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I am reading Arveson's A Short Course on Spectral Theory, in which the author states that Toeplitz operators are very important without giving references on their applications. After some searching, I understand the importance of Toeplitz matrices. They are related to the so-called Toeplitz systems.

But this does not justify why so much effort has been put into studying Toeplitz operators, which are much harder than their matrices counterpart.

Can someone explain why Toeplitz operators are important? Thanks!

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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.

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It is also worth mentioning that the $C^*$-algebraic Bott periodicity can be proven with the help of Toeplitz operators. The idea is roughly that there is a sequence involving all compact operators as ideal in the Toeplitz algebra, its quotient will be isomorphic to $C(S^1)$. You can tensor this sequence with an arbitrary $C^*$-algebra, and the connecting homomorphism in $K$-theory will be an inverse to the Bott map. I think this is a pretty cool application of the algebra of Toeplitz operators. –  mland Jan 3 '13 at 11:26
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@mland: Many thanks for your comments! Yes, that application slipped my mind; I believe it should be more well-known. –  Haskell Curry Jan 3 '13 at 11:42
    
Thanks very much! I did not expect that my naive question could receive such a beautiful answer. Neither did I expect Toeplitz could be connected to so many important parts of modern mathematics (although I do not quite understand the last paragraph). –  Hui Yu Jan 3 '13 at 13:10
    
@mland Thanks very much! –  Hui Yu Jan 3 '13 at 13:12
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@OtavioKaminski: Hi Otavio. When I said, “... if $ f $ is invertible, then ...” I didn’t mean that $ f $ is a bijection. I actually meant that $ \dfrac{1}{f} $ is a well-defined function on $ \mathbb{S}^{1} $, which is the case when $ 0 \notin f[\mathbb{S}^{1}] $. Hence, you should interpret ‘invertible’ as ‘multiplicatively invertible’. My apologies if my choice of words caused you much confusion. –  Haskell Curry Jan 12 '13 at 4:45

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