Toeplitz-Pimsner algebra - Imported Knowlpedia Content

Definition

Let $X$ be a $C^*$ -correspondence over $A$ . The Toeplitz–Pimsner algebra $\mathcal T_X$ is the universal $C^*$ -algebra generated by a Toeplitz representation $(i_X,i_A)$ of $X$ : for every Toeplitz representation $(t,\pi)$ of $X$ in a $C^*$ -algebra $B$ , there is a unique $*$ -homomorphism

\pi\times t:\mathcal T_X\longrightarrow C^*(\pi(A),t(X))

satisfying $(\pi\times t)(i_A(a))=\pi(a)$ and $(\pi\times t)(i_X(\xi))=t(\xi)$ . Thus $\mathcal T_X$ imposes exactly the Toeplitz relations, with no Cuntz–Pimsner covariance relation.

Fock realization

The creation operators on the Fock module $\mathcal F(X)$ , together with the diagonal left action of $A$ , generate a concrete copy of $\mathcal T_X$ . This realization proves existence of the universal algebra and makes tensor degree visible Pimsner, Fock-space construction. The canonical map $i_A$ is injective because the degree-zero summand carries the faithful left-multiplication representation of $A$ .

Gauge action and quotient

There is a circle action $\gamma:\mathbb T\to\operatorname{Aut}(\mathcal T_X)$ determined by

\gamma_z(i_A(a))=i_A(a),\qquad \gamma_z(i_X(\xi))=z\,i_X(\xi).

The Cuntz–Pimsner algebra is obtained by quotienting $\mathcal T_X$ by the ideal generated by the appropriate covariance relations. Keeping $\mathcal T_X$ separate records all Toeplitz representations, including ones that do not descend to that quotient.

Examples and conventions

For $A=X=\mathbb C$ , the Fock module is $\ell^2(\mathbb N)$ , the creation operator is the unilateral shift, and $\mathcal T_X$ is the classical Toeplitz algebra. For $X=\mathbb C^d$ , it is generated by $d$ isometries with pairwise orthogonal ranges, without requiring their range projections to sum to the identity.

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: the universal Toeplitz algebra and its Fock representation.
Takeshi Katsura, “On C-algebras associated with C-correspondences,” Journal of Functional Analysis 217 (2004), 366–401. DOI record. Relevant: representations of correspondences, gauge actions, and universal algebras.