This paper presents two decomposition theorems for Elliott dimension drop interval algebras, which are crucial for classifying a broad class of $AH$ algebras with no dimension growth.
Contribution
It introduces new decomposition theorems that advance the classification of $AH$ algebras with the ideal property and no dimension growth.
Findings
01
Two key decomposition theorems for Elliott dimension drop interval algebras
02
Progress towards classifying all $AH$ algebras with no dimension growth
03
Foundational results for the structure of $C^*$-algebras in classification theory
Abstract
Elliott dimension drop interval algebra is an important class among all C∗-algebras in the classification theory. Especially, they are building stones of AHD algebra and the latter contains all AH algebras with the ideal property of no dimension growth. In this paper, we will show two decomposition theorems related to the Elliott dimension drop interval algebra. Our results are key steps in classifying all AH algebras with the ideal property of no dimension growth.
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Taxonomy
TopicsAdvanced Operator Algebra Research · Advanced Topics in Algebra · Advanced Banach Space Theory
Full text
ON THE DECOMPOSITION THEOREMS FOR C∗-ALGEBRAS
Chunlan Jiang1 Liangqing Li2, and Kun Wang3∗
††footnotetext:
1
College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, 050024, China.
**Abstract
Elliott dimension drop interval algebra is an important class among all C∗-algebras in the classification theory.
Especially, they are building stones of AHD algebra and the latter contains all
AH algebras with the ideal property of no dimension growth.
In this paper, we will show two decomposition theorems related to the Elliott dimension drop interval algebra.
Our results are key steps in classifying all AH algebras with the ideal property of no dimension growth.
**
Keywords: C∗-algebra, Elliott dimension drop interval algebra, decomposition theorem, spectral distribution property
Classification theorems have been obtained for AH algebras—the inductive limits of cut downs of matrix algebras over compact metric spaces by projections—and AD algebras—the inductive limits of Elliott dimension drop interval algebras in two special cases:
1. Real rank zero case: all such AH algebras
with no dimension growth and such AD algebras (See [4], [12], [7], [8], [13], [1], [14]-[17], [3], and [2]);
2. Simple case: all
such AH algebras with no dimension growth (which includes all simple AD algebras by [11]) (See [5], [6], [33], [42], [43], [26]-[29], [18], and
[9]).
In [9], the authors pointed out two important possible next steps after the completion of classification of simple AH algebras (with no dimension
growth). One of these is the classification of simple ASH algebras—the simple inductive limits of subhomogeneous algebras (with no dimension growth). The
other is to generalize and unify the above-mentioned classification theorems for simple AH algebras and real rank zero AH algebras by classifying
AH algebras with the ideal property. In this article, we have achieved several key results for the second goal by providing two decomposition theorems.
As in [8], let TII,k be the 2-dimensional connected simplicial complex with H1(TII,k)=0 and H2(TII,k)=Z/kZ, and
let Ik be the subalgebra of Mk(C[0,1]) defined by
[TABLE]
This algebra is called an Elliott dimension drop interval algebra. Denote by HD the class of algebras consisting of direct sums of building
blocks of the forms Ml(Ik) and PMn(C(X))P, with X being one of the spaces {pt}, [0,1], S1, and TII,k, and with P∈Mn(C(X)) being a
projection. (In [2], this class is denoted by SH(2), and in [23], this class is denoted by B). We will call a C∗-algebra an AHD
algebra, if it is an inductive limit of algebras in HD.
In [20], [21], [29], and [24], it is proved that
all AH algebras with the ideal property of no dimension growth are inductive limits of algebras in the class HD—that is, they are AHD algebras.
By this reduction theorem, to classify AH algebras with the ideal property, we must study
the properties of homomorphisms between those basic building blocks.
In the local uniqueness theorem for classification, it requires the homomorphisms involved to satisfy a certain spectral distribution property, called the sdp
property (more specifically, sdp(η,δ) property introduced in [18] and [9] for some positive real numbers η and δ).
This property automatically
holds for the homomorphisms ϕn,m (provided that m is large enough) giving rise to a simple inductive limit procedure.
But for the case of general inductive limit C∗-algebras
with the ideal property, to obtain this sdp property, we must pass to certain good quotient algebras which corresponding to
simplicial sub-complexes of the original spaces; a uniform uniqueness theorem, that does not depend on the choice of simplicial sub-complexes involved, is
required.
For the case of an interval, whose simplicial sub-complexes are finite unions of subintervals and points, such a uniform uniqueness theorem is proved
in [22] (see [27] and [5] also).
But for the general case, there are no uniqueness theorem for the general case
involving arbitrary finite subsets of Mn(C(TII,k)) (or Ml(Ik)).
In this paper, we prove decomposition theorems between such building blocks or between
a building block of this kind and a homogeneous building block.
And we will compare the decompositions of two different homomorphisms in the last part of chapter 4.
Such decomposition and comparison results will be used in the proof of the uniqueness
theorem for AH algebras with the ideal property in [19] by Gong, Jiang and Li.
2 Notation and terminology
In this section, we will introduce some notation and terminology.
Definition 2.1**.**
Let X be a compact metric space and ψ:C(X)→PMk1(C(Y))P (with rank(P)=k) be a unital homomorphism. For any point y∈Y, there are k mutually orthogonal rank 1 projections p1,p2,⋯,pk with i=1∑kpi=P(y) and {x1(y),x2(y),⋯,xk(y)}⊂X (may be repeat) such that
[TABLE]
*We denote the set {x1(y),x2(y),⋯,xk(y)} (counting multiplicities), by Spψy. We shall call Spψy **the spectrum of *** ψat the pointy.
2.2*.*
For any f∈Ik⊂Mk(C[0,1])=C([0,1],Mk(C)) as in 3.2 of [13], let function f:[0,1]⟶C⊔Mk(C) (disjoint union) be defined by
[TABLE]
That is, f(t) is the value of irreducible representation of f corresponding to the point t. Similarly, for f∈Ml(Ik), we can
define f:[0,1]⟶Ml(C)⊔Mlk(C), by
[TABLE]
2.3*.*
Suppose that ϕ:Ik⟶PMn(C(Y))P is a unital homomorphism. Let r=rank(P). For each y∈Y, there are t1,t2,⋯,tm∈[0,1] and a unitary
u∈Mn(C) such that
[TABLE]
and
[TABLE]
for all f∈Ik.
2.4*.*
Let ϕ be the homomorphism defined by the equation (2.1) above with t1,t2,⋯,tm as appeared in the diagonal of the matrix. We define the set Spϕy to be the points t1,t2,⋯,tm with possible
fraction multiplicity. If ti=0 or 1, we will assume that the multiplicity of ti is k1; if 0<ti<1, we will assume that the multiplicity of ti is 1. For example if we assume
[TABLE]
then
Spϕy={0∼k1,0∼k1,0∼k1,t4,t5,⋯,tm−2,1∼k1,1∼k1}, which can also be written
as
[TABLE]
Here we emphasize that, for t∈(0,1), we do not allow the multiplicity of t to be non-integral. Also for 0 or 1, the multiplicity must be multiple of k1 (other fraction numbers are not allowed).
Let ψ:C[0,1]⟶PMn(C(Y))P be defined by the following composition
[TABLE]
where
the first map is the canonical inclusion. Then we have Spψy={Spϕy}∼k—that is, for each element t∈(0,1), its multiplicity in Spψy is exactly k times of the multiplicity in ϕy.
2.5*.*
(a)
we use ♯(.) to denote the cardinal number of a set. Very often, the sets under consideration will be sets with multiplicity, in which case we shall
also count multiplicity when we use the notation ♯.
The set may also contain fractional point. For example,
[TABLE]
2. (b)
We shall use a∼k to denote \underbrace{a,a,\cdots a}\limits_{k}. For example {a∼3,b∼2}={a,a,a,b,b}.
3. (c)
For any metric space X, any x0∈X and c>0, let Bc(x0)≜{x∈X∣d(x,x0)<c}, the open ball with radius c and center x0.
4. (d)
Suppose that A is a C∗-algebra, B⊂A a subset (often a subalgebra), F⊂A is a finite subset and ε>0. If for each element f∈F, there is an element g∈B such
that ∥f−g∥<ε, then we shall say that F is approximately contained in B to within ε, and denote this by F⊂εB.
5. (e)
Let X be a compact metric space. For any δ>0, a finite set {x1,x2,⋯,xn} is said to be δ-dense in X if for
any x∈X, there is xi∈{x1,x2,...,xn} such that dist(x,xi)<δ.
6. (f)
We shall use ∙ or ∙∙ to denote any possible positive integers.
7. (g)
For any two projections p,q∈A, by [p]≤[q] we mean that p is unitarily equivalent to a sub-projection of q. And we
use p∼q to denote that p is unitarily equivalent to q.
2.6*.*
Let A=Ml(Ik). Then every point t∈(0,1) corresponds to an irreducible representation πt, defined by πt(f)=f(t). The representations π0 and π1 defined by
[TABLE]
are no longer irreducible. We use 0 and 1 to denote the corresponding points for the irreducible representations. That is,
[TABLE]
Or we can also write f(0)≜f(0) and f(1)≜f(1). Then the equation (∗) could be written as
[TABLE]
where some of ti may be 0 or 1. In this notation, up to unitary equivalence,
f(0) is equal to diag(\underbrace{f(\underline{0}),f(\underline{0}),\cdots,f(\underline{0})}\limits_{k}) .
Under this notation, we can also write 0∼k1 as 0. Then the example of Spϕy in 2.4 can be written as
[TABLE]
2.7*.*
For a homomorphism ϕ:A⟶Mn(Ik), where A=Ik or C(X), and for any t∈[0,1], define Spϕt=Spψt, where
ψ is defined by the composition
[TABLE]
Also Spϕ0=Sp(π0∘ϕ). Hence, Spϕ0={Spϕ0}∼k.
2.8*.*
Let ϕ:Mn(A)⟶B be a unital homomorphism. It is well known (see 1.34 and 2.6 of [8]) that there is an identification of B
with (ϕ(e11)Bϕ(e11))⊗Mn(C) such that
[TABLE]
where e11 is the matrix unit of upper left corner of Mn(A) and ϕ1=ϕ∣e11Mn(A)e11:A⟶ϕ(e11)Bϕ(e11).
If we further assume that A=Ik or C(X) (with X being a connected CW complex) and B is either QMn(C(Y))Q or Ml(Ik1), then for any
y∈SpB, define Spϕy≜Sp(ϕ1)y.
Here, we use the standard notation that if B=PMm(C(Y))P then SpB=Y; and if B=Ml(Ik), then Sp(B)=[0,1].
2.9*.*
Let A and B be either of form PMn(C(X))P (with X path connected) or of form Ml(Ik). Let ϕ:A⟶B be a unital
homomorphism, we say that ϕ has property sdp(η,δ) (spectral distribution property with respect to η and δ) if for any η-ball
[TABLE]
and any point y∈Sp(B),
[TABLE]
counting multiplicity. If ϕ is not unital, we say that ϕ has sdp(η,δ) if the corresponding
unital homomorphism ϕ:A⟶ϕ(1A)Bϕ(1A) has property sdp(η,δ).
2.10*.*
Set P^{n}X=\underbrace{X\times X\times\cdots\times X}\limits_{n}/\thicksim, where the equivalence relation ∼ is defined by
[TABLE]
if there is a permutation σ of {1,2,⋯,n} such that
xi=xσ(i)′ for each 1≤i≤n. A metric d on X can be extended to a metric on PnX by
[TABLE]
where σ is taken from the set of all permutations, and [x1,x2,⋯,xn] denote the equivalence class of (x1,x2,⋯,xk) in PkX.
2.11*.*
Let X be a metric space with metric d. Two k-tuple of (possible repeating) points {x1,x2,⋯,xn}⊂X and
{x1′,x2′,⋯,xn′}⊂X are said to be paired withinη if there is a permutation σ such that
[TABLE]
This is equivalent to the following statement. If one regards [x1,x2,⋯,xn] and
[x1′,x2′,⋯,xn′] as points in PnX, then
[TABLE]
2.12*.*
For X=[0,1], let P(n,k)X, where n,k∈Z+\{0}, denote the set of kn elements from X, in which only 0 or 1 may appear
fractional times. That is, each element in X is of the form
[TABLE]
with 0<t1≤t2≤⋯≤tm<1 and kn0+m+kn1=kn.
An element in P(n,k)X can always be written as
[TABLE]
where 0≤k0<k, 0≤k1<k, 0≤t1≤t2≤⋯≤ti≤1 and kk0+i+kk1=kn. (Here ti
could be 0 or 1.) In the above representations 2 and 3, we know that
[TABLE]
Let
[TABLE]
and
[TABLE]
with k0,k1,k0′,k1′∈{0,1,⋯,k−1}.
We define dist(y,y′) as the following: if k0=k0′ or k1=k1′, then dist(y,y′)=1; if k0=k0′ and k1=k1′ (consequently i=i′), then
[TABLE]
as we order the {tj} and {tj′} as
t1≤t2≤⋯≤ti and t1′≤t2′≤⋯≤ti′, respectively.
Note that P(n,1)X=PnX with the same metric. Let ϕ,φ:Ik⟶Mn(C) be two unital homomorphisms. Then Spϕ and
Spψ define two elements in P(n,k)[0,1]. We say that Spϕ and Spψ can be paired within η, if dist(Spϕ,Spψ)<η.
Note that
if dist(Spϕ,Spψ)<1, then KK(ϕ)=KK(ψ).
2.13*.*
Let A=PMk(C(X))P, or Ml(Ik) and X1⊂Sp(A) be a closed subset—that is, X1 is a closed subset of X or of [0,1]. We define A∣X1 to be the quotient algebra A/I, where I={f∈A,f∣X1=0}.
Evidently Sp(A∣X1)=X1.
If B=QMk(C(Y))Q, ϕ:A⟶B is a homomorphism, and Y1⊂Sp(B)(=Yor[0,1]) is a closed subset, then we use ϕ∣Y1 to denote the composition.
[TABLE]
If Sp(ϕ∣Y1)⊂X1∪X2∪⋯∪Xk, where X1,X2,⋯,Xk are mutually disjoint closed subsets of X, then the
homomorphism ϕ∣Y1 factors as
[TABLE]
We will use ϕ∣Y1Xi to denote the part of ϕ∣Y1 corresponding to the map A∣Xi⟶B∣Y1. Hence
ϕ∣Y1=i⨁ϕ∣Y1Xi.
3 Decomposition Theorem I
In this section, we will prove the following theorem.
Theorem 3.1**.**
Let F⊂Ik be a finite set, ε>0.
There is an η>0, satisfying that if
[TABLE]
is
a unital homomorphism such that for any x∈X,
♯(Spϕx′∩[0,4η])≥k* and ♯(Spϕx′∩[1−4η],1])≥k,*
where
[TABLE]
then there are three mutually orthogonal projections
Let η>0 (and η<1) be such that if ∣t−t′∣<η, then ∥f(t)−f(t′)∥<6ε for all f∈F.
We will prove that this η is as desired. Let a unital homomorphism ϕ:Ik→PM∙C(X)P satisfy that ♯(Spϕx∩[0,4η])≥k and ♯(Spϕx∩[1−4η,1])≥k for each x∈X, we will prove such ϕ
has the decomposition as desired.
3.3*.*
Let rank(P)=n. And let ei,j∈Mn(C) be the matrix units. For any closed set Y⊂[0,1], define
hY∈C[0,1]⊂Ik (considering C[0,1] as in the center of Ik) as
[TABLE]
Define H′={hY∣Y is closed}∪{hYeij∣Y⊂[12nη,1−12nη] is closed}. Note that for a closed set Y⊂[12nη,1−12nη], hY(0)=hY(1)=0, and therefore hYeij∈Ik. Note also that the family H′ is equally continuous. There is a finite set H⊂H′ satisfying that for any h′∈H′, ∃h∈H such that
[TABLE]
For finite set H∪F, ε>0, and ϕ:Ik→PM∙(C(X))P, there is a τ>0 such that the following are true:
(a) For x,x′∈X with dist(x′,x)<τ, Spϕ∣x and Spϕ∣x′ can be paired within
24n2η. This is equivalent to the condition that Spϕ′∣x can be paired with Spϕ′∣x′ to within 24n2η (since KK(ϕ∣x)=KK(ϕ∣x′)), where ϕ′=ϕ∘ is as the above.
(b) For x,x′∈X with dist(x′,x)<τ,
[TABLE]
regarding ϕ(h)(x)∈P(x)M∙(C)P(x)⊂M∙(C) and ϕ(h)(x′)∈P(x′)M∙(C)P(x′)⊂M∙(C). In particular, ∥P(x)−P(x′)∥<12(n+1)2ε since 1∈H.
3.4*.*
Choose any simplicial decomposition on X such that for any simplex Δ⊂X, the set
Star(△)=∪{Δ′∘∣Δ′ is a simplex of X with Δ′∩Δ=∅}
has diameter at most 2τ, where Δ′∘ is the interior of the simplex Δ′.
3.5*.*
We will construct the homomorphism ψ:Ik→PM∙(C(X))P which is of the form
[TABLE]
as described in the theorem. Our construction will be carried out simplex by simplex.
First, define the restriction of map ψ to PM∙(C(X))P∣v=P(v)M∙(C)P(v) for each vertex v∈X. The homomorphism is denoted by
[TABLE]
(Here and below, we refer the reader to 2.13 for the notation ψ∣X1 for a subset X1⊂X.)
Next, we will define, for each 1-simplex [a,b]⊂X, the homomorphisms
[TABLE]
which will give the same maps as the previously defined maps ψ∣{a} and ψ∣{b} on the boundary {a,b}. Finally, we will define, for each 2-simplex Δ⊂X, the homomorphism
[TABLE]
such that ψ∣∂△ should be the same as what previously defined.
3.6*.*
For each simplex Δ of any dimension, let CΔ denote the center of the simplex. That is, if Δ is a vertex v, then CΔ=v; if Δ is a 1-simplex identified with [a,b], then CΔ=2a+b; and if Δ is a 2-simplex identified with a triangle in R2 with vertices {a,b,c}⊆R2, then CΔ=3a+b+c∈R2 which is barycenter
of Δ.
3.7*.*
According to each simplex Δ (of possible dimensions 0, 1, or 2), we will divide the set Spϕ′∣Δ⊂[0,1] into pieces, where ϕ′:C[0,1]↪IkϕPM∙C(X)P.
(Recall Spϕ′∣x={Spϕ∣x}∼k, and Spϕ′∣x has no fractional multiplicity). So for each x∈X,
Spϕ′∣x=n=rank(P) (counting multiplicity).
If we order Spϕ′∣x as
[TABLE]
then all functions λi are continuous functions. By path connectedness of simplex Δ, the set Spϕ∣Δ can be written as
[TABLE]
with
[TABLE]
(Note that, if ai=bi, then [ai,bi]={ai} is a degenerated interval.)
We will group the above intervals into groups T0∪T1∪⋯∪Tlast such that Spϕ∣Δ=∪Tj, with the condition that for any λ∈Tj,μ∈Tj+1, we have λ<μ, according to the following procedure:
(i) Spϕ∣Δ∩[0,4η+12nη]⊂T0, that is, all the above intervals [ai,bi] with ai≤4η+12nη should be in the group T0; and Spϕ∣Δ∩[1−(4η+12nη),1]⊂Tlast, that is all [ai,bi] with bi≥1−(4η+12nη) will be grouped into the last group Tlast;
(ii) If ai−bi−1≤12nη, then [ai−1,bi−1] and [ai,bi] are in the same group, say Tj;
(iii) If ai−bi−1>12nη, ai>4η+12nη and bi−1<1−(4η+12nη), then [ai−1,bi−1] and [ai,bi] are in different groups, say, Tj and Tj+1.
Denote Tlast by TlΔ (i.e., lΔ=last) — if there is no confusion, we will call TlΔ by Tl. Let t0=0, s0=max{4η,maxT0}, tl=min{1−4η,minTl}, sl=1; and for 1<i<l, let ti=minTi, and si=maxTi. Then Ti⊂[ti,si]. With the above notation, we have the following lemma.
Lemma 3.8**.**
*With the above notation, we have the following
(a) length [t0,s0]≤4η+6η;
(b) length [tl,sl]≤η/4+η/6;
(c) length [ti,si]≤η/6 for i∈{1,2,⋯,l−1};
(d) ti+1−si>12nη for i∈{0,1,2,⋯,l−1}.*
Proof.
From (ii) of 3.7, we know that minTi+1−maxTi>12nη; and from (i), we know that minT1>4η+12nη and maxTl−1<1−(4η+12nη). Hence (d) holds.
The following fact is well known.
Fact: For any two sequences 0≤λ1≤λ2≤⋯≤λn≤1 and 0≤μ1≤μ2≤⋯≤μn≤1, {λi}i=1n and {μi}i=1n can be paired within σ if and only if ∣λi−μi∣<σ for all i∈{1,2,⋯,n}.
Note that Δ is path connected and Spϕ∣Δ=i=1⋃k′[ai,bi] with [ai,bi]∩[aj,bj]=∅ if i=j. We conclude that for any z,z′∈Δ and i,
[TABLE]
counting multiplicity. In our construction, we know that Spϕ∣z and Spϕ∣z′ can be paired within η/24n2, using the above mentioned fact. We know also that
Spϕ∣z∩[ai,bi] and Spϕ∣z′∩[ai,bi]
can be paired within η/24n2. Consequently
[TABLE]
where CΔ is the center of simplex Δ. Note that Spϕ∣CΔ∩[ai,bi] is a finite set with at most n points in [0,1] and η/24n2-neighborhood of each point is a closed interval of length at most (η/24n2)⋅2=η/12n2. Hence we have
[TABLE]
Furthermore, each Tj contains at most n intervals [ai,bi]. And for each consecutive pair of intervals in Tj (0<j<l), we have
[TABLE]
and the distance between them ai+1−bi≤η/12n. That is, the gap between them is at most η/12n. Hence for each i∈{1,2,⋯,l−1}, the length of [ti,si] is at most
[TABLE]
(at most n possible intervals and n−1 gaps).
Also,
[TABLE]
and
[TABLE]
∎
3.9*.*
For each simplex Δ with face Δ′⊂Δ, we use Ti(Δ) and Tj(Δ′) to denote the sets [ti(Δ),si(Δ)] or [tj(Δ′),sj(Δ′)] as in 3.7, corresponding to Δ and Δ′.
Then evidently, the decomposition
[TABLE]
is a refinement of the decomposition Spϕ∣Δ=∪(Ti(Δ)∩Spϕ∣Δ)— that is, if two elements λ,μ∈Spϕ∣Δ′ are in the set Tj(Δ′) for a same index j, then they are in the set Ti(Δ) for a same index i.
3.10*.*
For each simplex Δ, consider the homomorphism
[TABLE]
Since Spϕ∣Δ⊂⋃j=0lTj(Δ)=j=1⋃l[tj,sj],
ϕ factors through as
[TABLE]
Let Pj(x)=ϕj(1k∣[tj,sj])(x) for each x∈Δ. Then Pj(x) are mutually orthogonal projections satisfying
[TABLE]
By the assumption of Theorem 3.1, we have rank(P0)≥k and rank(Pl)≥k.
3.11*.*
Now we define ψ:Ik→A∣Δ simplex by simplex, starting with vertices — the zero dimensional simplices.
where 0=t0<s0<t1≤s1<⋯<tl−1≤sl−1<tl<sl=1, with
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Recall that ϕ∣{v}:Ik→P(v)M∙(C)P(v) (as in 3.10) can be written as
[TABLE]
where ϕi=ϕ∣{v}[ti,si]:Ik→Ik∣[ti,si]→PiM∙(C)Pi and P(v)=∑i=0lPi. (Here and below, we refer the reader to 2.13 for the notation ϕ∣X1Zj (X1⊂X), which makes sense, provided that Sp(ϕ∣X1)⊂⋃jZj, where {Zj} are mutually disjoint closed subsets of the spectrum of the domain algebra of ϕ.)
From now on, we will use diag0≤i≤l(ϕi) to denote diag(ϕ0,ϕ1,⋯,ϕl).
Define ψi:Ik∣[ti,si]→PiM∙(C)Pi by
ψi=ϕi if 1≤i≤l−1,
(That is, we do not modify ϕi for 1≤i≤l−1.)
For i=0 (the case i=l is similar) we do the following modification. There is a unitary u∈M∙(C) such that
[TABLE]
where ξi∈(0,s1], 0<ξ1≤ξ2≤⋯≤ξ∙∙≤s1. Or write it as
[TABLE]
If 0<j≤k then we do not do any modification and just let ψ0=ϕ0. If j>k, then write j=kk′+j′ with 0<j′≤k, choose ξ′∈(0,ξ1), and define
[TABLE]
That is, change kk′ terms of f(0) in the diagonal of the definition of ϕ0 to k′ terms of the form f(ξ′).
If j=0, then we change ξ1 to 0, that is,
[TABLE]
Since ∣ξ′−0∣<2η and ∣ξ1−0∣<2η, we have ∥ϕ0(f)−ψ0(f)∥<6ε, for all f∈F (see 3.2).
We modify ϕl in a similar way to define ψl. Let
[TABLE]
where ψi=ψ∣{v}[ti,si]. Then ∥ϕ(f)−ψ(f)∥<6ε for all f∈F.
Remark 3.12*.*
Let us emphasize that the homomorphisms ψi are the same as ϕi for
i∈{1,2,⋯,l{v}−1}. But we do modify ϕ0 and ϕl (l=l{v}) to get ψ0 and ψl.
Also, we have
[TABLE]
Furthermore, ψi(1)=ϕi(1) for any i, and consequently ψ(1)=ϕ(1).
3.13*.*
Now consider 1-simplex Δ=[a,b]⊂X. We need to define ψ∣Δ=ψ∣[a,b] from previously defined ψ∣{a} and ψ∣{b}. According to 3.7, write Spϕ∣Δ=j=1⋃lΔSpϕ∣Δ∩Tj(Δ) with T0(Δ)=[0,s0(Δ)] and TlΔ(Δ)=[tlΔ(Δ),1]. Recall that in the definition of ψ∣{a}, ψ∣{b}, we use the decomposition
[TABLE]
and
[TABLE]
and only modified ϕ0=ϕ∣{a}[0,s0{a}] (or ϕ∣{b}[0,s0{b}]) and ϕl{a}=ϕ∣{a}[tl{a}({a}),1] (or ϕ∣{b}[tl{b}({b}),1]).
For Δ=[a,b], let us consider the decomposition
[TABLE]
From the above, we know that for any 0<j<lΔ, the definition of ψ∣{a}[tj(Δ),sj(Δ)] is the same as (ϕ∣Δ[tj(Δ),sj(Δ)])∣{a}, since the decomposition
[TABLE]
is finer than the decomposition
[TABLE]
(see 3.9) and only partial maps involving [0,s1{a}] (⊂[0,s1(Δ)]) and [tl{a}({a}),1] (⊂[tlΔ(Δ),1]) are modified. The same is true for ϕ∣{b} and ψ∣{b}. Therefore, we can define the partial maps
[TABLE]
for 0<j<lΔ.
The only parts need to be modified are ϕ∣Δ[0,s0(Δ)] and ϕ∣Δ[tl(Δ),1].
3.14*.*
Now denote ϕ∣Δ[0,s0(Δ)](Δ=[a,b]) by ϕ0 and ϕ∣Δ[tl(Δ),1] by ϕl, and s0(Δ) by s0, tl(Δ)(Δ) by tl. Now we have two unital homomorphisms
[TABLE]
and
[TABLE]
where P0, Pl are defined as in 3.10. We will do the modification of ϕ0 to get ψ0 (the one for ϕl is completely the same).
We already have the definitions of ψ0∣{a} and ψ0∣{b}. Note that P0∈M∙(C(Δ)) can be written as ϕ(h[0,s0]), where h[0,s0] is the test function appeared in 3.3, which is equal to 1 on [0,s0] and [math] on [s0+12nη,1].
(Note that ϕ(h[0,s0]) is a projection since Spϕ⊂[0,s0]∪[t1,1] and t1>s0+12nη.) Consequently,
As mentioned in 3.13, when we modify ϕ∣[a,b] to obtain ψ∣[a,b], we only need to modify ϕ0=ϕ∣[a,b][0,s0] and ϕl=ϕ∣[a,b][tl,1]. The modifications of ϕl to ψl are the same as the one from ϕ0 to ψ0. Thus we have the definition of ψ∣[a,b]=diag0≤i≤l(ψi).
3.17*.*
Let us estimate the difference of ϕ∣[a,b] and ψ∣[a,b] on the finite set F⊂Ik. Note that
[TABLE]
and ϕi=ψi, for 0<i<l. So we only need to estimate ∥ϕ0(f)−ψ0(f)∥ and ∥ϕl(f)−ψl(f)∥.
Note that ϕ0 and ψ0 are from Ik∣[0,s0] to P0M∙(C[a,b])P0, where P0 is as in 3.14. And both AdW∘ϕ0 and AdW∘ψ0 can be regarded as ∘ϕ′ and ∘ψ′ for
[TABLE]
where
[TABLE]
is given by
[TABLE]
Claim: Let α:Ik∣[0,s0]→Mrank(P0)(C[a,b]) be any unital homomorphism. Then we have
[TABLE]
In fact, for each x∈[a,b], there exist ux∈U(Mrank(P0)(C)) and k′∈{1,2,⋯,k} and 0≤ξ1≤ξ2≤⋯≤ξ∙∙≤s0 such that
[TABLE]
It follows that
[TABLE]
[TABLE]
Thus, the claim is true.
It follows from the claim that
[TABLE]
for all t∈[a,b], and f∈F, as ∣s0−0∣<2η and ∣tl−1∣<2η.
Hence we have the definition of ψ on the 1-skeleton X(1)⊂X satisfying
[TABLE]
for all t∈X(1) and f∈F.
3.18*.*
Now fix a 2-simplex Δ⊂X. We will define
[TABLE]
based on the previous definition of
[TABLE]
Again, write
[TABLE]
where
[TABLE]
and Pi are projections defined on Δ with
[TABLE]
For each face Δ′⊂∂Δ, we know that the decomposition
[TABLE]
is finer than the decomposition
[TABLE]
Consequently,
[TABLE]
Note that when we define ψ∣Δ′ by modifying ϕ∣Δ′, we only modify the parts of
ϕ∣Δ′[0,s0(Δ′)] and
ϕ∣Δ′[tl(Δ′),1] — that is,
[TABLE]
where δ∈(0,12nη). Hence
[TABLE]
since
[TABLE]
and
[TABLE]
Because Δ′⊂∂Δ is an arbitrary face, we have
[TABLE]
Therefore similar to what we did on 1-simplexes, define
[TABLE]
for j∈{1,2,⋯,l(Δ)−1}. Then we only need to modify ϕ∣Δ[0,s0(Δ)]=ϕ0 and ϕ∣Δ[tl(Δ),1]=ϕl. We will only do it for ϕ0.
3.19*.*
We have the definition of unital homomorphism
[TABLE]
such that
[TABLE]
for any x∈∂Δ, where Δ is a 2-simplex and ψ0
is defined as the composition
[TABLE]
We need to extend it to a homomorphism
[TABLE]
such that ♯(Spψ0∣Δ∩{0})=k′ for all x∈Δ. Once this extension is obtained, as in 3.17, we can use the claim in 3.17 to prove that ϕ∣Δ[0,s0] and ψ∣Δ[0,s0] are approximately equal to within 3ε for all f∈F.
(Note that in the argument of 3.17, the estimation is true which do not depend on the choice of the extension. It only uses ∣s0−0∣<η/2<η, and ∥f(t)−f(t′)∥<6ε whenever ∣t−t′∣<η.)
There is a W∈U(M∙(C(Δ))) such that
[TABLE]
for all x∈Δ. Again, if we can extend
[TABLE]
to
[TABLE]
then we can set ψ0∣Δ=AdW∗∘α∣Δ to obtain our extension. But (AdW∘ψ0)∣∂Δ (or α∣Δ) should be regarded as a homomorphism from Ik∣[0,s0] to Mrank(P0)(C(∂Δ)) (or to Mrank(P0)(C(Δ)). Hence the construction of ψ0∣Δ follows from the following lemma.
Lemma 3.20**.**
Let β:Ik∣[0,s0]→Mn′(C(S1)) be a unital homomorphism such that for any x∈S1,
[TABLE]
for some fixed k′ (not depending on x), where :C[0,s0]→Ik∣[0,s0]. Then there is a homomorphism
[TABLE]
(where D is the disk with boundary S1)
such that
[TABLE]
for all x∈D and π∘β=β, where
[TABLE]
is the restriction.
Proof.*
Let h(t)=t⋅1k be the function in the center of Ik∣[0,s0]. Then β(h) is a self adjoint element in Mn′(C(S1)). For each z∈S1, write the eigenvalue of β(h)(z) in increasing order*
[TABLE]
Then λ1,λ2,⋯,λn′ are continuous functions from S1 to [0,s0]. From the assumption, we know that
λ1(z)=λ2(z)=⋯=λk′(z)=0 and for all j>k′, λj(z)>0. (Note that each λj(j>k′) repeats some multiple of k times.) Consequently, there is ξ∈(0,s0] such that λj(z)≥ξ for all j>k′. Hence β factors through as
[TABLE]
where
[TABLE]
and
[TABLE]
with
[TABLE]
Note that rank(Q0)=k′, and rank(Q1)=n′−k′, which is a multiple of k. Write rank(Q1)=n′−k′=kk′′. There is a unitary u∈Mn(C(S1)) such that
[TABLE]
Hence
[TABLE]
with
[TABLE]
and
[TABLE]
Evidently,
[TABLE]
For β1′, there exist β′′:C[ξ,s0]→Mk′′(C(S1)) and a unitary V∈Mkk′′(C(S1)) such that
Vβ1′(f)V∗=β′′⊗idk(f),∀f∈Mk(C[ξ,s0]).
Let
[TABLE]
Then
[TABLE]
Let m be the winding number of the map
S1∋z⟼* det*(W(z))∈T⊆C.**
*Then W∈U(Mn′(C(S1))) is homotopic to W′∈Mn′(C(S1)) defined by
[TABLE]
Let {wr}21≤r≤1 be a unitary path in Mn′(C(S1)) with
[TABLE]
Evidently the homomorphism
[TABLE]
is homotopic to the homomorphism
[TABLE]
defined by
[TABLE]
— that is, β′′′(f)(eiθ) is the constant matrix f(ξ)1k′′ (which does not depend on θ). There is a path {βr}0≤r≤21 of homomorphisms
[TABLE]
such that β21=β′′ and β0=β′′′.
Finally, regard D={reiθ,0≤r≤1}, and define β:Ik∣[0,s0]→Mn′(C(D)) by
[TABLE]
This homomorphism is as desired.
3.21*.*
Proof of Theorem 3.1
From 3.3—3.20, we have constructed
[TABLE]
with the property
[TABLE]
for all f∈F. And importantly, for each x∈X, ♯(Spψ∣x∩{0})
is a constant k′∈{1,2,⋯,k} and ♯(Spψ∣x∩{1}) is also a constant k1′∈{1,2,⋯,k}, where ψ is the composition
[TABLE]
Let h(t)=t⋅1k∈Ik be the canonical function in the center of Ik. Then ψ(h)∈PM∙(C(X))P is a self adjoint element. For each x∈X, denote the eigenvalues of
ψ(h)(x) by
[TABLE]
Then all λi(x) are continuous functions from X to [0,1]. Furthermore,
[TABLE]
[TABLE]
and
[TABLE]
Let
ξ1=x∈Xminλk′+1(x)>0 and ξ2=maxλrank(P)−k1′(x)<1.
Then
[TABLE]
That is, ψ factors through as
[TABLE]
where we identify Ik∣{0}=C
and Ik∣{1}=C.
Let Q0=α0(1),Q1=α1(1) and P1=ψ1(1Mk(C([ξ1,ξ2]))). Finally, regarding ψ1 as
From the definition of ψ in the above procedure, for every x∈X, the map
[TABLE]
is defined when the construction of
[TABLE]
is carried out for the unique simplex Δ such that x∈Δ∘ (the interior of Δ). And when we define ψ∣Δ by modifying ϕ∣Δ, the only modifications are made on the two parts ϕ∣Δ[0,s0(Δ)] and ϕ∣Δ[tl(Δ),1]. Consequently,
[TABLE]
as sets with multiplicity. On the other hand for any simplex Δ, s0(Δ)<2η and tl(Δ)(Δ)>1−2η. Hence
[TABLE]
If we further assume that ϕ has property sdp(η/4,δ), then ψ has property sdp(η,δ). As a consequence, we can use the decomposition theorem for
[TABLE]
to study the homomorphisms ϕ,ψ:Ik→PM∙(C(X))P.
Note that the homomorphisms f↦f(0)Q0 and f↦f(1)Q1 factor through the C∗-algebra C.
3.23*.*
Lemma 3.20 is not true for the case k′=0. In fact, there exists a unital homomorphism α:Mk(C)→Mk(C(S1)),
which can not be extended to a homomorphism α:Mk(C)→Mk(C(D)).
Let πs0:Ik∣[0,s0]→Mk(C) be the map defined by evaluating at the point s0. Then β=α∘πs0:Ik∣[0,s0]→Mk(C(S1)) can not be extended to β:Ik∣[0,s0]→Mk(C(D)) such that ♯Sp(β∘)x⋂{0}=k′=0 for all x∈D, where is the canonical map from Mk(C) to Ik∣[ζ,s0] for some 0<ζ<s0.
4 Decomposition Theorem II
Our next task is to study the possible decomposition of ϕ:C(X)→Ml(Ik2) for X being [0,1],S1 or TII,k. The cases of [0,1],S1 are more or less known (see [4] and [11]). Let us assume X is a 2-dimensional connected simplicial complex.
The following lemma is essentially due to H. Su (See [40]). The case of X=\noindentgraph was stated in [26].
Lemma 4.1**.**
For any connected simplicial complex X, a finite set F⊂C(X) which generates C(X), η>0 and a positive interger n>0, there is a δ>0, such that for any two unital homomorphisms ϕ,ψ:C(X)→Mn(C), if ∥ϕ(f)−ψ(f)∥<δ for all f∈F,
then Sp(ϕ) and Sp(ψ) can be paired within η.
This is a consequence of Lemma 2.2 and Lemma 2.3 of [40]; also see the argument 2.1.3 in [26]. For the case of graphs, it was stated in 2.1.9 of [26].
Lemma 4.2**.**
For any connected simplicial complex X, a finite generating set F⊂C(X), ε>0 and positive integer n>0, there is δ>0 with the following property: If x1,x2,⋯,xn∈X are n points (possibly repeating), u,v∈Mn(C) are two unitaries such that
[TABLE]
for all f∈F, then there is a path of unitaries ut∈Mn(C) connecting u and v (i.e, u0=u,u1=v) with the property that
[TABLE]
for all f∈F and t,t′∈[0,1] (of course δ depends on both ε and n).
This was proved in step 2 and step 3 of the proof of Theorem 3.1 of [40].
The following lemma reduces the study of ϕ:C(X)→Ml(Ik) to the study of homomorphism ϕ1:C(Γ)→Ml(Ik),
where Γ⊂X is 1-skeleton of X under a certain simplicial decomposition. Since Γ is a graph, then we will apply the technique in [26] and [27] to obtain the decomposition of ϕ1.
Lemma 4.3**.**
Let X be a 2-dimensional simplicial complex. For any F⊂C(X),ε>0,η>0,
and any unital homomorphism ϕ:C(X)→Ml(Ik),
there is a simpicial decomposition of X with 1-skeleton X(1)=Γ and a homomorphism
ϕ1:C(Γ)→Ml(Ik) such that:
1. ∥ϕ(f)−ϕ1∘π(f)∥<ε, where π:C(X)→C(Γ) is given by π(f)=f∣Γ;
2. For any t∈[0,1], Spϕ∣t and Sp(ϕ1∘π)t can be paired within η.
Proof.
By Lemma 4.1, we only need to prove that there exists a homomorphism ϕ1 to satisfy condition (1). Without loss of generality, we assume that F generates C(X). By 4.2, there is an ε′>0 such that for any x1,x2,⋯,xkl∈X and unitaries u,v∈Mkl(C), if
[TABLE]
then there is a continuous path ut with u0=u,u1=v satisfying that
[TABLE]
Recall for the simplicial complex, a continuous path {x(t)}0≤t≤1 is called piecewise linear if there are a sequence of points
[TABLE]
such that {x(t)}ti≤t≤ti+1 fall in the same simplex of X and are linear there. Note that the property of piecewise linear is preserved under any subdivision of the simplicial complex. For the simplicial complex X, we endow the standard metric on X, briefly described as below (see [18, 1.4.1] for detail). Identify each n-simplex with an n-simplex in Rn whose edges are of length 1, preserving affine structure of the simplexes. Such identifications give rise to a unique metric on the simplex Δ. For any two points x,y∈X, d(x,y) is defined to be the length of the shortest path connecting x and y. (The length is measured in individual simplex, by breaking the path into small pieces). With this metric, if x0,x1∈X with d(x0,x1)=d, then there is a piecewise linear path x(t) with length d such that x(0)=x0,x(1)=x1. Furthermore, d(x(t),x(t)′)≤d for all t,t′∈[0,1]. In fact, we can choose x(t), such that
[TABLE]
There is an η′<4η such that the following is true: For any x,x′∈X with d(x,x′)<2η′,
[TABLE]
Let δ>0, such that if ∣t−t′∣≤δ, then
[TABLE]
and Spϕt and Spϕt′ can be paired within η′.
Dividing the interval [0,1] into pieces 0=t0<t1<t2<⋯<t∙=1, with ∣ti+1−ti∣<δ.
We first define ψ:C(X)→Ml(Ik) such that ψ is close to ϕ on F to within 3ε, Spϕt and Spψt can be paired within η′, and with extra property that on each interval [ti,ti+1]; Spψt={α1(t),α2(t),⋯,αlk(t)} with all αj:[ti,ti+1]→X being piecewise linear.
Set ψ∣{ti}=ϕ∣{ti}, for each ti (i=0,1,2,⋯,∙)— that is,
[TABLE]
And we will define ψ∣{t} for t∈(ti,ti+1) by interpolating the definitions between ψ∣{ti} and ψ∣{ti+1}. (Note that we do not change the definitions of ϕ∣{0} and ϕ∣{1}, hence ψ is a homomorphism into Ml(Ik) instead of Mlk(C[0,1]).)
Let
[TABLE]
[TABLE]
Since Spψ∣{ti} and Spψ∣{ti+1} can be paired within η′, we can assume dist(αi,βi)<η′. There exist two unitaries u,v∈Mlk(C) such that
[TABLE]
Note that ∥f(αj)−f(βj)∥<3ε′ for each j, we have
[TABLE]
Combining with ∥ψ(f)(ti)−ψ(f)(ti+1)∥<3ε′, we get
[TABLE]
Since ε′ is the number δ in Lemma 4.2 for 3ε, applying Lemma 4.2, there is a unitary path u(t), ti≤t≤2ti+ti+1
with u(ti)=u,u(2ti+ti+1)=v such that
[TABLE]
for all t,t′∈[ti,2ti+ti+1].
There are piecewise linear paths ri(t) with ri(2ti+ti+1)=αi and ri(ti+1)=βi such that
[TABLE]
Define ψ(f) as follows:
For t∈[ti,2ti+ti+1],
[TABLE]
for t∈[2ti+ti+1,ti+1],
[TABLE]
Then {Spψt,t∈[ti,ti+1]} is a collection of kl piecewise linear maps from [ti,ti+1] to X. (Note that for t∈[ti,2ti+ti+1], we use constant maps which are linear.)
Now subdivid the simplicial complex X so that each simplex of the subdivision has diameter at most η′, and so that all the points in
Spϕ∣{0}=Spψ∣{0} and Spϕ∣{1}=Spψ∣{1} are vertices. With this simplicial decomposition we have,
Spψ∩Δ⫋Δ, for every 2-simplex Δ. This is true because Spψ∣[ti,ti+1] is the union of the collection of images of kl piecewise linear maps from [ti,ti+1] to X, and a finite union of line segments must be 1-dimensional. Hence for each simplex Δ
of dimension 2, we can choose a point xΔ∈Δ∘, such that xΔ∈Spψ.
There is a σ>0 such that Spψ has no intersection with
Bσ(xΔ)={x∈X,dist(x,xΔ)≤σ}
for all Δ.
Let Y=X\backslash\big{(}\cup\{B_{\sigma}(x_{\Delta})~{}|\Delta\mbox{ is 2-simplex}\}\big{)}. Then Spψ⊂Y. That is, ψ factors through C(Y) as
[TABLE]
Let α:Y→X(1) be the standard retraction defined as a map sending Δ\{xΔ} to ∂Δ for each simplex Δ. Then d(x,α(x))<η′.
Let ϕ1:C(X(1))→Ml(Ik) be defined by
[TABLE]
Evidently ϕ1 is as desired.
∎
Corollary 4.4**.**
Suppose that ϕ:C(X)→Ml(Ik) is a unital homomorphism. For any finite set F⊂C(X), ε>0, and η>0, there is a unital homomorphism
[TABLE]
*such that
(1) ϕ(f)(0)=ψ(f)(0),ϕ(f)(1)=ψ(f)(1) for all f∈C(X);
(2) ∥ϕ(f)−ψ(f)∥<ε for all f∈F;
(3) Spϕt and Spψt can be paired to within η;
(4) For each t∈(0,1), the maximal multiplicity of Spψt is one — that is, ψ∣{t} has distinct spectra.*
Proof.
Applying Lemma 4.3, we reduce the case of C(X) to the case of C(X(1)), where X(1) is a 1-dimensional simplicial complex.
The corollary of this case is almost the same as the special case of [26, Theorem 2.1.6] (where we let Y=[0,1]).
Note that from the proof of Theorem 2.1.6 in [26], if we do not require the homomorphism ψ to have distinct spectrum at the end points 0 and 1, then we do not need to modify the original homomorphism ϕ at these two end points. The proof goes the same way as the proof there with some small modifications.
We briefly describe them as below. One divides the interval Y=[0,1] into small pieces [0,1]=∪i=0m−1[yi,yi+1] with y0=0<y1<y2⋯<ym=1, as in the proof of [26, Theorem 2.1.6]. Define ψ∣yi with 1≤i≤m−1, by slightly modifying ϕ∣yi so that ψ∣yi has distinct spectra; but define ψ∣0=ϕ∣0 and ψ∣1=ϕ∣1 (no modification are made at the ending points). Therefore, in our case, ψ∣0 and ψ∣1 do not have distinct spectra—this is the only difference from [26, Theorem 2.1.6]. For all intervals [yi,yi+1] with 1≤i≤m−2, the constructions of ψ∣[yi,yi+1] are the same as in the proof of [26, Theorem 2.1.6]. For the constructions of ψ∣[0,y1] and ψ∣[ym−1,1], we need to modify [26, Lemma 2.1.1] and [27, Lemma 2.1.2] accordingly, in an obvious way, and then apply these modifications. For example, [26, Lemma 2.1.1] should be modified to the following case: among two l-element sets X0={x10,x20,⋯,xl0} and
X1={x11,x21,⋯,xl1} — only one of them is distinct. That is, the following statement is true with the same proof:
Let X=X1∨X2∨⋯∨Xk be a bunch of k intervals Xi=[0,1](1≤i≤k) and Y=[0,1]. Suppose that
[TABLE]
with xi1=xj1 if i=j.
Then there are l continuous functions f1,f2,⋯,fl:Y→X such that
(1) as sets with multiplicity, we have
[TABLE]
(2) for each t∈(0,1]⊂Y and i=j, we have
[TABLE]
∎
Remark 4.5*.*
In Corollary 4.4, we can further assume that Spψ∣{0} and Spψ∣{1} have eigenvalue multiplicity just k as homomorphisms from C(X) to Mlk(C[0,1]), or equivalently, both maps
[TABLE]
have distinct spectrum. To do this, we first extend the definition of the original ϕ to a slightly larger interval [−δ,1+δ] as below.
Find u∈Ml(C) and x1,x2,⋯,xl∈X such that
[TABLE]
Since X is path connected and X={pt}, there are functions αi:[−δ,0]→X such that {αi(−δ)}i=1l
is a set of distinct l points, αi(0)=xi, and dist(αi(t),αi(0)) are as small as we want. Define
Similarly, we can define ϕ(f)(t) for t∈[1,1+δ], so that ϕ∣1+δ as a homomorphism from C(X) to Mkl(C) has multiplicity exactly k and ϕ(f)(1+δ)∈Ml(C)⊗1k. One can reparemetrize [−δ,1+δ] to [0,1] so that ϕ∣0 and ϕ∣1 as homomorphisms from C(X) to Mkl(C)
have multiplicity exactly k. Then we apply the corollary to perturb ϕ to ψ without changing the definition at the end points.
Remark 4.6*.*
The same argument can be used to prove the following result. Let X={pt} be a connected finite simplicial complex of any
dimension. Let Y be a 1-dimensional simplicial complex. Then any homomorphism ϕ:C(X)→Mn(C(Y)) can be approximated arbitrarily well by a homomorphism ψ with distinct spectrum. This is a strengthened form of [18, Theorem 2.1] for the case dim(Y)=1.
The following Theorem for X=\noindentgragh, is a slight modification of [28, Theorem 2.7].
Theorem 4.7**.**
Let X be a connected simplicial complex of dimension at most 2, and G⊂C(X) be a finite set which generates C(X). For any ε>0, there is an η>0 such that the following statement is true.
Suppose that ϕ:C(X)→Ml1l2+r(Ik) is a unital homomorphism satisfying the following condition:
There are l1 continuous maps
[TABLE]
such that for every y∈[0,1], Spϕy (considered as a homomorphism from C(X) to M(l1l2+r)k(C[0,1])) and Θ(y) can be paired within η, where
[TABLE]
*It follows that there are l1 mutually orthogonal projections p1,p2,⋯,pl1∈Ml1l2+r(Ik) such that
(i) for all g∈G and y∈Y*
[TABLE]
*where p0=1−∑i=1l1pi;
(ii) rank(pi)=(l2−3)k for 1≤i<l1,
rank(pl1)=(l2+r−3)k (as projections in M(l1l2+r)k(C[0,1])) and rank(p0)=3l1k.*
Proof.
We will apply [28, Theorem 2.7] (using map ai to replace map b∘ai as in [28, Remark 2.8]) and its proof (see 2.9-2.16 of [28]) for the case Y in [28, Theorem 2.7] being [0,1]. As a matter of fact, in the proof of [28, Theorem 2.7], Li does use that X to be graph, for only one property that any homomorphism from C(X) to MnC(Y) (Y graph) can be approximated arbitrarily well by homomorphisms with distinct spectra.
By Remark 4.6, [28, Theorem 2.7]
holds for the case X={pt} being any connected simplicial complex and Y, a graph.
For finite set G⊂C(X), and ε>0, choose η>0 such that dist(x1,x2)≤η
implies ∣g(x1)−g(x2)∣<4ε for all g∈G, as in [28, 2.16]. Without lose of generality, we can assume that the Spϕ∣t is distinct for any t∈(0,1) and Spϕ∣0 and Spϕ∣1 have multiplicities exact k as in Corollary 4.4 and Remark 4.5 above. When we go through Li’s proof in [28], we need
to make the projections pi to satisfy the extra condition:
[TABLE]
We will repeat part of the proof of [28, Theorem 2.7] and point out how to modify it.
As in the proof of [28, Theorem 2.7], we can choose an open cover U0,U1,⋯,U∙ of [0,1]
with
[TABLE]
[TABLE]
We will define PUi(i=1,2,⋯,l1) as same as in [28, 2.12] for U=Ui(0<i<∙)—note that Spϕy, for y∈(a1,b∙−1)⊂(0,1), are distinct. For U0 and U∙, a special care is needed as follows. We will only do it for U0
(it is the same for U∙).
Write Spϕ∣0={λ1∼k,λ2∼k,⋯,λq∼k} with q=l1l2+r. Then {λ1,λ2,⋯,λq} can be paired with {a0(0)∼l2,a2(0)∼l2,⋯,al1−1(0)∼l2,al1(0)∼(l2+r)} (note ∼l2k is changed to ∼l2 here) to within η. We can divide {λ1,λ2,⋯,λq} into groups {λ1,λ2,⋯,λq}=⋃j=1l1E′j (where ∣E′j∣=l2
if 1≤j≤l1−1, and ∣E′j∣=l2+r if j=l1) such that dist(λi,aj(0))<η, for all λi∈Ej.
Let σ′ satisfy the following conditions:
(1) σ′<min{dist(λi,λj),i=j};
(2) σ′<η−max{dist(λi,aj(0)),λi∈Ej}.
We can choose b1(>b0>a1>0) being so small that for any y∈[0,b1], Spϕy and Spϕ0 can be paired to within 2σ′
and dist(aj(y),aj(0))<2σ′. Then for each y∈[0,b1], Spϕy can be written as a set of
[TABLE]
with λij(0)=λi. Then let Ej(y) be the set {λii′(y); λi∈E′j}. In this way we have, if λii′∈Ej, then
[TABLE]
Let both PU0j(y) and PU1j(y) (defined on U0=[0,b0) and U1=(a1,b1)) be the spectral projections corresponding to Ej(y). In particular, PU0j(0)∈Ml1l2+r(C)⊗1k. We can define pj(y) as a subprojection of PUj(y)
(for U∋y) as in 2.9-2.16 of [28] for each y∈[b0,a∙] but with rank (pj(y))=(l2−3)k (instead of l2−3 in [28]) for 1≤j≤l1−1 and rank(pl1(y))=(l2+r−3)k (instead of l2+r−3 in [28]). Also we can choose an arbitrary sub projection pj(0)<PU0j(0)∈Ml1l2+r(C)⊗1k of form
pj(0)=pj′(0)⊗1k∈Ml1l2+r(C)⊗1k
with rank(pj′(0))=l2−3 for 1≤j≤l1−1, and
rank(pl1′(0))=l2+r−3. Consequently,
[TABLE]
Finally, connect pj(0) and pj(b0) by pj(y) for y∈[0,b0] inside PU0j(y). As one can see from 2.16 of [28], if the projections pj(y) are subprojections of PUj(y), then all the estimations in that proof hold. After we do similar modifications for PU∙j(y) and pj(y) near point 1, we will get pj(y)∈Ml1l2+r(Ik)
instead of M(l1l2+r)k(C[0,1]). (This method was also used in the proof of [13, Theorem 3.10].)
∎
The following result is a generalization of [18, Proposition 4.42].
Theorem 4.8**.**
Let X be a connected finite simplicial complex of dimension at most 2, ε>0 and F⊂C(X), a finite set of generators. Suppose that η∈(0,ε) satisfies that if dist(x,x′)≤2η, then ∥f(x)−f(x′)∥<4ε for all f∈F.
For any δ>0 and positive integer J>0, there exist an integer L>0 and a finite set H⊆AffTC(X)(=CR(X)) such that the following holds.
*If ϕ,ψ:C(X)→B=MK(Ik) (or B=PM∙(C(Y))P) are unital homomorphisms with the properties:
(a) ϕ has sdp(η/32,δ);
(b) K≥L (or rank(P)≥L);
(c) ∥AffTϕ(h)−AffTψ(h)∥<4δ, for all h∈H,
then there are three orthogonal projections Q0,Q1,Q2∈B,
two homomorphisms ϕ1∈Hom(C(X),Q1BQ1)1 and ϕ2∈Hom(C(X),Q2BQ2)1, and a unitary u∈B such that
(1) 1B=Q0+Q1+Q2;
(2) \|\phi(f)-\big{(}Q_{0}\phi(f)Q_{0}+\phi_{1}(f)+\phi_{2}(f)\big{)}\|<\varepsilon and
\|(Adu\circ\psi)(f)-\big{(}Q_{0}(Adu\circ\psi)(f)Q_{0}+\phi_{1}(f)+\phi_{2}(f)\big{)}\|<\varepsilon, for all f∈F;
(3) ϕ2 factors through C[0,1];
(4) Q1=p1+⋯+pn with (rank(Q0)+2)J<rank(pi)(i=1,2,⋯,n), where rank: K0(B)→Z is the map induced on K0 by the evaluation map at 0 or 1. (which is rankpi(0) for B=MK(Ik), where rankpi(0) is regarded as projections in MK(C) not MK(Mk(C))), and ϕ1 is defined by*
[TABLE]
where p1,p2,⋯,pn are mutually orthogonal projections and
{x1,x2,⋯,xn}⊂X is an ε-dense subset of X.
Proof.
For the case B=PM∙(C(Y))P, this is [18, Proposition 4.42].
The proof for the case B=MK(Ik) is almost the same as the proof of [18, Proposition 4.42], replacing [18, Theorem 4.1] by
Theorem 4.7 above. The only thing one should notice is that, in [18, Lemma 4.33], rankϕ(1)=K, the K should be corresponding to K in our theorem (not Kk) and Θ(y) should be defined as
[TABLE]
(Note in the above, we use ∼L2k and ∼(L2+L1)k to replace ∼L2 and ∼(L2+L1) in [18].) In the proof of this version of [18, Lemma 4.33], one can choose the homomorphism ψ′:C(X)→Mk(C[0,1]) (not to MKk(C[0,1])) as the map ψ there, with
[TABLE]
as in [18, Lemma 4.33]. Then let ψ=ψ′⊗k, where k:C→Mk(C) is defined by k(λ)=λ⋅1k. With this modification, we have Spψy′ being
[TABLE]
and Spψy being
[TABLE]
as desired. All other parts of the proof are exactly the same.
∎
For the proof of uinqueness theorem in [19], it is important to have a simultaneous decomposition for two homomorphisms as below.
Theorem 4.9**.**
Let X be a connected finite simplicial complex of dimension at most 2, ε>0 and F⊂C(X), a finite set of generators. Suppose that η∈(0,ε) satisfies that if dist(x,x′)≤2η, then ∥f(x)−f(x′)∥<4ε for all f∈F. Let κ be a fixed simplicial structure of X.
For any δ>0 and positive integer J>0, there exist an integer L>0 and a finite set H⊆AffTC(X)(=CR(X)) such that the following holds.
*If X1 is a connected sub-complex of (X,κ), and if ϕ,ψ:C(X1)→B=MK(Ik) (or B=PM∙(C(Y))P) are unital homomorphisms with the following properties:
(a) ϕ has sdp(η/32,δ);
(b) K≥L (or rank(P)≥L);
(c) ∥AffTϕ(h∣X1)−AffTψ(h∣X1)∥<4δ, for all h∈H,
then there are three orthogonal projections Q0,Q1,Q2∈B, two homomorphisms ϕ1∈Hom(C(X1),Q1BQ1)1 and ϕ2∈Hom(C(X1),Q2BQ2)1, and a unitary u∈B such that
(1) 1B=Q0+Q1+Q2;
(2) \|\phi(f|_{X_{1}})-\big{(}Q_{0}\phi(f|_{X_{1}})Q_{0}+\phi_{1}(f|_{X_{1}})+\phi_{2}(f|_{X_{1}})\big{)}\|<\varepsilon and
\|(Adu\circ\psi)(f|_{X_{1}})-\big{(}Q_{0}(Adu\circ\psi)(f|_{X_{1}})Q_{0}+\phi_{1}(f|_{X_{1}})+\phi_{2}(f|_{X_{1}})\big{)}\|<\varepsilon for all f∈F;
(3) ϕ2 factors through C[0,1];
(4) Q1=p1+⋯+pn with (rank(Q0)+2)J<rank(pi)(i=1,2,⋯,n),
and ϕ1 is defined by*
[TABLE]
where p1,p2,⋯,pn are mutually orthogonal projections and
{x1,x2,⋯,xn}⊂X1 is an ε-dense subset of X1.
Proof.
Suppose that {Xi}i are all connected sub-complexes of (X,κ) (there are finitely many of them for a fixed simplicial structure of a finite complex). Apply Theorem 4.7 to each Xi to obtain Li and Hi⊆AffT(C(Xi)) as in the theorem. By Tietze Extension Theorem, there are finite sets H~i⊆AffT(C(X)) such that Hi⊆{h∣Xi∣h∈H~i}. Evidently L=maxi{Li} and H=∪iH~i are as desired.
∎
Acknowledgement The authors would like to express our special thanks of gratitude to Professor Guihua Gong who suggested us to do this interesting problem.
We also benefit a lot from discussions with him.
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