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Inner product space over a Hilbert C*-module

  1. Mar 10, 2013 #1
    I am working on this problem, and I am having difficulties with a certain part of a proof:

    If [itex]A[/itex] is a [itex]C^*[/itex]-algebra. And [itex]X[/itex] is a Hilbert [itex]A[/itex]-module. Can we say that [itex]\langle X,X \rangle[/itex] has an approximate identity [itex]e_\alpha = \langle u_\alpha , v_\alpha \rangle[/itex] such that [itex]u_\alpha , v_\alpha[/itex] are norm bounded by 1. Where [tex]\langle \cdot , \cdot \rangle [/tex] denotes the inner product.

    I know that any [itex]C^*[/itex]-algebra has a bounded approximate identity, and [itex]\langle X,X \rangle[/itex] is a [itex]C^*[/itex]-algebra. So we can have [itex]\| e_\alpha \| = \| \langle u_\alpha , v_\alpha \rangle \| \leq 1[/itex]
  2. jcsd
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