Importance of Toeplitz operators?

Toeplitz operators are very useful for proving index theorems in the framework of non-commutative geometry. Let us look at an example.

Identify $ \mathbb{S}^{1} $ as $ \mathbb{R}/\mathbb{Z} $, and let $ {L^{2}}(\mathbb{S}^{1}) $ denote the Hilbert space of square-integrable functions on $ \mathbb{S}^{1} $. Consider the orthonormal basis $ \left\{ e^{i(2n \pi \bullet)} ~ \Big| ~ n \in \mathbb{Z} \right\} $ of $ {L^{2}}(\mathbb{S}^{1}) $, which consists of eigenfunctions of the differential operator $ D \stackrel{\text{def}}{=} \dfrac{1}{i} \dfrac{d}{dx} $ on $ \mathbb{S}^{1} $. Let $ \mathcal{H} $ be the closed subspace of $ {L^{2}}(\mathbb{S}^{1}) $ that is generated by the eigenfunctions of $ D $ corresponding to non-negative eigenvalues. The space $ \mathcal{H} $ is called the Hardy space of $ \mathbb{S}^{1} $. Let $ P $ denote the orthogonal projection of $ {L^{2}}(\mathbb{S}^{1}) $ onto $ \mathcal{H} $.

For each continuous function $ f: \mathbb{S}^{1} \to \mathbb{C} $, define the Toeplitz operator with symbol $ f $, denoted by $ T_{f} $, to be the compression $ P M_{f} P $ of the operator of pointwise multiplication by $ f $. If $ f $ and $ g $ are continuous functions on $ \mathbb{S}^{1} $, then it is a basic fact that $ T_{f} T_{g} - T_{fg} $ is a compact operator on $ \mathcal{H} $. It now follows from Atkinson's Theorem that if $ f $ vanishes nowhere (so $ \dfrac{1}{f} $ is well-defined), then $ T_{f} $ is a Fredholm operator. This implies that the Fredholm index $ \text{Index}(T_{f}) $ is well-defined.

The foregoing discussion serves as a build-up to the following theorem, which can be viewed as the simplest case of the Atiyah-Singer Index Theorem applied to odd-dimensional manifolds.

Baby Index Theorem Let $ f: \mathbb{S}^{1} \to \mathbb{C} \setminus \{ 0 \} $ be a continuous function. Then $$ - \text{Index}(T_{f}) = \deg(f), $$ where $ \deg(f) $ denotes the degree of $ f $.

In this theorem, just like in the Atiyah-Singer Index Theorem, we see a relationship between two types of invariants: an analytic invariant and a topological one. The Fredholm index $ \text{Index}(T_{f}) $ is an analytic invariant that is constructed from a differential operator, and $ \deg(f) $ is clearly a topological invariant. Invariants are very useful tools in topology and geometry, and we see here that Toeplitz operators offer us a way to construct analytic invariants.

The development of the Atiyah-Singer Index Theorem from the viewpoint of non-commutative geometry is made possible by the study of Toeplitz operators. For example, in non-commutative geometry, one is usually interested in developing differential geometry on the leaf space of a foliated manifold $ (M,\mathcal{F}) $; one can formulate foliation index theorems for differential operators on a foliated manifold with the help of Toeplitz operators. There is an important result, by Douglas, Hurder and Kaminker, that uses Toeplitz operators to formulate an index theorem for the manifold $ (\mathbb{R} \times (\mathbb{R}/\mathbb{Z}))/\mathbb{Z} $ with the Kronecker foliation. The details are quite overwhelming, so I shall have to content myself with providing a few references below.

References

• Connes, A. Non-commutative Differential Geometry, Parts I and II, IHÉS Publ. Math., 62 (1985), pp. 257-360.

• Douglas, R.G; Hurder, S; Kaminker, J. The Longitudinal Cocycle and the Index of Toeplitz Operators, Journal of Functional Analysis, 101 (1991), pp. 120-144.