such that if B is a C ∗ -algebra with b1 , . . . bn ∈ B and
kp (b1 , . . . , bn )k < δ, then there exist c1 , . . . , cn ∈ B such that p (c1 , . . . , cn ) = 0 and
kck − bk k < ε for 1 ≤ k ≤ n.
In other words, if some tuple is close to satisfying the equation, it is near to
something that does satisfy the equation.
2000 Mathematics Subject Classification. Primary 46Lxx; Secondary 20Fxx.
Key words and phrases. tracial ultraproduct, tracially stable, tracial norms, almost commuting
matrices.
1

2

DON HADWIN AND TATIANA SHULMAN

”Stability under small perturbations” questions depend very much on the norm
we consider and the class of C*-algebras B we allow.
A folklore ”stability” result is related to projections. If x = x∗ and x − x2 <
ε < 1/4, with the norm being
√ the usual operator norm, then there is a projection
p ∈ C ∗ (x) with kp − xk < ε. There are easily proved similar results for isometries
2
2
1−x∗ x = 0 and unitaries (1 − x∗ x) +(1 − xx∗ ) = 0 using the polar decomposition.
∗
∗
For the property of being normal, x x−xx = 0, the famous question of stability
for finite matrices was asked by Halmos ([17]). He asked whether an almost normal
contractive matrix is necessarily close to a normal contractive matrix. This is
considered independently of the matrix size and ”almost” and ”close” are meant
with respect to the operator norm. This question was answered positively by Lin’s
famous theorem [20] (see also [7] and [19]).
However the result does not hold when matrices are replaced by operators. A
classical example is the sequence {Sn } of weighted unilateral shifts with weights,
1 2
n−1
, ,...,
, 1, 1, . . . .
n n
n
Each Sn is a compact perturbation of the unweighted shift and Fredholm index
arguments show that the distance from Sn to the normal operators is exactly 1,
but
lim kSn∗ Sn − Sn Sn∗ k = 0.

n→∞

In other words, the relation
kxk ≤ 1, xx∗ − x∗ x = 0
is stable with respect to the class of matrix algebras but is not stable with respect
to the class of all C ∗ -algebras. Thus ”stability” questions depend on the class of
C ∗ -algebras you are considering.
The property of being stable with respect to the class of all C ∗ -algebras and the
operator norm is called weak semiprojectivity ([3]). An excellent exposition of weak
semiprojectivity can be found in Loring’s book ([21]).
Although almost normal operators need not be close to normal, however, using
the remarkable distance formula of Kachkovsky and Safarov in [19], the first author
and Ye Zhang [15] proved that there is a constant C such that, for every Hilbertspace operator T , the distance from T ⊕ T ⊕ · · · to the normal operators is at most
1/2
C kT ∗ T − T T ∗k .
Another famous Halmos stability question ([17]) asks whether two almost commuting unitary matrices are necessarily close to two exactly commuting unitary
matrices. It was answered by Voiculescu in the negative [26] (see [5] for a short
proof). However if the operator norm is replaced by Hilbert-Schmidt norm, then
things change dramatically. In the Hilbert-Schmidt norm almost commuting unitary matrices turn out to be close to commuting ones, and almost commuting
self-adjoint ones to be close to commuting self-adjoint ones as was shown in [11]
by the first author and Weihua Li. Several quantitative results (estimating δ(ǫ))
for almost commuting k-tuples of self-adjoint, unitary, and normal matrices with
respect to this norm have been obtained in [8], [24] and [6] . Much more generally,
it was proved in [11] that any polynomial equation of commuting normal variables
is stable with respect to the tracial norms on diffuse von Neumann algebras.

TRACIALLY STABLE

3

However nothing is known for polynomial relations in non-commuting variables.
In this paper we initiate a study of stability of non-commutative polynomial relations, which we translate into lifting problems for noncommutative C ∗ -algebras.
We consider C*-algebras B that have a tracial state ρ, and we measure ”almost”
and ”close” in terms of the 2-(semi)norm on B given by
kxk2 = ρ (x∗ x)

1/2

.

Thus we will address a ”Hilbert-Schmidt” type of stability that we call tracial
stability.
The original ǫ-δ-definition of norm stability can be reformulated in terms of apα
Y
proximate liftings from ultraproducts
Ai of C*-algebras Ai . Similarly tracial
i∈I

stability can be reformulated in terms of approximate liftings from tracial ultraα
Y
products
(Ai , ρi ) of tracial C*-algebras (Ai , ρi ). These ideas are made precise
i∈I

in section 2.
Suppose C is a class of unital C*-algebras that is closed under isomorphisms.
We say that a separable unital C*-algebra A is C-tracially stable if every unital
∗-homomorphism from A into a tracial ultraproduct of C ∗ -algebras from the class
C is approximately liftable.
If A is the universal C ∗ -algebra of a relation p, this definition is equivalent to
the ǫ − δ definition above with the norm being a tracial norm and C ∗ -algebras B
being from the class C.
We will be interested here in matricial tracial stability, II1 -factor tracial stability,
W ∗ -factor-tracial stability, RR0-tracial stability (that is when C is the class of real
rank zero C ∗ -algebras), and C ∗ -tracial stability (that is when C is the class of all
C ∗ -algebras).
All previous results ([11], [8], [24] and [6]) on matrices which almost commute
w.r.t. the Hilbert-Schmidt norm can be reformulated as matricial tracial stability
of separable commutative C ∗ -algebras.
In fact first results related with RR0-tracial stability appeared in [22] and [23],
where there were proved some stability results for projections almost commuting
with matrix units, and this was applied to deducing the tracial Rokhlin property for
an automorphism of a C ∗ -algebra from the Rokhlin property for the corresponding
automorphism of an associated von Neumann algebra.
In section 2 of this paper we extend substantially all the previous results about
matrices almost commuting with respect to the Hilbert-Schmidt norm.
Theorem 2.7. Suppose the class C ⊆ RR0 is closed under taking direct sums and
unital corners. If B is separable unital and C-tracially stable, and if X is a compact
metric space, then B ⊗ C (X) is C-tracially stable. In particular every separable
unital commutative C*-algebra is C-tracially stable.
Of course a natural obstruction for a C ∗ -algebra to be tracially stable can be
simply a lack of ∗-homomorphisms. Say, a C ∗ -algebra has enough almost homomorphisms to matrix algebras to separate points, but not enough actual ∗homomorphisms to matrix algebras to separate points, then of course it is not
matricially tracially stable. However it turns out to be not the only obstruction.

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DON HADWIN AND TATIANA SHULMAN

We show that a certain approximation property for traces have to hold for a C*algebra to be matricially tracially stable. For nuclear C ∗ -algebras (even tracially
nuclear, see definition in section 3) this property is also sufficient.
Theorem 3.10 Suppose A is a separable tracially nuclear C*-algebra with at least
one tracial state. The following are equivalent
(1) A is matricially tracially stable
(2) for every tracial state τ on A, there is a positive integer n0 and, for each
n ≥ n0 there is a unital ∗-homomorphism ρn : A → Mn (C) such that, for
every a ∈ A,
τ (a) = lim τn (ρn (a)) .
n→∞

(here τn is the usual tracial state on Mn (C))
(3) A is W*-factor tracially stable.
These conditions are stronger than just a property to have a separating family
of ∗-homomorphisms to matrix algebras. Namely,

Example 3.11 There exists a residually finite-dimensional (RFD) nuclear C ∗ algebra which has finite-dimensional irreducible representations of all dimensions
but is not matricially tracially stable.
However a Type I C ∗ -algebra is W ∗ -factor tracially stable (in particular, matricially tracially stable) when it has sufficiently many matrix representations (for
example, to have a 1-dimensional representation is enough):
Corollary 3.9 Suppose A is a type I separable unital C ∗ -algebra such that for all
but finitely many positive integers n it has a unital n-dimensional representation.
Then A is W ∗ -factor tracially stable. In particular, for any type I C ∗ -algebra A,
A ⊕ C is W ∗ -factor tracially stable.

In section 4 we find a close relationship between matricial tracial stability and
Voiculescu’s free entropy dimension δ0 . The following result shows that a matricially
tracially stable algebra may be forced to have a lot of non-unitarily equivalent
representations of some given dimension. Below by Rep(A, k) / ⋍ we denote the
set of all unital ∗-homomorphisms from A into Mk (C) modulo unitary equivalence.
Theorem 4.2 Suppose A =C*(x1 , . . . , xn ) is matricially tracially stable and τ is
an embeddable tracial state on A such that 1 < δ0 (x1 , . . . , xn ) . Then
lim sup
k→∞

log Card (Rep (A, k) / ⋍)
= ∞.
k2

This allows us to show that for non-(tracially) nuclear C ∗ -algebras there arise
new obstructions, other than in the (tracially) nuclear case, for being matricially
tracially stable.
Theorem 4.6 There exists an RFD C ∗ -algebra which has the approximation property from Theorem 3.10 but which is not matricially tracially stable.
In the last section we consider C ∗ -tracial stability. In contrast to RR0-tracial
stability, not all commutative C ∗ -algebras have this property. The main result
of section 5 is a characterization of C ∗ -tracial stability for separable commutative
C ∗ -algebras. For that we introduce approximately path-connected spaces. We say
that a topological space X is approximately path-connected if, for any finitely many

TRACIALLY STABLE

5

points x1 , . . . , xn , one can find arbitrarily close to them points x′1 , . . . , x′n which can
be connected by a continuous path.
Theorem 5.3 Suppose X is a compact metric space. The following are equivalent:
(1) C (X) is C ∗ -tracially stable.
(2) X is approximately path-connected.
As a final remark here it is interesting how things are reversed when norm stability is replaced by C ∗ -tracial stability. For example, being a projection is norm
stable, but not C ∗ -tracially stable. Indeed, if {fn } is any sequence of functions in
R1
C [0, 1] with trace ρ (f ) = 0 f (x) dx and 0 ≤ fn ≤ 1 and fn (x) → χ[0,1/2] (x)
a.e., then fn − fn2 2 → 0, but since 0, 1 are the only projections, the fn ’s are not
kk2 -close to a projection in C [0, 1] . On the other hand, for C ∗ -tracial stability,
the problems of normality, commuting pairs of unitaries, and commuting triples
of selfadjoint operators the answers are all affirmative, in contrast to the norm
stability.
Acknowledgements. The first author gratefully acknowledges a Collaboration
Grant from the Simons Foundation. The research of the second-named author was
supported by the Polish National Science Centre grant under the contract number DEC- 2012/06/A/ST1/00256 and from the Eric Nordgren Research Fellowship
Fund at the University of New Hampshire.
1. Preliminaries
1.1. Ultraproducts.
If a unital C*-algebra B has a tracial state ρ, we denote the 2-norm (seminorm)
given by ρ as k·k2 = k·k2,ρ defined by
1/2

kbk2 = ρ (b∗ b)

.

We also denote the GNS representation for the state ρ by πρ .
Suppose I is an infinite set and α is an ultrafilter on I. We say α is nontrivial
if there is a sequence {En } in α such that ∩n En = ∅. Suppose α is a nontrivial
ultrafilter on a set I and, for each i ∈ I, suppose Ai is a unital C*-algebra with a
α
Y
Y
tracial state ρi . The tracial ultraproduct
(Ai , ρi ) is the C*-product
Ai modulo
Y i∈I
the ideal Jα of all elements {ai } in
Ai for which

i∈I

i∈I

lim kai k22,ρi
i→α

= lim ρi (a∗i ai ) = 0.
i→α
Y
We denote the coset of an element {ai } in
Ai by {ai }α .
i∈I

Tracial ultraproducts for factor von Neumann algebras was first introduced by
S. Sakai [25] where he proved that a tracial ultraproduct of finite factor von Neumann algebras is a finite factor. More recently, it was shown in [11] that a tracial
α
Y
ultraproduct
(Ai , ρi ) of C*-algebras is always a von Neumann algebra with a
i∈I

faithful normal tracial state ρα defined by

ρa ({ai }α ) = lim ρi (ai ) .
i→α

6

DON HADWIN AND TATIANA SHULMAN

If there is no confusion, we will denote it just by ρ.
1.2. Background theorem.
The next theorem is a key tool for some of our results. It says, that two tuples that
are both close to the same tuple in a hyperfinite von Neumann algebra are nearly
unitarily equivalent to each other.
Theorem 1.1. ([16]) Suppose A = W ∗ (x1 , . . . , xs ) is a hyperfinite von Neumann
algebra with a faithful normal tracial state ρ. For every ε > 0 there is a δ > 0
and an N ∈ N such that, for every unital C*-algebra B with a factor tracial state τ
and a1 , . . . , as , b1 , . . . , bs ∈ B, if, for every ∗-monomial m (t1 , . . . , ts ) with degree at
most N ,
kτ (m (a1 , . . . as )) − ρ (m (x1 , . . . xs ))k2 < δ,
kτ (m (b1 , . . . bs )) − ρ (m (x1 , . . . xs ))k2 < δ,
then there is a unitary element u ∈ B such that
s
X
kuak u∗ − bk k2 < ε.
k=1

The following lemma is very well known.

Lemma 1.2. If γ is a ∗-homomorphism from A to a von Neumann algebra M
with a faithful normal trace τ , then γ(A)′′ ∼
= πτ ◦γ (A)′′ .
2. Tracial stability
α
Y
Suppose A is a unital C*-algebra and π : A →
(Ai , ρi ). We say that π is
i∈I

approximately liftable if there is a set E ∈ α and for every i ∈ E there is a unital
*-homomorphism πi : A → Ai such that, for every a ∈ A,
π (a) = {πi (a)}α

where we arbitrarily define πi (a) = 0 when i ∈
/ E.
It actually makes no difference how we define πi (a) when i ∈
/ E since the equivalence class {πi (a)}α does not change.
Suppose C is a class of unital C*-algebras that is closed under isomorphisms.
We say that a separable unital C*-algebra A is C-tracially stable if every unital
α
Y
∗-homomorphism from A into any tracial ultraproduct
(Ai , ρi ), with Ai ∈ C
i∈I

and ρi any trace on Ai , is approximately liftable.
Thus A is C*-tracially stable if every unital ∗-homomorphism from A into any
tracial ultraproduct is approximately liftable.
We say that A is matricially tracially stable if every unital ∗-homomorphism from
A into an ultraproduct of full matrix algebras Mn (C) is approximately liftable.
We say that A is finite-dimensionally tracially stable if every unital ∗-homomorphism
from A into a tracial ultraproduct of finite-dimensional C*-algebras is approximately liftable.
We say that A is W*-tracially stable if every unital ∗-homomorphism from A
into a tracial ultraproduct of von Neumann algebras is approximately liftable.
We say that A is W*-factor tracially stable if every unital ∗-homomorphism from
A into a tracial ultraproduct of factor von Neumann algebras is approximately
liftable.

TRACIALLY STABLE

7

Remark 2.1. One can also define tracial stability in the non-unital category. For
instance we can define the non-unital version of matricial tracial stability by saying that a separable unital C*-algebra A is matricially tracially stable if every
∗-homomorphism from A into an ultraproduct of full matrix algebras Mn (C) is
liftable. It is easy to show that A is matricially tracially stable in the non-unital
category iff Ã is matricially tracially stable in the unital category. Here Ã = A+
when A is non-unital and Ã = A⊕ C when A is unital. Thus the non-unital version
can be easily obtained from the unital one.
To see how this ultraproduct formulation represents the ε-δ definition of tracial
stability mentioned in the introduction, suppose A is the universal unital C*-algebra
generated by contractions x1 , . . . , xs subject to a relation p (x1 , . . . , xs ) = 0. Supα
Y
pose π : A →
(Ai , ρi ) is a unital ∗-homomorphism such that, for each 1 ≤ k ≤ s,
i∈I

π (xk ) = {xk (i)}α .

We then have
0 = kπ (p (x1 , . . . xs ))k2,ρα = kp (π (x1 ) , . . . π (xs ))k2,ρα = lim kp (x1 (i) , . . . , xs (i))k2,ρi .
i→α

Since α is a nontrivial ultrafilter on I, there is a decreasing sequence E1 ⊃ E2 ⊃ · · ·
in α such that ∩k∈N Ek = ∅. First suppose A is C*-tracially stable. Then, for each
positive integer m there is a number δm > 0 such that, when
kp (x1 (i) , . . . , xs (i))k2,ρi < δm
there is a unital ∗-homomorphism γm,i : A→Ai such that
max kxk (i) − γm,i (xk )k2,ρi < 1/m.

1≤k≤s

Since limi→α kp (x1 (i) , . . . , xs (i))k2,ρi = 0 we can find a decreasing sequence {An }
in α with An ⊂ En such that, for every i ∈ An
kp (x1 (i) , . . . , xs (i))k2,ρi ≤ δn .
For i ∈ An \An+1 we define πi = γn,i . We then have that {πi }i∈A1 is an approximate
lifting of π.
On the other hand, if A is not C*-tracially stable, then there is an ε > 0 such
that, for every positive integer n there is a tracial unital C*-algebra (An , ρn ) and
x1 (n) , . . . , xs (n) such that
kp (x1 (n) , . . . , xs (n))k2,ρn < 1/n,
but for every unital ∗-homomorphism γ : A → An

max kxk (n) − γ (xk )k2,ρn ≥ ε.

1≤k≤s

If we let α be any free ultrafilter on N, we have that the map
π (xk ) = {xk (n)}λ
extends to a unital ∗-homomorphism into
liftable.

α
Y

n∈N

(An , ρn ) that is not approximately

8

DON HADWIN AND TATIANA SHULMAN

The following result shows that pointwise k k2 -limits of approximately liftable
representations are approximately liftable.
Lemma 2.2. Suppose A = C ∗ ({b1 , b2 , . . .}), {(Ai , ρi ) : i ∈ I} is a family of tracial
α
Y
C*-algebras, α is a nontrivial ultrafilter on I, and π : A →
(Ai , ρi ) is a unital
∗-homomorphism such that, for each k ∈ N,

i∈I

π (bk ) = {bk (i)}α .
The following are equivalent:
(1) π is approximately liftable
(2) For every ε > 0 and every N ∈ N, there is a set E ∈ α and for every i ∈ E
there is a unital ∗-homomorphism πi : A → Ai such that, for 1 ≤ k ≤ N
and every i ∈ E,
kπi (bk ) − bk (i)k2,ρi < ε.
Proof. Obviously 1) implies 2). We need to prove the opposite implication. Since
α is nontrivial, there is a decreasing sequence {Bn } of elements of α such that
∩∞
n=1 Bn = ∅. For each n ∈ N, let N = n and ε = 1/n and let En ∈ α and, for each
i ∈ En , choose a unital ∗-homomorphism πn,i : A → Ai such that
kπn,i (bk ) − bk (i)k2,ρi < 1/n
for 1 ≤ k ≤ n. We define Fn = ∩nk=1 (Bk ∩ Ek ) and for S
i ∈ Fn+1 \Fn we define
∞
πi = πn,i . Since ∩∞
n=1 Fn = ∅, πi ’s are defined for each i ∈
n=1 Fn ∈ α. Clearly,
{πi (bk )}α = {bk (i)}α = π (bk )

for k = 1, 2, . . .. Since the set S of all a ∈ A such that {πi (a)}α = π (a) is a unital
C*-algebra, we see that S = A and that π is approximately liftable.

Although we consider all tracial ultraproducts, however when the algebra A
is separable, we need to consider only ultraproducts over N with respect to one
non-trivial ultrafilter.
Lemma 2.3. Suppose A = C ∗ (x1 , x2 , . . .) is a separable unital C*-algebra, C is a
class of unital C*-algebras closed under isomorphism and α is a nontrivial ultrafilter
on N. The following are equivalent:
(1) A is C-tracially stable
(2) If {(Bn , γn )} is a sequence of tracial C*-algebras in C and π : A →

α
Y

(Bi , γi )

i∈I

is a unital ∗-homomorphism, then π is approximately liftable.
(3) For every ε > 0, for every positive integer s, and for every tracial state ρ
on A, there is a positive integer N such that if B ∈ C and γ is a tracial
state on B and b1 , . . . , bN ∈ B such that kbk k ≤ 1 + kxk k for 1 ≤ k ≤ N
and
1
|ρ (m (x1 , . . . , xN )) − γ (m (b1 , . . . , bN ))| <
N

TRACIALLY STABLE

9

for all ∗-monomials m (t1 , . . . , tN ) with degree at most N , then there is a
unital ∗-homomorphism π : A → B such that
s
X
kπ (xk ) − bk k2,γ < ε.
k=1

Proof. (1) ⇒ (2). This is obvious.
(2) ⇒ (3). Assume (3) is false. Then there is an ε > 0, an integer s ∈ N and a
tracial state ρ on A and, for each N ∈ N, there is a BN ∈ C with a tracial state γN
and {bN,1, bN,2 , . . . , bN,N } ⊂ BN such that
kbN,k k ≤ kxk k + 1

for 1 ≤ k ≤ N < ∞, and

1
N
for all ∗-monomials m (t1 , . . . , tN ) with degree at most N , such that, for each N
there is no ∗-homomorphism π : A → BN such that
s
X
kπ (xk ) − bN,k k2,γN < ε.
|ρ (m (x1 , . . . , xN )) − γN (m (bN,1 , . . . , bN,N ))| <

k=1

For each 1 ≤ N < k let bN,k = 0 ∈ BN . For 1 ≤ k < ∞, let
bk = {bN,k }α ∈

Let γ be the limit trace on

α
Y

N ∈N

m (t1 , . . . , tk ) we have
(2.1)

α
Y

(BN , γN ) .

N ∈N

(BN , γN ). It follows that for every ∗-monomial

ρ (m (x1 , . . . , xk )) = γ (m (b1 , . . . , bk )) .

Define a unital ∗-homomorphism π : A →

α
Y

(BN , γN ) by

N ∈N

π(xi ) = bi .
Since γ is faithful, it follows from (2.1) that π is well-defined and ρ = γ ◦ π. By
(2), π is approximately liftable so there is an E ∈ α and, for each N ∈ E a unital
∗-homomorphism πN : A → BN such that, for 1 ≤ k < ∞
π (xk ) = {πN (xk )}α .

It follows that
lim

N →α

s
X

k=1

which means for some N ∈ N,
s
X
k=1

kbN,k − πN (xk )k2,γN = 0,

kbN,k − πN (xk )k2,γN < ε.

This contradiction implies (3) must be true.
(3) ⇒ (1). Suppose (3) is true and {(Bi , γi ) : i ∈ I} is a collection of tracial
unital C*-algebras with each Bi in C and suppose β is a nontrivial ultrafilter on I

10

and π : A →

DON HADWIN AND TATIANA SHULMAN
β
Y
i∈I

(Bi , γi ). Let γ be the limit trace along β and define ρ = γ ◦ π. If

k ∈ N, we let ε = k1 and s = k we can choose Nk as in (2). Since β is nontrivial
there is a decreasing sequence {Ej } in β whose intersection is empty. For each
k ∈ N write
π (xk ) = {bi,k }β
with kbi,k k ≤ kxk k for every i ∈ I. It follows, for every ∗-monomial m (t1 , . . . , tk )
that
lim γi (m (bi,1 , . . . , bi,k )) = ρ (m (x1 , . . . , xk )) .
i→β

Thus the set Fk consisting of all i ∈ I such that

1
Nk
for all ∗-monomials m with degree at most Nk must be in β. For each k ∈ N let
|ρ (m (x1 , . . . , xNk )) − γi (m (bi,1 , . . . , bi,Nk ))| <
Wk = Ek ∩ F1 ∩ · · · ∩ Fk .
For each i ∈ Wk \Wk+1 there is a unital ∗-homomorphism πi : A → Bi such that
k
X
j=1

kπi (xj ) − bi,j k2,γi <

1
.
k

n
o
It is clear for every k ∈ N that π (xk ) = {πi (xk )}β . Since a ∈ A : π (a) = {πi (a)}β
is a unital C*-subalgebra oof A, containing all x1 , x2 , . . ., we conclude that A =
n
a ∈ A : π (a) = {πi (a)}β and thus π is approximately liftable.

It is clear that C*-tracially stable implies W*-tracially stable implies factor tracially stable implies matricially tracially stable. Here are some slightly more subtle
relationships.
Lemma 2.4. Suppose A is a separable unital C*-algebra. Then A is matricially
tracially stable if and only if every unital ∗-homomorphism from A into a tracial ultraproduct of hyperfinite factors is approximately liftable. Also A is finitedimensionally tracially stable if and only if every unital ∗-homomorphism from A
into a tracial ultraproduct of hyperfinite von Neumann algebras is approximately
liftable.
Proof. We prove the first statement; the second follows in a similar fashion. The ”if”
part is clear since Mn (C) is a hyperfinite factor. Suppose A is matricially tracially
stable, and suppose {(Mi , τi ) : i ∈ I} is a family with each Mi a hyperfinite factor
and τi is the unique normal faithful tracial state on Mi . Suppose also that α is
α
Y
a nontrivial ultrafilter on I and π : A →
(Mi , τi ) is a unital ∗-homomorphism.
i∈I
α
Y

If E = {i ∈ I : dim Mi < ∞} ∈ α, then

i∈I

(Mi , τi ) =

α
Y

(Mi , τi ) is a tracial

i∈E

product of matrix algebras and π is approximately liftable. If I\E ∈ α, we can
assume that each Mi is a hyperfinite II1 -factor. Since α is nontrivial, there is a

TRACIALLY STABLE

11

decreasing sequence {En } in α such that E1 = I and ∩∞
n=1 En = ∅. Suppose {an }
is a dense sequence in A and, for each n ∈ N write
π (an ) = {an (i)}α

with kan (i)k ≤ kan k, for all n and i. If n ∈ N and i ∈ En \En+1 , we can find a
unital subalgebra Bi ⊂ Mi such that Bi is isomorphic to Mki (C) for some ki ∈ N
an such that there is a sequence {bm (i)} in Bi with kbm (i)k ≤ kam k for each m
and such that
1
kam (i) − bm (i)k2 ≤
n
for 1 ≤ m ≤ n. It follows that
α
Y
π (an ) = {an (i)}α = {bn (i)}α ∈
(Bi , τi ) .
i∈I

α
Y

It follows that π : A→
approximately liftable.

i∈I

(Bi , τi ). Since A is matricially tracially stable, π is


Recall that a C ∗ -algebra has real rank zero (RR0) if each its self-adjoint element
can be approximated by self-adjoint elements with finite spectrum.
Theorem 2.5. Suppose every algebra in the class C is RR0. Then every separable
unital commutative C*-algebra is C-tracially stable.
Proof. In [14] the authors proved that if J is a norm-closed two-sided ideal of a
real-rank zero C*-algebra M such that M/J is an AW*-algebra, then for any
commutative separable unital C*-algebra A, any ∗-homomorphism π : A → M/J ,
lifts to a unital ∗-homomorphism ρ : A → M. Since the direct product of realrank zero C*-algebras is a real-rank zero C*-algebra and a tracial ultraproduct of
C ∗ -algebras is a von Neumann algebra, the statement follows.

Proposition 2.6. Suppose C is a class such that C ⊕ B ∈ C whenever B ∈ C. Every
C-tracially stable separable unital C ∗ -algebra A with at least one representation
into a tracial ultraproduct of C ∗ -algebras in C must have a one-dimensional unital
representation.
Proof. Suppose π : A →

α
Y
i∈I

(Bi , γi ) is a unital ∗-homomorphism where each Bi ∈ C

and α is a nontrivial ultrafilter on I. Let Di = C ⊕ Bi for each i ∈ I. Since α is
nontrivial there is a decreasing sequence I = E1 ⊇ E2 ⊇ · · · in α whose intersection
is ∅. For i ∈ En \En+1 define a trace ρi on Di by


1
1
γi (B) .
ρi (λ ⊕ B) = λ + 1 −
n
n

Then

α
Y

i∈N

The rest is easy.

(Di , ρi ) =

α
Y

(Bi , γi ) .

i∈N



12

DON HADWIN AND TATIANA SHULMAN

Theorem 2.7. Suppose the class C ⊆ RR0 is closed under taking direct sums and
unital corners. If B is separable unital and C-tracially stable, and if X is a compact
metric space, then B ⊗ C (X) is C-tracially stable.
Proof. Suppose {(An , ρn )} is a sequence of C ∗ -algebras in the class C, α is a
α
Y
(An , ρn ) is a unital ∗non-trivial ultrafilter on N, and f : B ⊗ C(X) →
n∈N

homomorphism. By Lemma 2.2, it will be enough for any x1 , . . . , xk ∈ B ⊗ C(X),
ǫ > 0 to find unital ∗-homomorphisms f˜n : B ⊗ C(X) → An , such that
k{f˜n (xj )}α − f (xj )k2,ρ < 3ǫ,

j ≤ k.
(j)
(j)
There are bi ∈ B, φi ∈ C(X), N1 , . . . , Nk such that

j ≤ k. Let

Nj
X

kxj −

(2.2)

M=

i=1

(j)

(j)

bi ⊗ φi k ≤ ǫ,
(j)

max

1≤i≤Nj ,1≤j≤k

kf (bi ⊗ 1)k, N = max Nj .
j≤k

Since f (1 ⊗ C(X)) is commutative, it is isomorphic to some C(Ω). We can choose
a disjoint collection {E1 , . . . , Em } of Borel subsets of Ω whose union is Ω and
ω1 ∈ E1 , . . . , ωm ∈ Em such that
(2.3)

(j)

kf (1 ⊗ φi ) −

m
X
l=1

(j)

f (1 ⊗ φi )(wl )χEl k2,ρ <

ε
,
NM

for j ≤ k and i ≤ Nj . Since pairwisely orthogonal projections generate a commutative C ∗ -algebra, by Theorem 2.5 for each χEl , l ≤ m, we can find projections
Pl,n ∈ An such that χEl = {Pl,n }α and P1,n , . . . , Pm,n are pairwisely orthogonal,
for each n. Since the subalgebra f (1 ⊗ C(X)) is central in the algebra f (B ⊗ C(X)),
each χEl commutes with f (B ⊗ C(X)). Hence
!
!
α
m
α
m
Y
M
Y
X
(Pi,n An Pi,n , ρn ) .
Pi,n An Pi,n , ρn =
f (B ⊗ C(X)) ⊆
n∈N

i=1

i=1

n∈N

Since B is C-tracially stable, we can find unital ∗-homomorphisms ψn,i : B →
Pi,n An Pi,n , such that
(2.4)

{ψn,i (b)}α = pi ◦ f (b ⊗ 1),

for any b ∈ B. Here pi is the projection onto the i-th summand in
Hence for ψn = ⊕m
i=1 ψn,i : B →
(2.5)

Pm

i=1

Pi,n An Pi,n , we have

{ψn (b)}α = f (b ⊗ 1),

for any b ∈ B. Define a ∗-homomorphism δn : C(X) →
(2.6)

δn (φ) =

m
X
i=1

Pm

f (1 ⊗ φ)(wi )Pi,n .

i=1

Lm

i=1

α
Y

n∈N

!

(Pi,n An Pi,n , ρn ) .

Pi,n An Pi,n by

TRACIALLY STABLE

13

Define f˜n : B ⊗ C(X) → An by

f˜n (b ⊗ φ) = ψn (b)δn (φ).

Since δn (C(X)) and ψn (B) commute, f˜n is a ∗-homomorphism. By (2.2), (2.3),
(2.5), (2.6), for each j ≤ k we have
(2.7)
{f˜n (xj )}α − f (xj )

2,ρ

≤ 2ǫ+

Nj

= 2ǫ +

X
i=1

(j)

X
i=1





f˜n 
(j)

Nj
X
i=1




Nj

X
(j)
(j)
(j)
(j)
bi ⊗ φi 
bi ⊗ φi 
−f

α

(j)

f (bi ⊗ 1)

m
X
l=1

i=1

Nj

{ψn (bi )}α {δn (φi }α −

Nj

= 2ǫ +




X
i=1

(j)

(j)

f (bi ⊗ 1)f (1 ⊗ φi )

2,ρ
(j)

(j)

!

f (1 ⊗ φi )(wl )χEl − f (1 ⊗ φi )

2,ρ

ǫ
≤ 2ǫ + Nj M
≤ 3ǫ.
NM


Proposition 2.8. Suppose the class C ⊆ RR0 is closed under taking finite direct
sums. Then the class of C-tracially stable C*-algebras is closed under taking finite
direct sums.
Proof. We will show it for direct sums with 2 summands, and the general case
α
Y
is similar. Let A, B be C-tracially stable, π : A ⊕ B →
(Dn , ρn ) a unital ∗n∈N

homomorphism and all Dn ∈ C. Let

p = π(1A ), q = π(1B ).
Then p + q = 1. By Theorem 2.5 we can find projections Pn , Qn ∈ Dn such that
{Pn }α = p, {Qn }α = q, Pn + Qn = 1Dn .
Then
π(A) ⊂

α
Y

n∈N

(Pn Dn Pn , ρn ) , π(B) ⊂

α
Y

n∈N

(Qn Dn Qn , ρn ) .

Since A and B are C-tracially
stable, there are unital ∗-homomorphisms φ : A →
Q
Q
Pn Dn Pn and ψ : B → Qn Dn Qn such that
{φ(a)n }α = π(a), {ψ(b)n }α = π(b),

for all a ∈ A, b ∈ B. Then π̃, defined by π̃(a, b) = φ(a) + ψ(b), is an approximate
lift of π.

The following proposition is obvious.
Proposition 2.9. The class of C-tracially stable C*-algebras is closed under finite
free products in the unital category for any C.

2,ρ

14

DON HADWIN AND TATIANA SHULMAN

Remark. The preceding proposition cannot be extended to countable free products
in the unital category. For example, Bn = Mn (C) ⊕ Mn+1 (C) is matricially
tracially stable for each positive integer n by Lemma 3.5 and Corollary 3.9 in the
next section. Let α be a nontrivial ultrafilter containing the set {k! | k ∈ N}. Then,
α
Y
(Mi , τi ) and hence there is a unital
for each n ∈ N, Bn embeds unitally into
n∈N

∗-homomorphism from the unital free product ∗n∈N Bn into

α
Y

(Mi , τi ). However,

n∈N

∗n∈N Bn has no finite-dimensional representations. Hence ∗n∈N Bn is not matricially
tracially stable. However, if An is a separable tracially-C-stable C ∗ -algebra for each
n ∈ N, then the unitization of the free product of the An ’s in the nonunital category
is tracially-C-stable.
3. Matricial tracial stability
Notation: τn usual trace on Mn (C). The tracial ultraproduct is denoted
α
Y

(Mn (C) , τn ) .

n∈N

3.1. Embeddable traces. Definition Suppose A is a unital separable C*-algebra
and τ is a tracial state on A. We say τ is embeddable if there is a non-trivial
ultrafilter α on N and a unital ∗-homomorphism
π:A→

α
Y

(Mn (C) , τn )

n∈N

such that
τα ◦ π = τ.

If Connes’ embedding theorem holds, then every tracial state is embeddable.
A tracial state τ is finite-dimensional if there is a finite-dimensional C*-algebra
B with a tracial state τB and a unital *-homomorphism
π:A→B
such that
τB ◦ π = τ.

A tracial state τ is called matricial if there is a positive integer n and a unital
*-homomorphism
π : A → Mn (C)

such that

τ = τn ◦ π.
We say that a matricial tracial state is a factor matricial state if the π above
can be chosen to be surjective.

TRACIALLY STABLE

15

Lemma 3.1. A C ∗ -algebra A = C ∗ (D) with a tracial state τ admits a tracepreserving ∗-homomorphism into an ultraproduct of matrix algebras if and only if,
for every ǫ > 0, every finite tuple (a1 , ..., an ) of elements in D, and every finite set
F of ∗-monomials, there is a positive integer k and a tuple (A1 , ..., An ) of k × k
matrices such that
(1) kAj k ≤ kaj k + 1 for 1 ≤ j ≤ n,
(2) |τ (m(a1 , ..., an )) − τk (m(A1 , ..., An )| < ǫ for every m ∈ F .
Proof. The ”only if” part is obvious. For the other direction, let Λ be the set of all
triples λ = (ελ , Nλ , Eλ ) with ελ > 0, Eλ a finite subset of D, Nλ a positive integer,
partially ordered by(≥, ≤, ⊆) . By the hypothesis, for each λ ∈ Λ there is a positive
integer kλ and there is a function fλ : D → Mkλ with fλ (1) = 1 and such that,
i. kfλ (a)k ≤ kak + 1 for every a ∈ A,
ii. For all monomials m with degree at most Nλ , and all a1 , . . . , an ∈ Eλ ,
|τ (m(a1 , ..., an )) − τk (m(fλ (a1 ), ..., fλ (an ))| < ελ .
(We can define f (a) = 0 on A\Eλ .)
Y
Define F : D →
(Mkλ , τkλ ) by F (a) (λ) = fλ (a) . Let α be an ultrafilter on
λ∈Λ

Λ containing {λ : λ ≥ λ0 } for each λ0 ∈ Λ. Define
ρ:A→

α
Y

(Mkλ , τkλ )

λ∈Λ

by ρ (a) = [F (a)]α . Let A0 denote the unital ∗-algebra generated by D. It follows
from the definition of the fλ ’s that ρ extends to a unital ∗-homomorphism π0 :
α
Y
A0 →
(Mkλ , τkλ ) with τα ◦ π0 = τ . We can now uniquely extend, by continuity,
λ∈Λ

π0 to a unital ∗-homomorphism on A such that τ = τα ◦ π.



Corollary 3.2. If a separable
C ∗ -algebra with a tracial state admits a trace-preserving
Qα
∗-homomorphism into n∈N (Mn , τn ), for some non-trivial ultrafilter α, then it adQβ
mits a trace-preserving ∗-homomorphism into n∈N (Mn , τn ), for any non-trivial
ultrafilter β.
Proof.
Suppose A = C ∗ (a1 , a2 , . . .) with a tracial state τ , and suppose ρ : A →
Qα
n∈N (Mn , τn ) is a unital ∗-homomorphism such that τα ◦ ρ = τ . We can assume
kan k = 1 for all n, and we can let ρ (an ) = {bn,κ }α with kbn,κ k ≤ kan k = 1 for
all k ≥ 1. For each positive integer s there is a positive ks such that, for every
∗-monomial m with degree at most s,

1
.
2s
Suppose t > 2sks . Then by dividing t by ks we get t = ks q + r with 0 ≤ r < ks .
(q)
We define cj = bj,ks ⊕ 0r (where b(q) is a direct sum of q copies of b and 0r is an
r × r zero matrix. A simple computation gives, for any monomial m of degree at
most s that

r
.
τt (m (c1 , . . . , cs )) = τks (m (b1,ks , . . . , bs,ks )) 1 −
t
|τ (m (a1 , . . . , as )) − τks (m (b1,ks , . . . , bs,ks ))| <

16

DON HADWIN AND TATIANA SHULMAN

Since r/t < 1/2s and |τks (m (b1,ks , . . . , bs,ks ))| ≤ 1, we see that

1
.
s
Hence we get representatives like (c1 , . . . , cs ) for
Q all t > 2sks . From this we see
that there is a function F : {a1 , a2 , . . .} → n∈N (Mn , τn ) so that if f (an ) =
(dn,1 , dn,2 , . . .), we have
|τ (m (a1 , . . . , as )) − τt (m (c1 , . . . , cs ))| <

lim τn (m (dn,1 , dn,2 , . . .)) = τ (m (a1 , a2 , . . .))

n→∞

for any ∗-monomial m. Hence, for any free ultrafilter β,
lim τn (m (dn,1 , dn,2 , . . .)) = τ (m (a1 , a2 , . . .)) .

n→β

Thus the map π0 : {a1 , a2 , . . .} →

Qβ

n∈N (Mn , τn )

defined by

π0 (ak ) = F (ak )β
Qβ
extends to a unital ∗-homomorphism π : A → n∈N (Mn , τn ) such that τβ ◦ π =
τ.

It follows easily from Lemma 3.1 that the set of embeddable tracial states on
A is ∗-weak-compact. It is not hard to show that it is also convex, and contains
the finite-dimensional tracial states. The set of finite-dimensional tracial states is
the convex hull of the set of factor matricial states. Moreover, the set of finitedimensional tracial states is contained in the weak*-closure of the set of matricial
tracial states.
3.2. Matricial tracial stability – necessary conditions. Let A be a unital
C ∗ -algebra and let Jet (A) be the largest ideal of A that is annihilated by every
embeddable trace on A. If A has no embeddable traces, then A/Jet = {0} . More
generally, the embeddable tracial states of A/Jet separate the points of A/Jet , and
if A is separable, A/Jet always has a faithful tracial embeddable state. Clearly, A
is matricially tracially stable iff A/Jet is matricially tracially stable. C ∗ -algebras
without any embeddable tracial states are automatically matricially tracially stable.
Theorem 3.3. Suppose A is a separable unital matricially tracially stable C ∗ algebra with at least one embeddable trace. Then
(1) For all but finitely many positive integers n there is a unital representation
πn : A → Mn (C)
(2) The set of embeddable tracial states on A is the ∗-weak-closed convex hull
of the set of matricial tracial states.
Proof. (1) . Suppose, via contradiction, that the set E of positive integers n for
which there is no n-dimensional representation of A is infinite. Let α be a nonα
Y
trivial ultrafilter on N with E ∈ α. Let τ be the limit trace on
(Mn (C) , τn ).
n∈N

Then, if ρ is an embeddable trace on A, by Corollary 3.2 there is a representation
α
Y
π:A→
(Mn (C) , τn ) such that τ ◦ π = ρ. Since there A is matricially stable,
n∈N

the set N\E ∈ α, which is a contradiction.

TRACIALLY STABLE

17

(2) . If ρ is an embeddable trace and α is a non-trivial ultrafilter on N and
α
Y
π:A→
(Mn (C) , τn ) is a representation such that τ ◦π = ρ, then the matricial
n∈N

tracial stability of A implies there is a set F ∈ α and, for each n ∈ F , there is a
representation πn : A → Mn (C) such that the ∗-weak-limit along α of the finitedimensional traces τn ◦ πn .

Corollary 3.4. Suppose A is separable and matricially tracially stable. Then
A/Jet is RFD. If the collection of embeddable traces on A separates points of A,
then A is RFD.

The set of positive integers n for which there is an n-dimensional representation is
an additive semigroup (direct sums) that is generated by the k ∈ N for which there
is a k-dimensional irreducible representation. The following simple lemma, which
yields a reformulation of the first condition in Theorem 3.3, should be well-known.
Lemma 3.5. Suppose n1 < n2 < · · · < ns . The following are equivalent:

(1) GCD (n1 , . . . , nk ) = 1.
(2) There is a positive integer N such that every integer n ≥ N can be written
as
s
X
n=
ak nk
k=1

with a1 , . . . ak nonnegative integers.
Ps
Ps
Proof. (2) =⇒ (1) . If we write N = k=1 ak nk and N + 1 = k=1 bk nk , then
s
X

k=1

(bk − ak ) nk = 1,

which implies (1) .
(1) =⇒ (2) . If GCD (n1 , . . . , nk ) = 1, there are integers s1 , . . . , sk such that
s1 n1 + · · · + sk nk = 1.
Let m = 1 + n1 max (|s1 | , . . . , |sk |). Let N = mn1 + · · · + mnk . Suppose n ≥ N .
Using the division algorithm we can find integers q ≥ 0 and r with 0 ≤ r < n1 such
that
n − N = n1 q + r.

Thus

n = mn1 + · · · + mnk + qn1 + r (s1 n1 + · · · + sk nk ) =
(m + rs1 + q) n1 +

k
X

(m + rsj ) nj .

j=2

However, m + rsj ≥ m − n1 |sj | ≥ 1 for 1 ≤ j ≤ k.



Corollary 3.6. A unital C*-algebra satisfies condition (1) in Theorem 3.3 if and
only if {n ∈ N : A has an irreducible n-dimensional representation} has greatest common divisor equal to 1.

18

DON HADWIN AND TATIANA SHULMAN

Lemma 3.7. A separable unital C ∗ -algebra A satisfies statements (1) and (2)
in Theorem 3.3 if and only if, for every embeddable tracial state τ on A, there
is a positive integer n0 and, for each n ≥ n0 there is a unital ∗-homomorphism
ρn : A → Mn (C) such that, for every a ∈ A,
τ (a) = lim τn (ρn (a))
n→∞

Proof. The ”if” part is clear. Suppose statements (1) and (2) in Theorem 3.3 are
true. Let {a1 , a2 , . . .} be a norm dense subset if the unit ball of A. Let τ be an
embeddable tracial state on A, and suppose N is a positive integer. Then there is
a convex combination σ, with rational coefficients, of matricial states, and hence
even of factor matricial states, such that
|τ (aj ) − σ (aj )| < 1/2N

for 1 ≤ j ≤ N . Thus there exist a positive integer m and positive integers s1 , . . . , sν
with s1 +· · ·+sν = m and positive integers t1 , . . . , tν with surjective representations
πi : A → Mti (C) such that
ν
X
si
σ=
τti ◦ πi .
m
i=1

Let M ∈ N be such stii M is an integer, for all i ≤ ν. If ρ(k) denotes a direct sum of
k copies of a representation ρ, then let
s

( t i M)

π = ⊕i πi

i

: A → MmM (C) .

It is easy to check that σ = τmM ◦ π. Since by the assumption the statement
(1) in Theorem 3.3 holds, there is n0 such that for any n ≥ n0 there is a unital
representation into Mn (C). Since for a positive integer q we have σ = τqmM ◦ π (q) ,
we can assume that mM > n0 . Suppose n > 2mM . We can write n = amM + b
with a ≥ 2 and 0 ≤ b < mM . Since n0 ≤ mM + b, there is a unital representation
ρ : A → MmM+b (C). Let
Then

πN,n = π (a−1) ⊕ ρ : A → Mn (C) .

(a − 1)mM τMm ◦ π(ak ) + (mM + b)τmM+b ◦ ρ(ak )
=
n
mM + b
(a − 1)mM
σ(ak ) +
τm+b ◦ ρ(ak ) =
n
n
mM + b
mM + b
)σ(ak ) +
τm+b ◦ ρ(ak ).
(1 −
n
n
1
< 4N
. Then for
There is a positive integer kN such that if n ≥ kN , then mM+b
n
all n ≥ kN
τn ◦ πN,n (ak ) =

for 1 ≤ k ≤ N and hence

|τn ◦ πN,n (ak ) − σ (ak )| < 1/2N
|τ (ak ) − τn ◦ πN,n (ak )| < 1/N

for 1 ≤ k ≤ N . We can easily arrange k1 < k2 < · · · and we can define
ρn = πN,n if kN ≤ n < kN +1 .



TRACIALLY STABLE

19

3.3. Matricial tracial stability for tracially nuclear C*-algebras. Recall
that a unital C ∗ -algebra A is nuclear if and only if, for every Hilbert space H and
every unital ∗-homomorphism π : A → B (H) the von Neumann algebra π (A)′′ is
hyperfinite. The algebra A is tracially nuclear [11] if, for every tracial state τ on
′′
A, the algebra πτ (A) is hyperfinite, where πτ is the GNS representation for τ . It
is easy to show that if A is tracially nuclear, then every trace on A is embeddable.
Theorem 3.8. Suppose A is a unital separable tracially nuclear C*-algebra with
at least one tracial state. The following are equivalent
(1) A is matricially tracially stable
(2) A satisfies the conditions (1) and (2) in Theorem 3.3.
(3) A is W*-factor tracially stable.
Proof. 1) ⇒ 2) This follows from Theorem 3.3.
2) ⇒ 3). Suppose A = C ∗ (x1 , x2 , . . .), with all xi ’s being contractions, suppose
α
Y
(Mn , ρn ), where (Mn , ρn )
α is a free ultrafilter on N, and suppose σ : A →
n∈N

is a factor von Neumann algebra with trace ρn . Denote by ρα the limit trace on
α
α
Y
Y
(Mn , ρn ). Since ρα a faithful trace on
(Mn , ρn ) and since A is tracially

n∈N

n∈N

′′

nuclear, it follows that σ (A) is hyperfinite.
Since α is an ultrafilter, either {n ∈ N : dim Mn < ∞} ∈ α or {n ∈ N : dim Mn = ∞} ∈
α. We first consider the latter case. In this case we can assume that every Mn is
a II1 -factor. In this case, it follows from [11] that, for each n ∈ N there is a unital
′′
′′
∗-homomorphism γn : σ (A) → Mn such that, for every b ∈ σ (A)
lim ρn (γn (b)) = ρα (b) ,

n→α

and
b = {γn (b)}α .

Hence {γn ◦ σ}n∈N is an approximate lifting of σ.
Next suppose E := {n ∈ N : dim Mn < ∞} ∈ α and Mn = Mkn (C) and ρn =
τkn for each n ∈ E.
By assumption and Lemma 3.7, there is an n0 ∈ N and, for every n ≥ n0 there
is a unital ∗-homomorphism πn : A → Mn (C) such that, for every a ∈ A,
τα (a) = lim τn (πn (a)) .
k→∞

For each k ∈ N, write σ (xk ) = {xk (n)}α . For any s ∈ N and any ∗-monomial
m (t1 , . . . , ts ) we have
lim τn (m (x1 (n) , . . . , xs (n))) = τα (m (σ (x1 ) , . . . , σ (xs )))

n→α

and
lim τn (m (πn (x1 ) , . . . , πn (xs ))) = τα (m (σ (x1 ) , . . . , σ (xs ))) .

n→∞
′′

Since σ (A) is hyperfinite, it follows from Connes’ theorem that, for each s ∈ N,
′′
′′
σ (C ∗ (x1 , . . . , xs )) ⊂ σ (A) is also hyperfinite. It now follows from Theorem 1.1
that there is a sequence of unitaries Un ∈ Mkn (C) such that, for every s ∈ N,
lim kxs (n) − Un∗ πn (xs ) Un k2 = 0.

n→α

20

DON HADWIN AND TATIANA SHULMAN

Hence {Un∗ πn (·) Un } is an approximate lifting of σ.
3) ⇒ 1) is obvious.



Recall that a C ∗ -algebra A is called GCR (or type I ) if for any its irreducible
representation π : A → B(H), π(A) contains all the compact operators.
Corollary 3.9. Suppose A is a separable GCR unital C ∗ -algebra satisfying condition (1) in Theorem 3.3. Then A is W ∗ -factor tracially stable.

Proof. The extreme points of the tracial states are the factor tracial states ([12]).
A factor representation of a GCR C ∗ -algebra must yield a factor von Neumann
algebra of type I, which must be isomorphic to some B(H). If it has a trace, then H
must be finite-dimensional. Thus the factorial tracial states are finite-dimensional,
and the Krein-Milman theorem gives that all tracial states are in the weak*-closed
convex hull of the finite-dimensional states. Thus the condition (2) in Theorem 3.3
holds.

The next statement follows from Theorem 3.8, Theorem 3.3 and Lemma 3.7.
Theorem 3.10. Suppose A is a separable tracially nuclear C*-algebra with at least
one tracial state. The following are equivalent
(1) A is matricially tracially stable
(2) for every tracial state τ on A, there is a positive integer n0 and, for each
n ≥ n0 there is a unital ∗-homomorphism ρn : A → Mn (C) such that, for
every a ∈ A,
τ (a) = lim τn (ρn (a)) .
n→∞

(here τn is the usual tracial state on Mn (C))
(3) A is W*-factor tracially stable.
Below we give an example of an RFD nuclear C ∗ -algebra which has finitedimensional irreducible representations of all dimensions but is not matricially tracially stable.
Example 3.11. Suppose 0 < θ1 < θ2 <, 1 are irrational and {1, θ1 , θ2 } is linearly
independent over Q. Let λk = e2πiθk for k = 1, 2. Let Aθk be the irrational rotation
algebra generated by unitaries Uk , Vk satisfying Uk Vk = λk Vk Uk . We know that
each Aθk is simple nuclear and has a unique tracial state ρk . Let U = U1 ⊕ U2 and
V = V1 ⊕ V2 . Since U V U ∗ V ∗ = λ1 ⊕ λ2 , we see that 1 ⊕ 0 ∈ C ∗ (U, V ) , so
C ∗ (U, V ) = Aθ1 ⊕ Aθ2 .

By the linear independence assumption, there is an increasing sequence {nk } of
positive integers such that
(3.1)

λn1 k → 1 and λn2 k → 1

as k → ∞. We can assume also that

(3.2)

n1 = 2 and n2 = 3.

For each positive integer n and each λ ∈ C, let {e1 , . . . , en } be the standard
orthonormal basis for Cn , let Un,λ and Vn be the matrices defined by
Un,λ ej = λj−1 ej

TRACIALLY STABLE

21

and
Vn ej = ej+1 for 1 ≤ j < n; Vn en = e1 .

It follows from (3.1) that

kUnk ,λs Vnk − λs Vnk Unk ,λs k → 0
as k → ∞ for s = 1, 2. The simplicity of each Aθs implies that, for every ∗polynomial p and s = 1, 2, we have
kp (Unk ,λs , Vnk ) − p (Us , Vs )k → 0.

(3.3)

The uniqueness of the trace on each Aθs implies that, for every ∗-polynomial p
and s = 1, 2, we have
lim τnk (p (Unk ,λs , Vnk )) = ρs (p (Us , Vs ))
α

for any non-trivial ultrafilter α, and hence
(3.4)

τnk (p (Unk ,λs , Vnk )) → ρs (p (Us , Vs ))

as k → ∞. P
P⊕
(k−1)
(k−1)
⊕ Vnk , and let
Let Û = ⊕
k∈N Vnk
k∈N Unk ,λ1 ⊕ Unk ,λ2 and V̂ =


A = C ∗ Û , V̂ + K,
where

K=

X⊕

k∈N

Mknk (C).

It follows that K is nuclear and by (3.3) A/K is isomorphic to C ∗ (U, V ) = Aθ1 ⊕
Aθ2 , which is also nuclear. Hence, A is nuclear. Moreover, the only finite-dimensional
irreducible representations of A are the coordinate representations πk for k ∈ N.
However, by (3.4), for every ∗-polynomial p, we have


 

1
k−1
τnk (p(Unk ,λ1 , Vnk )) + τnk (p(Unk ,λ2 , Vnk ))
= lim
lim τknk (πk p Û , V̂
k→∞
k→∞
k
k
= lim τnk (p(Unk ,λ1 , Vnk )) = ρ1 (p (U1 , V1 )) .
k→∞


Hence the trace ρ on A that annihilates K and sends p Û , V̂ to ρ2 (p (U2 , V2 )) is
embeddable and cannot be approximated by finite-dimensional tracial states. Thus
A is nuclear and RFD and by (3.2) and Corollary 3.6 satisfies condition (1) in
Theorem 3.3, but does not satisfy condition (2), hence is not matricially tracially
stable.
4. Matricial tracial stability and free entropy dimension
There is a close relationship between matricial stability and D. Voiculescu’s free
entropy dimension [27], [28] (via the free orbit dimension in Hadwin-Shen [13]). D.
Voiculescu defined his free entropy dimension δ0 ([27], [28]), and he applied it to
show the existence of a II1 factor von Neumann algebra without a Cartan MASA,
solving a longstanding problem.
Suppose A is a unital C*-algebra with a tracial state τ . Suppose x1 , . . . , xn are
elements in A, ε > 0, R > max1≤j≤n kxj k and N, k ∈ N. Voiculescu [27] defines

22

DON HADWIN AND TATIANA SHULMAN

ΓR,τ (x1 , . . . , xn ; N, k, ε) to be the set of all n-tuples (A1 , . . . , An ) of matrices in
Mk (C) with norm at most R such that
|τ (m (x1 , . . . , xn )) − τn (m (A1 , . . . , An ))| < ε
for all ∗-monomials m (t1 , . . . , tn ) with degree at most N.
Connes’ embedding problem is equivalent to the assertion that, for every unital
C*-algebra B with a tracial state τ , every n-tuple (x1 , . . . , xn ) of selfadjoint contractions in B, for any N, for any R > max1≤j≤n kxj k and for every ε > 0 there is
a positive integer k such that ΓR,τ (x1 , . . . , xn ; N, k, ε) 6= ∅.
n
n
Let Mk (C) = ((A1 , . . . , An ) : A1 , . . . , An ∈ Mk (C)) and define kk2 on Mk (C)
by
n
X

2
τk A∗j Aj .
k(A1 , . . . , An )k2 =
j=1

n

If A = (A1 , . . . , An ) ∈ Mk (C) and U is unitary, we define
U ∗ AU = (U ∗ A1 U, . . . , U ∗ An U ) .

If ω > 0, we define the ω-orbit ball of A = (A1 , . . . , An ), denoted by U (A1 , . . . , An ; ω),
n
to be the set of all B = (B1 , . . . , Bb ) ∈ Mk (C) such that there is a unitary
U ∈ Mk (C) such that
kU ∗ AU − Bk2 < ω.
n

If E ⊂ Mk (C) , we define the ω-orbit covering number of E, denoted by ν (E, ω),
to be the smallest number of ω-orbit balls that cover E.
Let A =C*(x1 , . . . , xn ) and for each positive integer k, let Rep(A, k) / ⋍ denote
the set of all unital ∗-homomorphisms from A into Mk (C) modulo unitary equivalence. If π1 , π2 ∈Rep(A, k) with corresponding images [π] , [ρ] ∈ Rep(A, k) / ⋍ we
define a metric
dk ([π] , [ρ]) = min k(π (x1 ) , . . . , π (xn )) − (U ∗ ρ (x1 ) U, . . . , U ∗ ρ (xn ) U )k2
as U ranges over all of the k × k unitary matrices.
For each 0 < ω < 1, we define
νdk (Rep (A, k) / ⋍, ω)
to be the minimal number of dk -balls of radius ω it takes to cover Rep(A, k) / ⋍.
Lemma 4.1. Suppose A =C*(x1 , . . . , xn ) is matricially tracially stable and τ is an
embeddable tracial state on A. Let R > max1≤j≤n kxj k. For each 0 < ω < 1 there
exists an mω ∈ N such that, for all integers k, N ≥ mω and every 0 < ε < 1/mω

Card (Rep (A, k) / ⋍) ≥ νdk (Rep (A, k) / ⋍, ω/4) ≥ ν (ΓR,τ (x1 , . . . , xn ; N, k, ε) , ω) .
Proof. Let 0 < ω < 1. It is easy to deduce from the ǫ − δ-definition of matricial
tracial stability that, there is a positive integer mω such that, for every k ∈ N
and every N ≥ mω and 0 < ε < 1/mω , we have for each B = (b1 , . . . , bn ) ∈
ΓR,τ (x1 , . . . , xn ; N, k, ε) a representation πB ∈ Rep (A, k) such that
kB − (πB (x1 ) , . . . , πB (xn ))k2 < ω/4.
Now suppose N ≥ mω and 0 < ε < 1/mω . It follows from the definition of
s = ν (ΓR,τ (x1 , . . . , xn ; N, k, ε) , ω) that there is a collection
{Bj ∈ ΓR,τ (x1 , . . . , xn ; N, k, ε) : 1 ≤ j ≤ s}

TRACIALLY STABLE

23

so that, for any k × k unitary U and 1 ≤ i 6= j ≤ j,

kU ∗ Bi U − Bj k2 ≥ ω.

For each j, 1 ≤ j ≤ s. It easily follows that, for 1 ≤ i 6= j ≤ s,


dk [πBi ] , πBj ≥ ω/2.



Hence every dk -ball with radius ω/4 contains at most one of πBj , 1 ≤ j ≤ s.
Hence,

Card (Rep (A, k) / ⋍) ≥ νdk (Rep (A, k) / ⋍, ω/4) ≥ ν (ΓR,τ (x1 , . . . , xn ; N, k, ε) , ω) .


In [13] the first named author and J. Shen defined the free-orbit dimension
K1 (x1 , . . . , xn ; τ ). First let
K (x1 , . . . , xn ; τ, ω) = inf lim sup
ε,N

and let

k→∞

log ν (ΓR,τ (x1 , . . . , xn ; N, k, ε) , ω)
,
k 2 |log ω|

K1 (x1 , . . . , xn ; τ ) = lim sup K (x1 , . . . , xn ; τ, ω) .
ω→0+

If A =C*(x1 , . . . , xn ) is matricially tracially stable and τ is an embeddable tracial
state on A, then by Lemma 4.1
log Card (Rep (A, k) / ⋍)
1
≥ K (x1 , . . . , xn ; τ, ω) .
lim sup
|log ω| k→∞
k2

= ∞.
It follows that if K1 (x1 , . . . , xn ; τ ) > 0, then lim supk→∞ log Card(Rep(A,k)/⋍)
k2
If δ0 (x1 , . . . , xn ; τ ) denotes D. Voiculescu’s free entropy dimension, we know from
[28] that
δ0 (x1 , . . . , xn ; τ ) ≤ 1 + K1 (x1 , . . . , xn ; τ ) .
This gives the following result, which shows that a matricially tracially stable algebra may be forced to have a lot of representations of each large finite dimension.
Theorem 4.2. Suppose A =C*(x1 , . . . , xn ) is matricially tracially stable and τ is
an embeddable tracial state on A such that 1 < δ0 (x1 , . . . , xn ) . Then
lim sup
k→∞

log Card (Rep (A, k) / ⋍)
= ∞.
k2

Below using this theorem we give an example which shows that for not tracially
nuclear C ∗ -algebras, the conditions 1) and 2) in Theorem 3.3 are not sufficient for
being matricially tracially stable.
Lemma 4.3. The set {(U, V ) ∈ Un × Un : C ∗ (U, V ) = Mn (C)} is norm dense in
Un × Un .
Proof. Suppose (U, V ) ∈ Un × Un . Perturb U by an arbitrary small amount so that
it is diagonal with no repeated eigenvalues with respect to some orthonormal basis
{e1 , . . . , en } and perturb V by an arbitrary small amount
Pn so that its eigenvalues
are not repeated and one eigenvector has the form k=1 λk ek with λk 6= 0 for
1 ≤ k ≤ n.

The following lemma states that every irreducible representation of (π1 ⊕ π2 ) (A)
must either factor through π1 or through π2 .

24

DON HADWIN AND TATIANA SHULMAN

Lemma 4.4. If π = π1 ⊕ π2 is a direct sum of unital representations of a unital
C*-algebra A and ρ is an irreducible representation of A such that ker π ⊂ ker ρ,
then either ker π1 ⊂ ker ρ or ker π2 ⊂ ker ρ.
Proof. Suppose ρ : A → B (H) is irreducible. Assume, via contradiction, ai ∈ ker πi
and ai ∈
/ ker ρ for i = 1, 2. Then a1 Aa2 ⊆ ker π ⊆ ker ρ, so ρ (a1 ) ρ (A) ρ (a2 ) = {0} ,
−wot
and since ρ (A)
= B (H) , we have ρ (a1 ) B (H) ρ (a2 ) = {0} , but ρ (a1 ) 6= 0 6=
ρ (a2 ) , which is a contradiction.

Lemma 4.5. (Hoover [18]) If π = π1 ⊕ π2 is a direct sum of unital representations
of a unital C*-algebra A, then π (A) = π1 (A) ⊕ π2 (A) if and only if there is no
irreducible representation ρ of A such that ker π1 ⊂ ker ρ and ker π2 ⊂ ker ρ.
We will give a proof of it for the reader’s convenience.
Proof. Let J1 = {b ∈ π1 (A) : b ⊕ 0 ∈ π (A)} and J2 = {b ∈ π2 (A) : 0 ⊕ b ∈ π (A)}.
Clearly,
π (A) = π1 (A) ⊕ π2 (A) if and only if 1 ∈ J1 if and only if 1 ∈ J2 if and only
if 1 ⊕ 1 ∈ J (where J = J 1 ⊕ J2 ). Hence, if π (A) 6= π1 (A) ⊕ π2 (A), J is a
proper ideal of π (A) and Jk is a proper ideal of πk (A) for k = 1, 2. Moreover, if
π (a) = π1 (a) ⊕ π2 (a), we have:
π (a) ∈ J if and only if π1 (a) ∈ J1 if and only if π2 (a) ∈ J2 . Thus the algebras
π (A) /J , π1 (A) /J1 and π2 (A) /J2 are isomorphic to a unital C*-algebra D. Thus
there are surjective homomorphisms ρ : π (A) → D and ρk : πk (A) → D such that
ρ ◦ π = ρ1 ◦ π1 = ρ2 ◦ π2 .
If we choose an irreducible representation γ of D and replace ρ, ρ1 , ρ2 with γ ◦ ρ, γ ◦
ρ1 , γ ◦ ρ2 we get irreducible representations.

Theorem 4.6. There exists a C ∗ -algebra which satisfies the conditions 1) and 2)
of Theorem 3.3 but is not matricially tracially stable.
Proof. Let Cr∗ (F2 ) be the reduced free group C*-algebra. Then Cr∗ (F2 ) has a
unique tracial state τ and two canonical unitary generators U, V . D. Voiculescu
[27] proved that
δ0 (U, V ; τ ) = 2.
It was also shown by U. Haagerup and S. Thorbjørnsen [9] that there are sequences
{Uk } , {Vk } of unitary matrices with Uk , Vk ∈ Mk (C) for each k ∈ N such that, for
every ∗-polynomial p (x, y)
(4.1)

lim kp (Uk , Vk )k = kp (U, V )k .

k→∞

It follows from the uniqueness of a trace on Cr∗ (F2 ) that, for every ∗-polynomial
p (s, t),
(4.2)

lim τk (p (Uk , Vk )) = τ (p (U, V )) .

k→∞

By Lemma 4.3 we can assume that each pair (Uk , Vk ) is irreducible, i.e., C*(Uk , Vk ) =
Mk (C). Let U∞ = U1 ⊕U2 ⊕· · · and V∞ = V1 ⊕V2 ⊕· · · , and let A = C ∗ (U∞ , V∞ ).
Clearly A is an RFD C*-algebra. Moreover, for each k ∈ N, A has, up to unitary
equivalence, exactly one irreducible representation πk of dimension k, namely the
one with πk (U∞ ) = Uk and πk (V∞ ) = Vk . Since each k-dimensional representation

TRACIALLY STABLE

25

of A is unitarily equivalent to a direct sum of at most k irreducible representations
of dimension at most k, it follows that
Card (Rep (A, k) / ⋍) ≤ k k .

Thus

log k
log Card (Rep (A, k) / ⋍)
≤ lim sup
= 0.
k2
k
k→∞
However, there is a unital ∗-homomorphism π : A → Cr∗ (F2 ) such that π (U∞ ) = U
and π (V∞ ) = V . Thus τ ◦ π is an embeddable tracial state on A and since
lim sup
k→∞

δ0 (U∞ , V∞ ; τ ◦ π) = δ0 (U, V ; τ ) = 2,

it follows from Theorem 4.2 that A is notPmatricially tracially stable.
⊕
Claim 1: πn does not factor through 0≤k6=n<∞ πk .
Proof: Assume, via contradiction, the claim is false. Since (Un , Vn ) is an irreP
ducible pair for n ∈ N, it follows that πn does not factor through ⊕
0≤k<N,k6=n πk for
each
positive
integer
N
.
Hence,
by
Lemma
4.4,
it
follows
that
π
n factors through
P⊕
N ≤k6=n<∞ πk , for each positive integer N . However, since by (4.1), for every
a ∈ A, we have
kπk (a)k → kπ (a)k ,
we see that, for every a ∈ A,
kπn (a)k ≤ kπ (a)k .

This means that πn factors through π, which contradicts the fact that Cr∗ (F2 ) has
no finite-dimensional
P∞representations. Thus Claim 1 must be true.
Claim 2: J = k=1 Mk (C) ⊂ A.

Idk if k = n
Proof: It is sufficient to show that the sequence
belongs to
0
otherwise
A. Since idA = ⊕πk , the claim follows from Claim 1 and Lemma 4.5.

Clearly A/J is isomorphic to Cr∗ (F2 ) . Thus any factor representation of A must
either be finite-dimensional or be factorable through the representation π. Since
the extreme tracial states are the factor tracial states [12], we see that the extreme
tracial states on A are
{τ ◦ π} ∪ {τk ◦ πk : k ∈ N} .
By (4.2) we see that both conditions in Theorem 3.3 are satisfied, but A is not
matricially tracially stable.

5. C ∗ -tracial stability
The following lemma must be very well known. We give a proof of it because of
lack of convenient references.
Lemma 5.1. Suppose (X, d) is a compact metric space and τ is a state on C (X).
Let σ be the state on C [0, 1] given by
Z 1
σ (f ) =
f (t) dt.
0

Then, for any non-trivial ultrafilter α on N there is a unital ∗-homomorphism π :
α
Y
C (X) →
(C [0, 1] , σ) , such that σα ◦ π = τ.
n∈N

26

DON HADWIN AND TATIANA SHULMAN

Proof. We know from [11] that
α
Y

n∈N

(C [0, 1] , σ) =

α
Y

α
Y

(C [0, 1] , σ) is a von Neumann algebra and that

n∈N

(L∞ [0, 1] , σ). We know that there is a probability Borel

n∈N

measure µ on X such that, for every f ∈ C (X),
Z
τ (f ) =
f dµ.
X

Choose a dense subset {f1 , f2 , . . .} of C (X). For each n ∈ N we can find a Borel
partition {En,1 , . . . , En,kn } of X so that each En,j has sufficiently small diameter
and points xn,j ∈ En,j for 1 ≤ j ≤ kn such that, for 1 ≤ m ≤ n and for every
x ∈ X,


kn
X
1
fm (xn,j ) χEn,j  (x) ≤ .
fm (x) − 
n
j=1
For each n ∈ N we can partition [0, 1] into intervals {In,j : 1 ≤ j ≤ kn } so that

σ χIn,j = µ (En,j ) .
For each n ∈ N we define a unital ∗-homomorphism πn : C (X) → L∞ [0, 1] by
πn (f ) =

kn
X

f (xn,j ) χIn,j .

j=1

We define π : C (X) →

α
Y

(L∞ [0, 1] , σ) by

n∈N

It is clear, for m ∈ N, that

π (fm ) = {πn (fm )}α

σα (π (fm )) = lim σ (πn (fm )) = lim
n→α

n→α

kn
X

fm (xn,j ) µ(En,j ) = τ (fm ) .

j=1

Since {f1 , f2 , . . .} is dense in C (X), we see that τ = σα ◦ π.



We say that a topological space X is approximately path-connected if, for each
collection {V1 , . . . , Vs } of nonempty open subsets of X there is a continuous function
γ : [0, 1] → X such that
γ [(0, 1)] ∩ Vk 6= ∅
for 1 ≤ k ≤ s.
Equivalently, traversing the above path γ back and forth a finite number of times,
given 0 < a1 < b1 < · · · < as < bs < 1 we can find a path γ such that
γ ((ak , bk )) ⊂ Vk

for 1 ≤ k ≤ s. Indeed, we first find a path γ1 and 0 < c1 < d1 < · · · < cs < ds < 1
such that
γ1 ((ck , dk )) ⊂ Vk
for 1 ≤ k ≤ s, and then compose γ1 with a homeomorphism on [0, 1] sending ak to
ck and bk to dk for 1 ≤ k ≤ s.

TRACIALLY STABLE

27

To prove that X is approximately path-connected when X is Hausdorff, it clearly
suffices to restrict to the case in which {V1 , . . . , Vs } is disjoint (Choose xk ∈ Vk for
each k and chose an open set Wk ⊆ Vk with xk ∈ Wk such that xk = xj ⇒ Wj =
Wk and such that {Wk : 1 ≤ k ≤ s} (without repetitions) is disjoint, then consider
{Wk : 1 ≤ k ≤ s}.)
The following facts are elementary:
(1) Every approximately path-connected space is connected
(2) A continuous image of an approximately path-connected space is approximately path-connected
(3) A cartesian product of approximately path-connected spaces is approximately path-connected.
An interesting example of a compact approximately path connected metric space
in R2 is


 
1
A=
: 0 < x ≤ 1 ∪ {(0, y) : 1− ≤ y ≤ 1} .
x, sin
x

Note that A∪B with B = {(x, 0) : −1 ≤ x ≤ 0} is not approximately path-connected.
In particular, A and B are approximately path-connected and A∩B 6= ∅, but A∪B
is not approximately path-connected.
For compact Hausdorff spaces we have a characterization of approximate pathconnectedness which later will be used to characterize C ∗ -tracial stability for commutative C ∗ -algebras.
Theorem 5.2. Suppose X is a compact Hausdorff space. The following are equivalent:
(1) X is approximately path-connected.
(2) If A is a unital C*-algebra, B is an AW*-algebra, η : A → B is a surjective
unital ∗-homomorphism, and π : C (X) → B is a unital ∗-homomorphism,
then there is is a net {πλ } of unital ∗-homomorphisms from C (X) to A
such that, for every f ∈ C (X) ,
(η ◦ πλ ) (f ) → π (f )

in the weak topology on B.
(3) If A is a unital C*-algebra, B is an W*-algebra, η : A → B is a surjective
unital ∗-homomorphism, and π : C (X) → B is a unital ∗-homomorphism,
then there is is a net {πλ } of unital ∗-homomorphisms from C (X) to A
such that, for every f ∈ C (X) ,
(η ◦ πλ ) (f ) → π (f )

in the ultra*-strong topology on B.
(4) For every regular Borel probability measure ν on X, there is a net {ρω }
of unital ∗-homomorphisms from C (X) into C [0, 1] such that, for every
f ∈ C (X) ,
Z
Z 1
f dν = lim
ρω (f ) (x) dx.
X

ω

0

Proof. (1) ⇒ (2). Suppose X is approximately path-connected and suppose A, B, η,
and π are is in (2). Let Λ be the collection of all tuples of the form λ = (Sλ , Eλ , ελ )
where Sλ is a finite set of states on B, Eλ is a finite subset of C (X) of functions from
X to [0, 1], and ελ > 0. Suppose λ ∈ Λ. Clearly, π (C (X)) is an abelian selfadjoint

28

DON HADWIN AND TATIANA SHULMAN

C*-subalgebra of B, and therefore there is a maximal abelian C*-subalgebra M of B
such that π (C (X)) ⊆ M. Since B is an AW*-algebra, there is a commuting family
P of projections in B such that M = C ∗ (P). Since C ∗ (Eλ ) is a separable unital
C*-subalgebra of C (X), the maximal ideal space of C ∗ (Eλ ) is a compact metric
space Xλ and there is a surjective continuous function ζλ : X → Xλ . It follows that
Xλ is approximately path-connected. Note that if we identify C ∗ (Eλ ) with C (Xλ )
we can view a function f ∈ C ∗ (Eλ ) as both a function on X and on Xλ and we
have f = f ◦ ζλ . Now π (C ∗ (Eλ )) = π (C (Xλ )) is a separable subalgebra of C ∗ (P)
so there is a countable subset Pλ ⊆ P such that π (C (Xλ )) ⊆ C ∗ (Pλ ). If follows
from von Neumann’s argument that there is a wλ = wλ∗ with σ (wλ ) is P
a totally
disconnected subset of (0, 1) such that C ∗ (wλ ) = C ∗ (Pλ ). Let ψλ =
ψ∈Sλ ψ
and let nλ = ♯(Sλ ). Then there is a measure µλ on σ (wλ ) such that, for every
h ∈ C (σ (wλ )) , we have
Z
hdµλ = ψλ (h (wλ )) .
σ(wλ )

For each f ∈ Eλ there is a function hf ∈ C (σ (wλ )) such that π (f ) = hf (wλ ) and
there is a function ĥf ∈ C [0, 1] with 0 ≤ ĥf ≤ 1 such that ĥf |σ(wλ ) = hf . We can
view µλ as a Borel measure on n[0, 1] by defining
µλ ([0, 1] \σ (wλ )) = 0. Clearly,
o

µλ ([0, 1]) = ψλ (1) = nλ . Since ĥf : f ∈ Eλ is equicontinuous and [0, 1] \σ (wλ )
is dense in [0, 1], we can find 0 < a1 < b1 < a2 < b2 < · · · < as < bs < 1 and
r1 , . . . , rs ∈ σ (wλ ) such that, for every f ∈ E and 1 ≤ j ≤ s,
ĥf (t) − ĥf (rj ) < ελ /4nλ when t ∈ (aj , bj )
and such that

µλ [0, 1] \ ∪sj=1 (aj , bj ) < ελ /4.

We know ĥf (rj ) = hf (rj ) for all f ∈ Eλ and 1 ≤ j ≤ s. Since π : C (Xλ ) →
C (σ (wλ )), there is a yj ∈ Xλ for 1 ≤ j ≤ s such that, for every f ∈ Eλ , f (yj ) =
hf (rj ) . On the other hand, we see that there is an xj ∈ X for 1 ≤ j ≤ s such that
ζλ (xj ) = yj . Hence, we have
ĥf (t) − f (xj ) < ελ /4nλ when t ∈ (aj , bj ) .
We next choose an open set Vj ⊆ X with xj ∈ Vj such that, for every f ∈ Eλ and
every x ∈ V, we have
|f (x) − f (xj )| < ελ /4nλ .
We now use the fact that X is approximately path-connected to find a continuous
function γλ : [0, 1] → X such that, for 1 ≤ j ≤ s,
γλ ((aj , bj )) ⊆ Vj .
We have, for each f ∈ Eλ , each 1 ≤ j ≤ s, and each t ∈ (aj , bj )
ĥf (t) − f ◦ γλ (t) < ελ /2nλ
and for t ∈ [0, 1] \ ∪sj=1 (aj , bj ) ,
ĥf (t) − f ◦ γλ (t) ≤ 2.

TRACIALLY STABLE

29

Hence, for every f ∈ Eλ
|ψλ (π (f )) − ψλ ((f ◦ γλ ) (wλ ))| ≤

Z

ĥf − (f ◦ γλ ) (wλ ) dµλ <


(ελ /2nλ ) µλ ([0, 1]) + 2µλ [0, 1] \ ∪sj=1 (aj , bj ) < ελ .

P
Since ψλ = ψ∈Sλ ψ, the measure µλ is a sum of probability measures, one corresponding to each ψ ∈ Sλ . We therefore have, for every f ∈ Eλ and every ψ ∈ Sλ ,
Z
ĥf − (f ◦ γλ ) (wλ ) dµλ < ελ .
|ψ (π (f )) − ψ ((f ◦ γλ ) (wλ ))| ≤

We can choose vλ ∈ A with 0 ≤ vλ ≤ 1 such that η (vλ ) = wλ . We define a unital
∗-homomorphism πλ : C (X) → A by πλ (f ) = (f ◦ γλ ) (vλ ). Hence, for every
f ∈ C (X),
(η ◦ πλ ) (f ) = η ((f ◦ γλ ) (vλ )) = (f ◦ γλ ) (wλ ) .
Hence, for every f ∈ Eλ and every ψ ∈ Sλ we have
|ψ ((η ◦ πλ ) (f )) − ψ (π (f ))| < ελ .

It follows, for every f ∈ C (X) with 0 ≤ f ≤ 1 and every state ψ on B, that
lim ψ ((η ◦ πλ ) (f )) = ψ (π (f )) .
λ

Since every g ∈ C (X) is a linear combination of f ’s with 0 ≤ f ≤ 1 and every
continuous linear functional on B is a linear combination of states, we see, for every
f ∈ C (X), that (η ◦ πλ ) (f ) → π (f ) in the weak topology on B.
(2) ⇒ (3) . Since every W*-algebra is an AW*-algebra, it is clear that we can
find a net {πλ } as in (2). Thus, for every f ∈ C (X), (η ◦ πλ ) (f ) → π (f ) and
∗

∗

(η ◦ πλ ) (f ) (η ◦ πλ ) (f ) = (η ◦ πλ ) (f ∗ f ) → π (f ) π (f )

ultraweakly. Hence we have (η ◦ πλ ) (f ) → π (f ) in the ultra*-strong topology on
B.
(3) ⇒ (4). By Lemma 5.1 there exists a unital ∗-homomorphism π : C(X) →
α
Y
(C [0, 1] , σ) such that
n∈N

σα ◦ π(f ) =

(5.1)

Z

f dν,

X

R1
for each f ∈ C(X). Here a state σ on C[0, 1] is given by σ(g) = 0 gdx. Let
α
Y
Q
η :
C[0, 1] →
(C [0, 1] , σ) be the canonical surjection. By 3) there is a
n∈N
Q
net {πλ } of unital ∗-homomorphisms from C (X) to C[0, 1] such that, for every
f ∈ C (X) ,
(η ◦ πλ ) (f ) → π (f )
ultra*-strongly. By Lemma 2.2 there exist unital ∗-homomorphisms ρn : C(X) →
C[0, 1] such that
π(f ) = η((ρn (f ))n∈N ),
for each f ∈ C(X). By (5.1)
Z 1
Z
f dν = lim
ρn (f )dx.
X

α

0

30

DON HADWIN AND TATIANA SHULMAN

R1
R1
The sequence 0 ρn (f )dx contains
a subnet 0 ρnω (f )dx which is an ultranet. This
R
ultranet has to converge to X f dν. Indeed, if limα tn = t and tnω is an ultranet,
then limω tnω = t. (Proof: for any ǫ > 0 the set {nω | |tnω − t| < ǫ} is infinite,
/ α and we would have ∅ = {nω | |tnω − t| ≥
otherwise
{nω | |tnω − t| < ǫ} ∈
T
ǫ} {n | |tn − t| < ǫ} ∈ α. Hence t is an accumulation point for {tnω } and since
{tnω } is an ultranet, t is its limit.) Thus we have
Z 1
Z
f dν = lim
ρnω (f )dx.
ω

X

0

(4) ⇒ (1). Assume (4) is true. Suppose V1 , V2 , . . . , Vs are nonempty open
subsets of X. There is no harm in assuming {V1 , . . . , Vs } is disjoint. For each
1 ≤ j ≤ s we can choose xk ∈ Vk and a continuous
function hj : X → [0, 1] such that
Ps
hj (xj ) = 1 and hj |X\Vj = 0. Let µ = 1s j=1 δxj . Then µ is a probability measure
R
with X hj dµ = 1s . It follows from (4) that there is a unital ∗-homomorphism
ρ : C (X) → C [0, 1] such that
Z 1
ρ (hj ) (x) dx 6= 0
0

for 1 ≤ k ≤ s. However, there must be a continuous map γ : [0, 1] → X such that
R1
π (f ) = f ◦ γ for every f ∈ C (X) . For each 1 ≤ j ≤ s, 0 6= 0 (hj ◦ γ) (x) dx implies
that there is a tj ∈ [0, 1] such that hj (γ (tj )) 6= 0. Thus, by the definition of hj , we
have γ (tj ) ∈ Vj for 1 ≤ j ≤ s. Therefore, X is approximately path-connected. 
Remark. In statement (2) in Theorem 5.2, if we view B ⊆ B (H) as the universal
representation (that is the direct sum of all irreducible representations), then the
weak operator topology on B is the weak (and the ultraweak) topology on B. Thus,
if (η ◦ πλ ) (f ) → π (f ) and
∗

∗

[(η ◦ πλ ) (f )] [(η ◦ πλ ) (f )] = (η ◦ πλ ) (f ∗ f ) → π (f ∗ f ) = π (f ) π (f )

weakly for every f ∈ C (X) implies, for each f ∈ C (X) that
(η ◦ πλ ) (f ) → π (f )
is the ∗-strong operator topology in B (H) .
As a corollary we obtain a characterization of when a separable commutative
C ∗ -algebra is C ∗ -tracially stable.
Theorem 5.3. Suppose (X, d) is a compact metric space. The following are equivalent:
(1) C (X) is C ∗ -tracially stable.
(2) X is approximately path-connected
(3) For every state τ on C (X) there is a sequence πn : C (X) → C [0, 1] such
that, for every f ∈ C (X) ,
Z 1
τ (f ) = lim
πn (f ) (x) dx.
n→∞

0

TRACIALLY STABLE

31

Proof. 2) ⇔ 3) by the equivalence of statements 1) and 4) in Theorem 5.2 and
separability of C(X).
Q
3) ⇒ 1): Let φ : C(X) → α
i∈I (Ai , ρi ) be a unital ∗-homomorphism. By [11]
Qα
(A
,
ρ
)
is
a
von
Neumann
algebra
and by the equivalence 3) ⇔ 4) in Theorem
i
i
i∈I
5.2
φ
is
a
∗-strong
pointwise
limit
of
liftable ∗-homomorphisms from C(X) →
Qα
(A
,
ρ
).
By
Lemma
2.2
φ
is
liftable.
i
i
i∈I
1) ⇒ 3): By Lemma 5.1 there is a unital ∗-homomorphism π : C (X) →
α
Y
(C [0, 1] , σ) such that σα ◦ π = τ. By 1) we can lift it and obtain a sequence

n∈N

πn : C (X) → C [0, 1] such that, for every f ∈ C (X) ,
Z 1
πn (f ) (x) dx.
τ (f ) = lim
α

0

Taking a subnet, which is an ultranet we obtain (by the same arguments as in the
proof of the implication 3) ⇒ 4) in Theorem 5.2) that for any f ∈ C(X)
Z 1
τ (f ) = lim
πnω (f ) (x) dx.
ω

0

Since C(X) is separable, we can pass to a subsequence.



Corollary 5.4. Suppose A is a unital commutative C ∗ -tracially stable C*-algebra
and A0 is a unital C*-subalgebra of A. Then A0 is C ∗ -tracially stable.
Proof. We can assume A = C (X) with X an approximately path-connected compact metric space. Also we can write A0 = C (Y ), and, since A0 embeds into
A, there is a continuous surjective map ϕ : X → Y. Thus Y is approximately
path-connected, which implies A0 is C*-tracially stable.

At this point there is little else we can say about C*-tracially stable algebras,
except that they do not have projections when there is a faithful embeddable tracial
state.
Theorem 5.5. Suppose A is a separable C*-tracially stable unital C*-algebra.
Then A/Jet has no nontrivial projections.
Proof. Without loss of generality we can assume Jet = {0}. Then A has a faithful
embeddable tracial state σ. Then there is a tracial embedding π of (A, σ) into
α
Y
(Cr∗ (F2 ) , τ ) for some non-trivial ultrafilter α on N. Indeed, Cr∗ (F2 ) has a

n∈N

unique trace τ and it is a subalgebra of the factor von Neumann algebra LF2 so
that LF2 = W ∗ (Cr∗ (F2 )). It follows from the Kaplansky density theorem that the
|| k2 -closure of the unit ball of C ∗ (F2 ) is the unit ball of LF2 which implies that
α
α
Y
Y
∗
(LF2 , τ ).
(C (F2 ), τ ) =
Qα
Q
(Mn (C), τn ) embeds into α (C ∗ (F2 ), τ ) =
Since LF2 contains Mn (C) for each n ∈ N,
Q
α
(LF2 , τ ).
Now since A is tracially stable, there is a sequence {πn } of unital ∗-homomorphisms
from A into Cr∗ (F2 ) such that, for every a ∈ A ,
σ (a) = lim τ (πn (a)) .
n→α

32

DON HADWIN AND TATIANA SHULMAN

Suppose P is a projection in A. Since Cr∗ (F2 ) contains no non-trivial projections,
for each n ∈ N πn (P ) = 0 or πn (P ) = 1. Since α is an ultrafilter, eventually
πn (P ) = 0 along α or eventually πn (P ) = 1 along α. Thus π (P ) = {πn (P )}α is
either 0 or 1. Hence P is either 0 or 1.

Remark. As was pointed out in [11], the tracial ultraproducts remain unchanged
when you replace the 2-norm by a p-norm (1 ≤ p < ∞). Therefore all results in
this paper remain valid if 2-norms are replaced by p-norms.
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University of New Hampshire
E-mail address: operatorguy@gmail.com
Institute of Mathematics of the Polish Academy of Sciences, Poland
E-mail address: tshulman@impan.pl

