Covariant representations of subproduct systems: Invariant subspaces and curvature

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Research Article

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New York Journal of Mathematics


Let X = (X(n))n∈Z+ be a standard subproduct system of C∗-correspondences over a C∗-algebra M. Let T = (Tn)n∈Z+ be a pure completely contractive, covariant representation of X on a Hilbert space H. If S is a closed subspace of H, then S is invariant for T if and only if there exist a Hilbert space D, a representation π of M on D, and a partial isometry Π: FX ⊗π D → H such that Π(Sn(ζ) ⊗ ID) = Tn(ζ)Π (ζ ∈ X(n), n ∈ Z+), and S = ran Π, or equivalently, PS = ΠΠ∗. This result leads us to a list of consequences including Beurling–Lax–Halmos type theorem and other general observations on wandering subspaces. We extend the notion of curvature for completely contractive, covariant representations and analyze it in terms of the above results.

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