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# Approximately diagonalizing matrices over *C*(*Y*)

Edited by Richard V. Kadison, University of Pennsylvania, Philadelphia, PA, and approved November 8, 2011 (received for review January 24, 2011)

## Abstract

Let *X* be a compact metric space which is locally absolutely retract and let *φ*: *C*(*X*) → *C*(*Y*,*M*_{n}) be a unital homomorphism, where *Y* is a compact metric space with dim *Y* ≤ 2. It is proved that there exists a sequence of *n* continuous maps *α*_{i,m}: *Y* → *X* (*i* = 1,2,…,*n*) and a sequence of sets of mutually orthogonal rank-one projections {*p*_{1,m},*p*_{2,m},…,*p*_{n,m}}⊂*C*(*Y*,*M*_{n}) such that This is closely related to the Kadison diagonal matrix question. It is also shown that this approximate diagonalization could not hold in general when dim *Y*≥3.

Over two decades ago, Richard Kadison proved that a normal element in , where is a von-Neumann algebra, can be diagonalized [(1) and (2)]. He showed that this cannot be done if is replaced by a unital *C*^{∗}-algebra in general. He then asked what topological properties of a compact metric space *Y* will guarantee that every normal element in *M*_{n}(*C*(*Y*)) can always be diagonalized. Karsten Grove and Gert K. Pedersen (3) showed that this could not go very far. They demonstrated that *Y* has to be sub-Stonean and dim *Y* ≤ 2 if every self-adjoint element can be diagonalized in *M*_{n}(*C*(*Y*)). Furthermore, they showed that, even for sub-Stonean spaces *Y* with dim *Y* ≤ 2, one still could not diagonalize a normal element in general. In fact, they showed that in order to have every normal element in *M*_{n}(*C*(*Y*)) diagonalized, one must have that every finite covering space over each closed subset of *Y* is trivial and every complex line bundle over each closed subset of *Y* is trivial, in addition to the requirements that *Y* is sub-Stonean and dim *Y* ≤ 2. So not every sub-Stonean space *Y* with dimension at most two has the property that every normal element can be diagonalized. Since sub-Stonean spaces are not the everyday topological spaces with dimension at most two, it seems that the question of diagonalizing normal elements in *M*_{n}(*C*(*Y*)) has a rather negative answer.

However, in the decades after the original question was raised and answered, it seems that approximately diagonalizing some normal elements or some commutative *C*^{∗}-subalgebras in *M*_{n}(*C*(*Y*)), where *Y* is a lower dimensional nice topological space, becomes quite useful and important. In this paper, instead of considering exact diagonalization of commutative *C*^{∗}-subalgebras in *M*_{n}(*C*(*Y*)), we study the problem whether a unital homomorphism *φ*: *C*(*X*) → *M*_{n}(*C*(*Y*)) can be approximately diagonalized. To be precise, we formulate as follows: Let *ϵ* > 0 and a compact set be given. Are there continuous maps *α*_{i}: *Y* → *X* (1 ≤ *i* ≤ *n*) and mutually orthogonal rank-one projections *p*_{1},*p*_{2},…,*p*_{n}∈*M*_{n}(*C*(*Y*)) such that [1]

The main result that we report here is that the answer to [**1**] is affirmative for any compact metric space *X* which is locally absolutely retract (see 6.1) and any compact metric space *Y* with dim *Y* ≤ 2. Moreover, we show that, the answer is negative for general compact metric space *Y* with dim ≥3. In fact, a unitary in *M*_{2}(*C*(*S*^{3})) may not be approximately diagonalized. We also show that if dim *Y* > 3, then there are self-adjoint elements with spectrum [0,1] in *M*_{n}(*C*(*Y*)) which cannot be approximately diagonalized. For more general compact metric space *X*, we show that, for any *ϵ* > 0, any compact subset , there is a unital commutative diagonal *C*^{∗}-subalgebra *B*⊂*M*_{n}(*C*(*Y*)) such that provided that dimY ≤ 2 (see 6.5).

## 2 Approximate Homomorphisms

Let *X* be a compact metric space and let *n*≥1 be an integer. Let be a unital homomorphism. Then there exists an infinite subsequence such that the induced homomorphism *H*^{′}: *C*(*X*) → *C*^{b}(*S*,*M*_{n})/*C*_{0}(*S*,*M*_{n}) has finite spectrum.

Put . Denote by the quotient map. Let *ξ*_{1}∈*X* be a point in the spectrum of *H*. Let Then is a *σ*-unital hereditary *C*^{∗}-subalgebra and it is not *A*_{0}, since *ξ*_{1} is in the spectrum of *H* and *H*(*I*_{1}) is a proper closed ideal of *H*(*C*(*X*)). Note that *A*_{0} is the corona algebra of the separable *C*^{∗}-algebra . It follows from a theorem of G.Pedersen [Th.15 of (7)] that . Since is a hereditary *C*^{∗}-subalgebra of *A*_{0} and *A*_{0} has real rank zero, there is a nonzero projection . It follows that [2]It is standard that there exists a sequence of projections {*p*_{1}(*m*)}⊂*M*_{n} such that *π*({*p*_{1}(*m*)}) = *p*_{1}. Let be the subsequence so that *p*_{1}(*m*) ≠ 0 for all *m*∈*S*_{1}. Note that *S*_{1} must be infinite. Let *A*_{1} = *A*_{0}/*J*_{1}, where One also has that *A*_{1} ≅ *C*^{b}(*S*_{1},*M*_{n})/*C*_{0}(*S*_{1},*M*_{n}). Let Φ_{1}: *A*_{0} → *A*_{1} be the quotient map and define *H*_{1} = Φ_{1}∘*H* If *ξ*_{1} is the only point in the spectrum of *H*_{1}, the lemma follows. Otherwise, let *ξ*_{2} ≠ *ξ*_{1} be another point in the spectrum of *H*_{1}. Let From the above argument, one obtains a nonzero projection . Then Φ_{1}(*p*_{1})*p*_{2} = 0 and [3]There exists a projection {*p*_{2}(*m*)}∈*C*^{b}(*S*_{1},*M*_{n}) such that *π*_{1}({*p*_{2}(*m*)}) = *p*_{2} and [4]for all *m*, where *π*_{1}: *C*^{b}(*S*_{1},*M*_{n}) → *A*_{1} is the quotient map. Let *S*_{2}⊂*S*_{1} be such that *p*_{2}(*m*) ≠ 0 for all *m*∈*S*_{2}. Then *S*_{2} is an infinite subset. Let Put *A*_{2} = *A*_{1}/*J*_{2} and let Φ_{2}: *A*_{1} → *A*_{2} be the quotient map. Note that *A*_{2} ≅ *C*^{b}(*S*_{2},*M*_{n})/*C*_{0}(*S*_{2},*M*_{n}). Moreover, [5]Define *H*_{2} = Φ_{2}∘*H*_{1}. Then *χ*_{1} *χ*_{2} are in the spectrum of *H*_{2}. If the spectrum of *H*_{2} contains only these two points, the lemma follows. Otherwise, we continue. However, since there can be no more than *n* mutually orthogonal nonzero projections in *M*_{n}, from [**4**] and [**5**], this process has to stop at the stage *n* or earlier. At that point, one obtains an infinite subset , for which *H*^{′}: *C*(*X*) → *C*^{b}(*S*,*M*_{n})/*C*_{0}(*S*,*M*_{n}) has finite spectrum.

Let *X* be a compact metric space, let *n*≥1 be an integer and let *M* > 0. For any *ϵ* > 0 and any finite subset , there exists *δ* > 0 and a finite subset satisfying the following: for any unital map *φ*: *C*(*X*) → *M*_{n} with ||*φ*(*f*)|| ≤ *M* for all *f*∈*C*(*X*) with ||*f*|| ≤ 1 such that [6][7]for all with |*λ*_{i}| ≤ 1 (*i* = 1,2) and *x*, , there exists a unital homomorphism *ψ*: *C*(*X*) → *M*_{n} such that [8]

## 3 Commutative *C*^{∗}-Subalgebras of Matrix Algebras

Let *X* be a compact metric space and let *n*≥1 be an integer. Then, for any *ϵ* > 0 and any finite subset , there exist a finite subset and *δ* > 0 satisfying the following. Let *φ*, *ψ*: *C*(*X*) → *M*_{n} be two unital homomorphisms for which [9]where *tr* is the normalized tracial state. Then there exists a unitary *u*∈*M*_{n} such that [10]

Let *X* be a compact metric space which is locally path connected and let *n*≥1. Then, for any *ϵ* > 0, *ϵ*_{1} > 0 and any finite subset , there exist a finite subset *C*(*X*) and *δ* > 0 satisfying the following. Let *φ*, *ψ*: *C*(*X*) → *M*_{n} be two unital homomorphisms for which [11]where *tr* is the normalized tracial state. Then there exists a unital homomorphism Φ: *C*(*X*) → *C*([0,1],*M*_{n}) such that [12]and there exists a unitary *u*∈*M*_{n} such that [13]Moreover, there are continuous maps *α*_{i}: [0,1] → *X* (*i* = 1,2,…,*n*) and mutually orthogonal rank-one projections {*p*_{1},*p*_{2},…,*p*_{n}}⊂*M*_{n} such that [14]and [15]

Let *ϵ* > 0, *n*≥1 be an integer and *M* > 0. There exists *δ* > 0 satisfying the following: For any finite subset with ||*a*|| ≤ *M* for all , and a unitary *u*∈*M*_{n} such that there exists a continuous path of unitaries {*u*(*t*):*t*∈[0,1]}⊂*M*_{n} with *u*(0) = *u* and *u*(1) = 1 such that Moreover,

Note that contains an arc with length at least 2*π*/*n*. Thus the lemma follows immediately from Lemma 2.6.11 of (12).

## 4 Commutative *C*^{∗}-Subalgebras in Matrix Algebras Over One-Dimension Spaces

Let *X* be a path connected finite CW complex, let *C* = *C*(*X*) and let *A* = *C*([0,1],*M*_{n}). For any *ϵ* > 0 and any finite subset , there exists a finite subset and *δ* > 0 satisfying the following: Let *φ*, *ψ*: *C* → *A* be two unital homomorphisms such that [16]

Then there exists a unitary *U*∈*A* such that [17]for all .

Moreover, if, in addition, *φ*(*f*)(0) = *ψ*(*f*)(0), or *φ*(*f*)(0) = *ψ*(*f*)(0) and *φ*(*f*)(1) = *ψ*(*f*)(1) for all *f*∈*C*(*X*) then one may assume that *U*(0) = 1_{Mn}, or *U*(0) = *U*(1) = 1, respectively.

Let *X* be a connected CW complex and let *n*≥1. Fix a unital homomorphism *h*_{0}: *C*(*X*) → *M*_{n} given by where {*ξ*_{1},*ξ*_{2},…,*ξ*_{m}} (*m* ≤ *n*) is a subset of *m* distinct points in *X* and {*e*_{1},*e*_{2},…,*e*_{m}} is a set of mutually orthogonal nonzero projections. For any *ϵ* > 0, there exists *δ* > 0 and a finite subset satisfying the following: Suppose that *Y* is a connected compact metric space, *φ*: *C*(*X*) → *C*(*Y*,*M*_{n}) is a unital homomorphism and *y*_{0}∈*Y* for which [18][19]and there are continuous maps *x*_{i}: *Y* → *X* (*i* = 1,2,…,*n*) and mutually orthogonal rank-one projections {*q*_{1},*q*_{2},…,*q*_{n}}⊂*C*(*Y*,*M*_{n}) such that [20]

Then, there is a partition {*S*_{1},*S*_{2},…,*S*_{m}} of {1,2,…,*n*} such that *ξ*_{i} = *x*_{j}(*y*_{0}) for some *j*∈*S*_{i}, dist(*ξ*_{i},*x*_{j}(*y*)) < *ϵ* for all *j*∈*S*_{i}, for all *j*∈*S*_{i}, [21]*i* = 1,2,…,*m*.

Let *X* be a compact metric space, let *n*≥1 be an integer and let *t*∈*X* be a point. In what follows, denote by *π*_{t}: *C*(*X*,*M*_{n}) → *M*_{n} the point-evaluation homomorphism defined by *π*_{t}(*f*) = *f*(*t*) for all *f*∈*C*(*X*).

Let *X* be a locally path connected compact metric space and let *n*≥1 be an integer. Then, for any *ϵ* > 0, *η* > 0, and any finite subset , there exist *δ* > 0 and a finite subset satisfying the following: if *φ*_{1}, *φ*_{2}: *C*(*X*) → *M*_{n} are two unital homomorphisms for which [22]then there is a unital homomorphism Φ: *C*(*X*) → *C*([0,1],*M*_{n}) and there are continuous maps *α*_{i}: [0,1] → *X* (1 ≤ *i* ≤ *n*) and mutually orthogonal rank-one projections {*p*_{1},*p*_{2},…,*p*_{n}}⊂*C*([0,1],*M*_{n}) such that *π*_{0}∘Φ = *φ*_{1}, *π*_{1}∘Φ = *φ*_{2}, [23]and [24]Moreover, [25]

Let *X* be a connected finite CW complex and let *n*≥1. Let *Y* be a finite CW complex of dimension 1 and let *φ*: *C*(*X*) → *C*(*Y*,*M*_{n}). Then, for any *ϵ* > 0 and any finite subset , there exist mutually orthogonal rank-one rank projections *p*_{1},*p*_{2},…,*p*_{n}⊂*C*(*Y*,*M*_{n}) and continuous maps *α*_{i}: *Y* → *X* such that [26]

Moreover, if {*y*_{1},*y*_{2},…,*y*_{L}} is fixed, then one can also require that [27]*l* = 1,2,…,*L*.

## 5 Approximate Homomorphisms and the Basic Homotopy Lemma

Let *X* be a locally path connected compact metric space and let *n*≥1 be an integer. Then, for any *ϵ* > 0 and any finite subset , there exist *δ* > 0 and a finite subset satisfying the following: for any unital map *φ*: *C*(*X*) → *C*([0,1],*M*_{n}) with ||*φ*(*f*)|| ≤ *M* for all ||*f*|| ≤ 1 such that [28][29]for all *λ*_{1}, with |*λ*_{i}| ≤ 1 (*i* = 1,2) and *x*, , there exists a unital homomorphism *ψ*: *C*(*X*) → *C*([0,1],*M*_{n}) such that [30]If, moreover, *π*_{0}∘*φ* is a unital homomorphism, or both *π*_{0}∘*φ* and *π*_{1}∘*φ* are unital homomorphisms, then *ψ* can be so chosen that *π*_{0}∘*ψ* = *π*_{0}∘*φ* (or *π*_{0}∘*ψ* = *π*_{1}∘*φ* and *π*_{1}∘*φ* = *π*_{1}∘*ψ*).

Let *X* be a compact metric space and let *n*≥1 be an integer. For any *ϵ* > 0 and any finite subset , there exist a finite subset and *δ* > 0 satisfying the following:

Suppose that *φ*: *C*(*X*) → *C*([0,1],*M*_{n}) is a unital homomorphism and *u*∈*C*([0,1],*M*_{n}) such that [31]Then there exists a continuous path of unitaries {*U*(*s*):*s*∈[0,1]} in *C*([0,1],*M*_{n}) with *U*(0) = *u* and *U*(1) = 1 such that [32]and for all *s*∈[0,1]. Moreover, if [33]for all *f*∈*C*(*X*), then one can choose *U* so that [34]for all *f*∈*C*(*X*).

## 6 Approximate Diagonalization

Let *Y* be a compact metric space. Recall that *Y* is a locally absolute retract, if, for any *y*∈*Y* and any *ϵ*_{1} > 0, there exist *ϵ*_{1} > *ϵ*_{2} > 0 and a closed neighborhood *Z* of *y* such that *B*(*y*,*ϵ*_{2})⊂*Z*⊂*B*(*y*,*ϵ*_{1}) and *Z* is an absolute retract.

Let *X* be a compact metric space which is locally absolutely retract and *n*≥1 be an integer. Let be the unit circle. Let *ϵ* > 0 and be a finite subset. Suppose that is a unital homomorphism satisfying the following:

, where are continuous maps and are mutually orthogonal rank-one projections;

, where {

*x*_{1},*x*_{2},…,*x*_{m}}⊂*X*are distinct points, {*e*_{1},*e*_{2},…,*e*_{m}} is a set of mutually orthogonal nonzero projections,There is a partition {

*S*_{1},*S*_{2},…,*S*_{m}} of {1,2,…,*n*} such that, for*s*∈*S*_{i}, [35]*B*(*x*_{i},*η*_{1})⊂*Z*_{i}⊂*B*(*x*_{i},*η*_{2}/4) and*Z*_{i}is a compact subset which is also absolutely retract,*i*= 1,2,…,*m*, where if dist(*x*,*x*^{′}) <*η*_{2}, and where*δ*_{0}(in place of*δ*) and associated with*ϵ*/2 (in place of*ϵ*) and required by 5.2;[36]

Then, there exist continuous maps

*γ*_{i}:*D*→*X*and mutually orthogonal rank-one projections {*q*_{1},*q*_{2},…,*q*_{n}}⊂*C*(*D*,*M*_{n}) such that, for any and any*y*∈*D*, [37][38]*i*= 1,2,…,*n*, where*D*is the unit disk.

Let *X* be a compact metric space which is a locally absolute retract and let *n*≥1. Suppose that *Y* is a compact metric space with dim *Y* ≤ 2 and suppose that *φ*: *C*(*X*) → *C*(*Y*,*M*_{n}) is a unital homomorphism. Then, for any *ϵ* > 0 and any finite subset , there exist continuous maps *α*_{i}: *Y* → *X* (1 ≤ *i* ≤ *n*) and mutually orthogonal rank-one projections *e*_{1},*e*_{2},…,*e*_{n}∈*C*(*Y*,*M*_{n}) such that [39]

Fix *ϵ* > 0 and a finite subset . Let *δ*_{0} > 0 (in place of *δ*) and (in place of ) be a finite subset associated with *ϵ*/16 and required by (5.1) (for the given *n*). One may assume that . Let *η* > 0 be such that [40]provided that dist(*x*,*x*^{′}) < *η*. Since *X* is locally absolute retract and compact, there exists *η*_{1} > 0 such that, for any *x*∈*X*, *B*(*x*,*η*_{1})⊂*Z*_{x}⊂*B*(*x*,*η*/2), where *Z*_{x} is a compact subset which is also an absolute retract. For each *y*∈*Y*, let *δ*_{1}(*y*) > 0 (in place of *δ*) and (in place of ) be a finite subset associated with min{*η*_{0}/3,*ϵ*/16} (in place of *ϵ*) and *π*_{y}∘*φ* required by (4.2).

For each *y*∈*Y*, let *δ*_{2}(*y*) > 0 (in place of *δ*) and (in place of ) be a finite subset associated with min{*η*_{0}/3,*ϵ*/6} (in place of *ϵ*) and *π*_{y}∘*φ* (in place of *φ*_{1}) required by (4.4). Without loss of generality, to simplify notation, we may assume that *δ*_{1}(*y*) ≤ *δ*_{2}(*y*) and .

For each *y*, there exists *d*^{′}(*y*) > 0 such that [41]provided that dist(*y*,*y*^{′}) < *d*^{′}(*y*).

Fix *r* > 0. For each *y*∈*Y*, let *d*(*y*) = *d*^{′}(*y*)*r*. Now ∪_{y∈Y}*B*(*y*,*d*(*y*)/12)⊃*Y*. Let *y*_{1},*y*_{2},…,*y*_{K}∈*Y* be a finite subset such that {*B*(*y*_{i},*d*(*y*_{i})/12):*i* = 1,2,…,*K*} covers *Y*. Moreover, one may assume that the order of the cover ≤ 2. One builds a simplicial complex as follows: *y*_{1},*y*_{2},…,*y*_{N} are vertices and 0-simplexes, and *y*_{i1}*y*_{i2} or *y*_{i1}*y*_{i2}*y*_{i3} form a 1-simplex (or 2-simplex) if and only if [42][and [43]].

Denote by the simplicial complexes constructed this way and by *S*(*r*) the underline polyhedra. Moreover, if *y*_{i}*y*_{j} is a 1-simplex, then [44]

If *y*_{j} is a vertex, then there are points , *k* = 1,2,…,*n*, and mutually orthogonal rank-one projections such that [45]

Denote by *I*_{i,j} the line segment defined by *y*_{i}*y*_{j}. Therefore, by applying (4.4), there are continuous maps , *k* = 1,2,…,*n*, and mutually orthogonal rank-one projections such that [46][47]where *s* = *i*, *j* and *s*^{′} = *i*, or *j* if max{*d*(*y*_{i})/6,*d*(*y*_{j})/6} = *d*(*y*_{i})/6, or max{*d*(*y*_{i})/6,*d*(*y*_{j})/6} = *d*(*y*_{j})/6.

Let *I*(*r*) = ∪*I*_{i,j} be the union of all 0-simplex and 1-simplexes in *S*(*r*). One obtains a unital homomorphism Φ^{′}: *C*(*X*) → *C*(*I*(*r*),*M*_{n}) defined by [48]if *t*∈*I*_{i,j}, and [49]Define by if *t*∈*I*_{i,j} and define projections in *C*(*I*(*r*),*M*_{n}) by if *t*∈*I*_{i,j}. Next one extends *π*_{t}∘Φ^{′} on *S*(*r*).

To do this, one assumes that *y*_{i1}*y*_{i2}*y*_{i3} is a 2-simplex. Then [50]and for one of some *j*^{′}∈{1,2,3}. Without loss of generality, one may assume that 3 = *j*^{′}.

One identifies the 2-polyhedron *K*_{i1,i2,i3} determined by *y*_{i1}*y*_{i2}*y*_{i3} with the unit disk and identifies *y*_{i1} with 1, *y*_{i2} with -1 and *y*_{i3} with . Here the line segments determined by *y*_{i1}*y*_{i2}, *y*_{i1}*y*_{i3} and *y*_{i2}*y*_{i3} with the arc with endpoints -1 and 1, the arc with endpoints 1 and , and the arc with endpoints and -1, respectively.

Let Ψ be the restriction of Φ^{′} on the unit circle (with the above-mentioned identification). Then it is clear that Ψ satisfies (1), (2) and (4) in (4.2) (by replacing *φ* by Ψ) for *ϵ*/4 (in place of *ϵ*) and . By the choice of each *δ*_{1}(*y*) and by (4.2), (3) is also satisfied (for Ψ). By applying (6.2), and identifying the unit dick *D* with *K*_{i1,i2,i3}, one obtains a unital homomorphism Φ_{i1,i2,i3}: *C*(*X*) → *C*(*K*_{i1,i2,i3},*M*_{n}), continuous maps and mutually orthogonal rank-one projections such that (where *K*_{ij,ij′} is the 1-simplex determined by *y*_{ij}*y*_{ij′}) [51][52]and [53]for all *t* in the boundary of *K*_{i1,i2,i3}, *s*∈*K*_{i1,i2,i3} and for all . Define *α*_{k}: *Y* → *S*(*r*) by *α*_{k}(*y*_{j}) = *y*_{j}, if *y* is in the polyhedron determined by *y*_{i}*y*_{j} and if *y*∈*K*_{i1,i2,i3}. Define *p*_{k}∈*C*(*Y*,*M*_{n}) by , if *y*∈*K*_{i,j} and if *y*∈*K*_{i1,i2,i3}. Define *ψ*: *C*(*X*) → *C*(*S*(*r*),*M*_{n}) by [54]Note that *ψ*(*f*)(*t*) = Φ_{i1,i2,i3}(*f*)(*t*) if *t*∈*K*_{i1,i2,i3} and if *t*∈*K*_{i,j}. Moreover, [55]and [56]and for some *j* so that *y* is in a simplex with *y*_{j} as one of the vertices.

Now one changes *r*. To simplify notation, one may assume that diam(*Y*) ≤ 1. One obtains a sequence of open covers [with *d*(*y*,*r*_{m}) = *d*^{′}(*y*)*r*_{m}] such that: (*i*) the order of the cover is at most 2, and,

(*ii*) [57]where *ϵ*_{m} is a Lebesque number for the cover . It follows from (*ii*) that (*iii*) holds: if , then there exists *k* ≤ *K*(*k*) such that , *s* = 1,2,…,*l*. For each *m* = 1,2,…, let be the simplicial complex constructed from points {*y*_{1},*y*_{2},…,*y*_{K(m)}} as above, and let *S*(*r*_{m}) be the underline polyhedra of dimension at most 2 (see [**42**] and [**43**]). Denote by *ψ*_{m}: *C*(*X*) → *C*(*S*(*r*_{m}),*M*_{n}) the unital homomorphism constructed above using *r* = *r*_{m}, *m* = 1,2,…..

To specify the map , for each *j* (≤ *K*(*m* + 1)), let be one of the vertex. By virtue of (*iii*) above, the family is nonempty. Since , the vertices of *S*_{m} which correspond to the members of span a simplex . Define [58]where *b*(*K*^{(j,m)}) denotes the barycenter of *K*^{(j,m)}. As in the proof 1.13.2 of (14), this implies that, for every simplex , the images of vertices of *S* under are contained in a simplex .

This construction leads to an inverse limits which is homeomorphic to *Y* [see the proof of 1.13.2 of (14)]. One identifies these two spaces. Denote by the continuous map induced by the inverse limit.

Denote by *J*_{m}: *C*(*S*(*r*_{m}),*M*_{n}) → *C*(*S*(*r*_{m+1}),*M*_{n}) the unital homomorphism defined by [59]*m* = 1,2,…. Denote by *J*_{m,∞}: *C*(*Y*,*M*_{n}) → *C*(*S*(*r*_{m}),*M*_{n}) the unital homomorphism induced by the inductive limit which can also be defined by for all *f*∈*C*(*Y*,*M*_{n}).

Fix *y*∈*Y* and *m*. There is a simplex such that and therefore there exists a vertex such that [60][see for example the proof of 1.13.2 of (14)]. Let *ψ*_{m}: *C*(*X*) → *S*(*r*_{m}) be the unital homomorphism construct above (by replacing *r* by *r*_{m}). So where *α*_{k} and *p*_{k} (*k* = 1,2,…,*n*) as constructed above (with *r* replaced by *r*_{m}).

One estimates, by [**35**], [**41**], [**S32**] and [**S33**], that [61]for all . Note that [62]for all *f*∈*C*(*X*). This completes the proof.

Let *Y* be a compact metric space and *C*⊂*C*(*Y*,*M*_{n}) be a unital *C*^{∗}-subalgebra. *C* is said to be *diagonalized* if there are mutually orthogonal rank-one projections {*p*_{1},*p*_{2}, ..,*p*_{n}}⊂*C*(*Y*,*M*_{n}) such that *p*_{i} commutes with every element in *C*, *i* = 1,2,…,*n*.

Let *X* be a compact metric space and let *n*≥1. Suppose that *Y* is a compact metric space with dim *Y* ≤ 2 and suppose that *φ*: *C*(*X*) → *C*(*Y*,*M*_{n}) is a unital homomorphism. Then, for any *ϵ* > 0 and any compact subset , there is a unital commutative *C*^{∗}-subalgebra *B*⊂*C*(*Y*,*M*_{n}) which can be diagonalized and [63]

Let *Y* be a compact metric space with dim *Y* ≤ 2, let *n*≥1 be an integer and let *x* be a normal element. Then, there are *n* sequences of functions in *C*(*Y*) (*k* = 1,2,…,*n*) and there is a sequence of sets of *n* mutually orthogonal rank-one projections such that Moreover, if *x* is self-adjoint, can be chosen to be real and if *x* is a unitary, can be chosen so that , *k* = 1,2,…,*n* and *m* = 1,2,….

## 7 Higher Dimensional Cases

In this section, we consider the cases that dim *Y*≥3. One would hope that the similar argument used in section 6 can repeat for higher dimensional space *Y*. In fact, a version of 5.1 and 5.2 can be proved for two dimensional spaces. However, the last requests in 5.1 and 5.2 cannot be improved, for example, in a generalized version of 5.2, *U*(*s*) cannot be chosen so it exactly commutes with *φ* on a given line segment even *u* can. The reason is that not every homomorphism to *C*([0,1],*M*_{n}) can be exactly diagonalized [see (3)]. This technical problem is fatal as one can see from the results of this section. Nevertheless, one has the following.

Let *X* be a zero-dimensional compact metric space, *n*≥1 and let *Y* be a compact metric space for which every minimal projection in *C*(*Y*,*M*_{n}) has rank one or zero at each point of *Y* (which is the case if dim *Y* ≤ 3—see 7.2).

Then any unital homomorphism *φ*: *C*(*X*) → *C*(*Y*,*M*_{n}) can be approximately diagonalized.

Let *Y* be a compact metric space for which *π*^{1}(*Y*) is trivial and *K*_{1}(*C*(*Y*)) ≠ {0}. Then there are unital homomorphisms from for some *n*≥2 which cannot be approximately diagonalized.

Since *K*_{1}(*C*(*Y*)) = {0}, there is an integer *n*≥2 and a unitary *u*∈*C*(*Y*,*M*_{n}) such that *u∉U*_{0}(*C*(*Y*,*M*_{n})). Define a unital homomorphism by *φ*(*f*) = *f*(*u*) for all . Suppose that there are continuous maps , *k* = 1,2,…,*n* and mutually orthogonal rank-one projections {*p*_{1},*p*_{2},…,*p*_{n}}⊂*C*(*Y*,*M*_{n}) such that [64]where *z* is the identity function on the unit circle . Note that *u* = *φ*(*z*). Since *π*^{1}(*Y*) = {0}, for each *k*, there is a continuous path of unitaries {*w*_{k}(*t*):*t*∈[0,1]}⊂*C*(*Y*) such that One defines a continuous path of unitaries {*U*(*t*):*t*∈[0,1]}⊂*U*(*C*(*Y*,*M*_{n})) by Then and *U*(1) = 1_{Mn}. So . By [**S37**], *u*∈*U*_{0}(*C*(*Y*,*M*_{n})). A contradiction.

Let *X* be a finite CW complex with dim *X*≥2 and let *Y* be a compact metric space for which *π*^{1}(*Y*) is trivial but *K*_{1}(*C*(*Y*)) is not trivial. Then there are unital homomorphism *φ*: *C*(*X*) → *C*(*Y*,*M*_{n}) for some *n*≥2 which cannot be approximately diagonalized.

By modifying 4.4 of (3), one has the following:

Let *Y* be a finite CW complex with dim *Y* > 3. Then there is a self-adjoint element *b*∈*C*(*Y*,*M*_{2}) with *sp*(*b*) = [-1,1] which cannot be approximately diagonalized.

Let *Y* be a finite CW complex with dim *Y* > 3 and let *n*≥2 be an integer. Then, for any finite CW complex *X* with dim *X*≥1, there exists a unital homomorphism *φ*: *C*(*X*) → *C*(*Y*,*M*_{n}) which cannot be approximately diagonalized.

## Acknowledgments

Most of this work was done when the author was in East China Normal University in the summer 2009. This work was partially supported by a National Science Foundation grant, Changjiang Professorship from East China Normal University, and Shanghai Priority Academic Disciplines.

## Footnotes

- ↵
^{1}E-mail: hlin{at}uoregon.edu.

Author contributions: H.L. designed research, performed research, contributed new reagents/analytic tools, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1101079108/-/DCSupplemental

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*C*(

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