### abstract

- Let E be a C*-correspondence over a C*-algebra A with non-degenerate faithful left action. We show that E admits sufficiently many essential representations (ie represen- tations ψ such that ψ(E)H = H) to recover the Cuntz–Pimsner algebra OE … Keywords: Hilbert modules; C*-correspondence; Pimsner algebras … Let A be a C ∗ -algebra. Recall that by a C ∗ -correspondence over A we mean a … Hilbert module E over A, along with a fixed left action of A on E via adjointable operators (ie a homomorphism A → B(E) — we suppress notation for this homomorphism). TE and OE denote the Pimsner–Toeplitz algebra and the Cuntz– Pimsner algebra, respectively (see [10]). We shall frequently identify A and E with their images in TE and OE. Throughtout this paper, we shall assume that E is full (ie 〈E,E〉 = A) … We will write ψ for both ψA and ψE, when it causes no confusion. Note that what we call “representation” is called “isometric …