Cuntz-Pimsner algebra

Let $X$ be a $C^*$ -correspondence over $A$ , with left action $\varphi_X$ , and write $\mathcal K_A(X)$ for its compact module operators. Define

J_X=\varphi_X^{-1}(\mathcal K_A(X))\cap(\ker\varphi_X)^\perp.

A Toeplitz representation $(t,\pi)$ is Cuntz–Pimsner covariant when

\pi(a)=t^{(1)}(\varphi_X(a))\qquad(a\in J_X),

where $t^{(1)}$ is its representation of the compact module operators. The Cuntz–Pimsner algebra $\mathcal O_X$ is the universal $C^*$ -algebra generated by such a covariant representation. Here $(\ker\varphi_X)^\perp=\{a\in A:ab=0\text{ for every }b\in\ker\varphi_X\}$ . This is Katsura’s definition for an arbitrary correspondence.

Covariance ideal and quotient

The ideal $J_X$ isolates precisely the part of $A$ on which the left action is compact while avoiding the kernel that would make the covariance relation collapse coefficient elements. Equivalently, $\mathcal O_X$ is the quotient of the Toeplitz–Pimsner algebra $\mathcal T_X$ by the ideal generated by

i_A(a)-i_X^{(1)}(\varphi_X(a)),\qquad a\in J_X.

This formulation extends the construction to noninjective left actions.

Standard examples

For $A=\mathbb C$ and $X=\mathbb C^d$ with $2\leq d<\infty$ , covariance adds $\sum_{j=1}^d S_jS_j^*=1$ to the Toeplitz relations, producing the Cuntz algebra $\mathcal O_d$ . A correspondence obtained from an automorphism of $A$ yields a crossed product by $\mathbb Z$ , with the orientation of the automorphism determined by the chosen module convention. These examples motivated Pimsner’s construction.

Conventions and scope

References

Takeshi Katsura, “On C-algebras associated with C-correspondences,” Journal of Functional Analysis 217 (2004), 366–401. DOI record. Relevant: the Katsura ideal, covariant representations, and the universal algebra.
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 original construction and motivating examples.