---
tag: 3SKRBF4KG1
kind: object
name: Toeplitz algebra
msc: [47L80, 47B35, 46L05]
uses: [BMDG4JQPEH, GQ0X3R3WM2, M2NEP1K5AE, 2WAZF1RV2Y, PNNNFJVF6W]
sources:
  - url: https://en.wikipedia.org/wiki/Toeplitz_algebra
same_as:
  wikidata: Q2835942
  wikipedia: Toeplitz algebra
---

## Description

Let $\ell^2(\N)$ be the Hilbert space of square-summable sequences, with orthonormal basis $e_0, e_1, e_2, \ldots$.
The unilateral shift is the operator $S$ with $S e_n = e_{n+1}$ for all $n$.
The *Toeplitz algebra* $\mathcal{T}$ is the C*-subalgebra of the [[algebra of bounded operators|BMDG4JQPEH]] $B(\ell^2(\N))$ generated by $S$.
It is the norm closure of the set of polynomials in $S$ and $S^*$.
It is a C*-algebra by construction.
Identify $\ell^2(\N)$ with the [[Hardy space|PNNNFJVF6W]] $H^2$ of the unit circle.
Then $\mathcal{T}$ is the C*-algebra generated by the [[Toeplitz operators|GQ0X3R3WM2]] $T_f$ with continuous symbol $f$.

Check that $\mathcal{T}$ contains the [[compact operators|M2NEP1K5AE]]: $S^* S = 1$, and $P = 1 - S S^*$ is the projection onto $\C e_0$.
Then $S^m P (S^*)^n$ is the rank-one operator sending $e_n$ to $e_m$ and the other basis vectors to $0$.
These matrix units lie in $\mathcal{T}$, and the norm closure of their span is the [[ideal of compact operators|2WAZF1RV2Y]] $K$.

## Remarks

The quotient $\mathcal{T} / K$ is isomorphic to the algebra $C(\mathbb{T})$ of continuous functions on the unit circle, by a map that sends the class of $T_f$ to $f$.
This gives the exact sequence $0 \to K \to \mathcal{T} \to C(\mathbb{T}) \to 0$, often called the Toeplitz extension.
Coburn's theorem states that the C*-algebra generated by any isometry that is not unitary is isomorphic to $\mathcal{T}$, by an isomorphism that sends the isometry to $S$.
The index of the Fredholm operator $T_f$, for a continuous nowhere-vanishing symbol $f$, is minus the winding number of $f$ around $0$.

## Explanations

### L1 human

The shift operator moves each entry of a sequence one place to the right and puts a zero in front.
It almost has an inverse, the backward shift, but shifting back and then forward loses the first entry.
The Toeplitz algebra is everything one can build from these two operators with sums, products and limits.
Modulo the compact operators, the shift becomes multiplication by $z$ on the circle, so the algebra links operators with functions on the circle and with winding numbers.