Equivariant band matrices and Fourier series

Recall that in the first post of this series we claimed that there exists an infinite matrix \(T\) which is in the closure (in operator norm) of the band matrices with uniformly bounded entries, but for which we have \(\|T^{(R)}\| \to \infty\). Here \[T^{(R)}_{m,n} := \begin{cases} T_{m,n} & \text{ if } |m-n| \le R\\ 0 & \text{ otherwise }\end{cases}\]

The goal of this post is to provide an example of such a matrix.

Equivariant band matrices

We will consider equivariant matrices, i.e., matrices \(T\) such that for all \(m,n \in \mathbb{Z}\) we have \(T_{m,n} = T_{m+k,n+k}\) for all \(k \in \mathbb{Z}\). Such matrices are completely determined by their entries \(T_{0,n}\) for all \(n \in \mathbb{Z}\).

Since an equivariant band matrix \(T\) only has finitely many non-zero values \(T_{0,n}\), we get a map \[\{\text{equivariant band matrices}\} \to \mathbb{C}[\mathbb{Z}], \quad T \mapsto \sum_{n \in \mathbb{Z}} T_{0,n} \cdot n,\] where \(\mathbb{C}[\mathbb{Z}]\) denotes the complex group ring of \(\mathbb{Z}\). This is actually an isomorphism of \(\mathbb{C}\)-algebras.

We can use the above map to define a norm on \(\mathbb{C}[\mathbb{Z}]\) by using the operator norm of the corresponding equivariant band matrix. The completion of \(\mathbb{C}[\mathbb{Z}]\) under this norm is called the reduced group \(C^*\)-algebra of \(\mathbb{Z}\) and denoted \(C_r^*(\mathbb{Z})\). Taking the completion corresponds to taking the closure of the equivariant band matrices in the space of all infinite matrices with bounded operator norm, i.e., elements of \(C_r^*(\mathbb{Z})\) can be written as equivariant infinite matrices.

Fourier series

Forming Fourier series can be thought of as the map \[\ell^2(\mathbb{Z}) \to L^2(S^1), \quad (c_n)_{n \in \mathbb{Z}} \mapsto \sum_{n \in \mathbb{Z}} c_n \cdot e^{2\pi i t \cdot n},\] which is continuous.

On \(\ell^2(\mathbb{Z})\) we can act with (equivariant) band matrices and on \(L^2(S^1)\) we can act with \(C(S^1)\), i.e., with continuous functions on the unit circle, by point-wise multiplication. These actions are intertwined with each other by the operation of forming Fourier series: the assignment \[\sum_{n \in \mathbb{Z}} T_{0,n} \cdot n \mapsto \text{ point-wise multiplication by } \sum_{n \in \mathbb{Z}} T_{0,n} \cdot e^{2\pi i t \cdot n}\] defines a map \(\mathbb{C}[\mathbb{Z}] \to C(S^1)\) and the operator norm on \(\mathbb{C}[\mathbb{Z}]\) corresponds under it to the sup-norm on \(C(S^1)\). The image of this map is dense in \(C(S^1)\) and so we get an isomorphism \(C_r^*(\mathbb{Z}) \cong C(S^1)\) by the Stone-Weierstrass theorem. Note that \(C_r^*(\mathbb{Z})\) is viewed here as a subalgebra of \(B(\ell^2(\mathbb{Z}))\) and \(C(S^1)\) is viewed as a subalgebra of \(B(L^2(S^1))\), where \(B(-)\) denotes the bounded, linear operators.

The counter-example

What does all the above help us in our goal of providing an infinite matrix \(T\) which is in the closure (in operator norm) of the band matrices with uniformly bounded entries, but for which we have \(\|T^{(R)}\| \to \infty\)?

We assume that \(T\) is from \(C_r^*(\mathbb{Z})\). Then it is in the closure (in operator norm) of the band matrices with uniformly bounded entries and can be represented as \(\sum_{n \in \mathbb{Z}} T_{0,n} \cdot n\). The operators \(T^{(R)}\) correspond then to \(\sum_{-R \le n \le R} T_{0,n} \cdot n\).

Now we apply the isomorphism \(C_r^*(\mathbb{Z}) \cong C(S^1)\) that we discussed above: the operator \(T\) becomes the function \(\sum_{n \in \mathbb{Z}} T_{0,n} \cdot e^{2\pi i t \cdot n}\), the operators \(T^{(R)}\) the functions \(\sum_{-R \le n \le R} T_{0,n} \cdot e^{2\pi i t \cdot n}\), and the operator norm becomes the sup-norm.

This means that if we can find a continuous function \(f\) on \(S^1\) with Fourier series \(\sum_{n \in \mathbb{Z}} c_n \cdot e^{2\pi i t \cdot n}\) such that the continuous functions \(f^{(R)}\) defined by \(\sum_{-R \le n \le R} c_n \cdot e^{2\pi i t \cdot n}\) satisfy \(\|f^{(R)}\|_\infty \to \infty\), then we have our counter-example: we just have to transform \(f\) back to an operator from \(C_r^*(\mathbb{Z})\) via the isomorphism \(C_r^*(\mathbb{Z}) \cong C(S^1)\).

An example of such a continuous function can be found in Section 3.2.2 of the book “Fourier Analysis” by Stein and Shakarchi.

I thank Rufus Willett for providing me the reference for the construction of such a continuous function with divergent Fourier series.