Application of Approximate Diagonalization of Commutative C*-Algebras to Invariants

The premise of my dissertation is founded on the idea that matrices over C*-algebras are important and that approximate diagonalization would make dealing with such matrices easier. I have yet to find any useful application of my own result, though I find some mildly amusing applications to results of matrices over commutative C*-algebras.

So we start with the main definition:

Definition. Let A be a C*-algebra and let n be a positive integer. A normal matrix a\in M_n(A) is approximately diagonalizable if for every \varepsilon > 0, there exist elements a_1, \dotsc, a_n\in A and a unitary u\in M_n(A) such that

\lVert uau^{*} - \mathrm{diag}(a_1,\dotsc,a_n) \rVert < \varepsilon.

Next, we present the theorem that we will be applying:

Theorem. Let X be a compact metrizable space. Every self-adjoint matrix in M_n(C(X)) is approximately diagonalizable if and only if \dim(X) \leq 2 and \check{H}^2(X) = 0.

The first mildly interesting application is that on the connection between K-theory and approximate diagonalization of matrices over commutative C*-algebras.

Theorem. (Theorem 3.4 and 4.1 of [3]) Let X be a compact metrizable space. If for every positive integer n, every projection in M_n(C(X)) is approximately diagonalizable, then K_0(C(X)) \cong C(X,\mathbb{Z}).

Proof. First, note that the K_0 class of any diagonal projection is an element of C(X,\mathbb{Z}). This is because projections in C(X) are characteristic (indictator) functions of clopen subsets and so the K_0-class of a diagonal projection is the sum of indicator functions of clopen subsets, which is a continuous, (non-negative) integer-valued function.

Given an approximately diagonalizable projection p, there exist projections p_1, p_2, \dotsc, p_n \in C(X) and a unitary u\in M_n(C(X)) such that

\lVert upu^{*} - \mathrm{diag}(p_1, p_2, \dotsc, p_n)\rVert < 1.

Note that p_1, p_2, \dotsc, p_n can be chosen to be projections by the stability of the projection relations (i.e. being self-adjoint and idempotent, see Lemma 2.5.4 of [1] for details). Since close projections (norm distance strictly less than 1) are unitary equivalent (see Lemma 2.5.1 of [1]), their K_0-classes are the same. So every projection has the same K_0-class as of a diagonal projection, and thus belongs to C(X, \mathbb{Z}). \Box

The observant reader can see that we actually proved every approximately diagonalizable projection is in fact diagonalizable. Combining this with Xue’s theorem, we get the corollary:

Corollary. Let X be a compact metrizable space such that \dim(X) \leq 2 and \check{H}^2(X) = 0. Then K_0(C(X)) \cong C(X,\mathbb{Z}).

Perhaps of more slightly more interest is the fact that we can use this method to prove the following:

Theorem. Let X be a compact metrizable space. If for every positive integer n, every positive matrix in M_n(C(X)) is approximately diagonalizable, then W(C(X)) \cong \mathrm{lsc}(X,\mathbb{N}\cup \{0\}).

Here W(A) = (A\otimes M_{\infty})/\sim denotes the Cuntz semigroup, where \sim denotes Cuntz equivalence. The proof of this theorem falls out in the same way as the previous theorem: the Cuntz class of positive elements correspond to characteristic functions of open sets, which sum to non-negative integer-valued lower semicontinuous functions and it is clear that every approximately diagonalizable matrix is Cuntz equivalent to a diagonal matrix.

When we combine this theorem with Xue’s theorem, we have the following corollary:

Collorary. Let X be a compact metrizable space such that \dim(X) \leq 2 and \check{H}^2(X) = 0. Then W(C(X)) \cong \mathrm{lsc}(X,\mathbb{N}).

This is finite and unital version of Theorem 1.3 of [2]. Perhaps the only real upside to this, is that this proof directly deals with positive elements in contrast to Robert’s proof using Hilbert modules.

I’m still looking for some applications of approximate diagonalization, especially my dissertation results. Any help in this respect would be appreciated.

[1] Lin, Huaxin. An Introduction to the Classification of Amenable C*-Algebras. World Scientific, 2001.

[2] Robert, Leonel. “The Cuntz semigroup of some spaces of dimension at most two” C. R. Math. Acad. Sci. Soc. R. Can. 35:1 (2013) pp. 22-32.

[3] Xue, Yifeng. “Approximate Diagonalization of Self-Adjoint Matrices over C(M)Funct. Anal. Approx. Comput. 2:1 (2010) pp. 53-65. [pdf]