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An alternate way to phrase this question might be, "How did the Toeplitz operators used in the definition of the Cuntz-Pimsner algebra come by their name?" or, "What's the relationship between the Toeplitz algebra generated by multiplication operators on the Hardy space $H^2$ and the Toeplitz algebra associated to a Hilbert C*-module?"

References would be greatly appreciated.


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This is not my area, and I don't really have a satisfactory answer, but in case it helps here's my own guess. A quick forage online suggests that the Toeplitz algebras used to define Cuntz-Pimsner algebras are generated by sets of shift operators on a Fock space; and Fock spaces are apparently good noncommutative versions of Hardy space. (Classical Toeplitz operators arise from the unilateral shift on $H^2.$) Hopefully someone who has a better handle on these things or knows some of the history will be able to elaborate on this or correct me. – Yemon Choi Feb 7 '12 at 2:07

After thinking about it some more, here's what I came up with -- but if anyone else has another explanation I'd love to hear it!

The classical Toeplitz operators $T_f$ are built from multiplication operators $M: C(\mathbb{T}) \to B(L^2(\mathbb{T}))$, where $M_f(\phi) = f \phi$ is given by pointwise multiplication. Note that $C(\mathbb{T})$ is dense in $L^2(\mathbb{T})$, so one can think of this as an action of $L^2(\mathbb{T})$ on itself by multiplication.

When building a Cuntz-Pimsner algebra (or generalized Toeplitz algebra), one starts with a Hilbert module $X$ instead of a Hilbert space; the algebra is generated by a set of Toeplitz operators $\{T_\xi\}_{\xi \in X}$, which act as multiplication operators on $X$ in the sense that $T_\xi(x) := \xi \otimes x \in X \otimes X$. However, without modifications, this definition would put $T_\xi \in \mathcal{L}(X, X \otimes X)$, which is not a $C^*$-algebra. In order to have $\{T_xi\}$ generate a $C^*$-algebra, we want to find a way to have $\{T_\xi\} \subseteq \mathcal{L}(\mathcal{E})$ for some (single) Hilbert module $\mathcal{E}$. Hence, we define $\mathcal{E} = \bigoplus_{n=0}^\infty X^{\otimes n}$, defining $T_\xi$ on elementary tensors by $T_\xi( x_1 \otimes \ldots \otimes x_n) = \xi \otimes x_1 \otimes \ldots \otimes x_n$.

In other words, the aspect of the classical Toeplitz algebras that I see the Cuntz-Pimsner algebras as generalizing is the idea of a Hilbert space acting on itself via multiplication operators.

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