Abstract :
Let H be a complex Hilbert space and let {Tn}n 1 be a sequence of commuting bounded operators
on H such that n 1 TnTn∗ IH. Let F(T¯) denote the space of all operators X in B(H) for which
n 1 TnXT ∗ n = X and suppose that F(T¯) = {0}. We will show that there exists a triple {K,Γ, {Un}n 1} where K is a Hilbert space, Γ : K →H is a bounded operator and {Un}n 1 ⊂ B(K) is a sequence of
commuting normal operators with n 1 UnU∗n = IK such that TnΓ = ΓUn for n 1, and for which
the mapping Y →Γ YΓ∗ is a complete isometry from the commutant of {Un}n 1 onto the space F(T¯).
Moreover we show that the inverse of this mapping can be extended to a ∗-homomorphism
π : C∗ IH,F(T¯) →{Un} n 1
from the unital C∗-algebra generated by F(T¯) onto the commutant of {Un}n 1. We also show that there
exists a ∗-homomorphism
Π : C∗ IH, {Tn}n 1 →C∗ IK, {Un}n 1
such that Π(Tn) = Un for n 1. In the particular case when {Tn}n 1 has only a finite number of non-zero
components, it turns out that {Un}n 1 is unitarily equivalent to the spherical unitary part of the standard
commuting dilation of {Tn}n 1.
© 2007 Elsevier Inc. All rights reserved.
