nLab C-star-correspondence

Idea

Roughly speaking, a $C^*$-correspondence between two $C^*$-algebras $A$ and $B$ is a generalised $C^*$-homomorphism from $A$ to $B$.

Definition

Definition

Let $A$ and $B$ be $C^*$-algebras. An $A,B$-correspondence is a pair $(E,\varphi)$ consisting of a right Hilbert $B$-module $E$ and a non-degerate $C^*$-homomorphism $\varphi \colon A\to \mathcal{L}_B(E)$; i.e., the image $\varphi(A)E$ is dense in $E$ (as Hilbert $B$-modules).

We write

$$A\stackrel{(E,\varphi)}{\to} B,$$

or just $A\stackrel{E}{\to}B$, and call this a $C^*$-correspondence.

Properties

Composition of correspondences

Definition

Let $A\stackrel{(E,\varphi)}{\longrightarrow}B$ and $B\stackrel{(F,\psi)}{\longrightarrow}C$ be $C^*$-correspondences. Then the internal tensor product $E\otimes_{\psi}F$ is a Hilbert right $C$-module. The composition

$$A \stackrel{(E,\varphi)\circ(F,\psi)}{\longrightarrow} C$$

is defined by the the pair $(E\otimes_{\psi}F,\varphi\otimes \psi)$, where

$$\begin{array}{lccc} \varphi\otimes\psi \colon & A & \to & \mathcal{L}_C(E\otimes_{\psi}F) \\ & a & \mapsto & \varphi(a)(\cdot)\otimes_{\psi}(\cdot) \end{array}$$

Unitary intertwiners

Definition

Let $A\stackrel{(E,\varphi)}{\longrightarrow}B$ and $A\stackrel{(F,\psi)}{\longrightarrow}B$ be $C^*$-correspondences. A unitary intertwiner

$$u\colon E\Rightarrow F$$

is a unitary $u\in \mathcal{L}_B(E,F)$ such that for all $a\in A$ the following diagram commutes

$$\array{E & \stackrel{\varphi(a)}{\longrightarrow} & E\\ ^u\downarrow && \downarrow^u\\ F & \stackrel{\psi(a)}{\longrightarrow} & F }$$

The $2$-category $\mathfrak{Corr}$

…

Related concepts

• Hilbert bimodule

References

• Alcides Buss, Chenchang Zhu, Ralf Meyer, A higher category approach to twisted actions on $C^*$-algebras, Proceedings of the Edinburgh Mathematical Society, 56 (2013), pp 387-426, doi:10.1017/S0013091512000259

  https://doi.org/10.1017/S0013091512000259), arXiv:0908.0455

Relation to KK-theory:

• El-kaioum Moutuou, Equivariant KK-theory for generalised actions and Thom isomorphism in groupoid twisted K-theory, Journal of K-theory, 13 (2014) pp83-113, doi:10.1017/is013010018jkt244

  https://doi.org/10.1017/is013010018jkt244), arXiv:1305.2495