Toeplitz representation of a C*-correspondence

Let $X$ be a $C^*$ -correspondence over $A$ , with left action $\varphi_X$ . A Toeplitz representation of $X$ in a $C^*$ -algebra $B$ is a linear map $t:X\to B$ and a $*$ -homomorphism $\pi:A\to B$ such that, for $a\in A$ and $\xi,\eta\in X$ ,

t(\xi a)=t(\xi)\pi(a),\qquad t(\varphi_X(a)\xi)=\pi(a)t(\xi),\qquad t(\xi)^*t(\eta)=\pi(\langle\xi,\eta\rangle_A).

The represented $C^*$ -algebra is $C^*(\pi(A),t(X))$ . No injectivity, nondegeneracy, or Cuntz–Pimsner covariance is part of the definition unless stated separately.

The induced compact-operator representation

Every Toeplitz representation determines a $*$ -homomorphism

t^{(1)}:\mathcal K_A(X)\longrightarrow B,\qquad t^{(1)}(\theta_{\xi,\eta})=t(\xi)t(\eta)^*,

on the compact Hilbert-module operators. The inner-product relation makes this formula multiplicative and independent of a chosen rank-one decomposition. This induced map is the object compared with $\pi$ in the Cuntz–Pimsner covariance relation.

Fock representation

On the Fock module $\mathcal F(X)$ , each $\xi\in X$ defines a creation operator $T_\xi(\eta)=\xi\otimes\eta$ , while $A$ acts diagonally on tensor powers. The pair $(T,\varphi_\infty)$ is a Toeplitz representation and gives the standard concrete model. Its relation $T_\xi^*T_\eta=\varphi_\infty(\langle\xi,\eta\rangle_A)$ is the operator form of the last defining axiom.

Conventions and nearby notions

Some authors call this simply a “representation” of $X$ , reserving “covariant representation” for a pair satisfying an additional quotient relation. Others use “Toeplitz-covariant representation” for the definition above. The phrase “covariant representation” alone is therefore ambiguous. The three displayed relations encode the module actions and inner product; they should not be replaced by the weaker requirement that $t$ is merely a bounded linear map.

References

Takeshi Katsura, “On C-algebras associated with C-correspondences,” Journal of Functional Analysis 217 (2004), 366–401. DOI record. Relevant: the opening definitions of representations, induced compact-operator maps, and covariance.
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: Toeplitz representations and the universal Toeplitz construction.