Fock module of a C*-correspondence

Let $X$ be a $C^*$ -correspondence over $A$ . Put $X^{\otimes 0}=A$ , $X^{\otimes 1}=X$ , and recursively $X^{\otimes n}=X\otimes_A X^{\otimes(n-1)}$ , using the internal tensor product. The Fock module of $X$ is the Hilbert $A$ -module

\mathcal F(X)=\bigoplus_{n=0}^{\infty}X^{\otimes n}.

Its left action $\varphi_\infty:A\to\mathcal L_A(\mathcal F(X))$ is diagonal: on degree zero it is left multiplication, and on degree $n\geq1$ it is $\varphi_X(a)\otimes 1$ . The tensor-degree summands are mutually orthogonal.

Creation operators

For $\xi\in X$ , the creation operator $T_\xi$ raises tensor degree by one:

T_\xi(a)=\xi a,\qquad T_\xi(\eta_1\otimes\cdots\otimes\eta_n) =\xi\otimes\eta_1\otimes\cdots\otimes\eta_n.

It is adjointable, and the operators satisfy

T_\xi^*T_\eta=\varphi_\infty(\langle\xi,\eta\rangle_A).

Consequently $(T,\varphi_\infty)$ is the Fock Toeplitz representation.

Grading and gauge action

Tensor degree gives $\mathcal F(X)$ an $\mathbb N$ -grading. For $z$ on the unit circle, the unitary $U_z$ acts as multiplication by $z^n$ on $X^{\otimes n}$ . Conjugation fixes the diagonal copy of $A$ and sends $T_\xi$ to $zT_\xi$ . This circle action is the concrete origin of the gauge action on the Toeplitz–Pimsner algebra.

Examples and scope

For $A=\mathbb C$ and $X=\mathbb C^d$ , $\mathcal F(X)=\bigoplus_{n\geq0}(\mathbb C^d)^{\otimes n}$ , the usual full Fock space. If $X=A$ with its standard correspondence structure, every tensor power is canonically $A$ , so the Fock module is the standard module $\ell^2(\mathbb N)\otimes A$ .

References

Michael V. Pimsner, “A Class of C-Algebras Generalizing Both Cuntz–Krieger Algebras and Crossed Products by Z,” in Free Probability Theory*, Fields Institute Communications 12, American Mathematical Society, 1997, 189–212. Bibliographic record. Relevant: tensor powers, Fock module, and creation operators.
Takeshi Katsura, “On C-algebras associated with C-correspondences,” Journal of Functional Analysis 217 (2004), 366–401. DOI record. Relevant: the Fock representation and gauge action.