A subspace M is called a wandering subspace of V if Vn(M) ⊥ Vm(M) for all n ≠ m. In particular, each Mi defined above is a wandering subspace of V. A sequence of isometries The decomposition above can be generalized slightly to a sequence of isometries, indexed by the integers. The C*-algebra generated by an isometry Consider an isometry V ∈ L(H). Denote by C*(V) the C*-algebra generated by V, i.e. C*(V) is the norm closure of polynomials in V and V*. The Wold decomposition can be applied to characterize C*(V). Let C(T) be the continuous functions on the unit circle T. We recall that the C*-algebra C*(S) generated by the unilateral shift S takes the following form C*(S) = {Tf + K | Tf is a Toeplitz operator with continuous symbol f ∈ C(T) and K is a compact operator}. In this identification, S = Tz where z is the identity function in C(T). The algebra C*(S) is called the Toeplitz algebra. Theorem (Coburn) C*(V) is isomorphic to the Toeplitz algebra and V is the isomorphic image of Tz. The proof hinges on the connections with C(T), in the description of the Toeplitz algebra and that the spectrum of a unitary operator is contained in the circle T. The following properties of the Toeplitz algebra will be needed: The semicommutator is compact. The Wold decomposition says that V is the direct sum of copies of Tz and then some unitary U: So we invoke the continuous functional calculus f → f(U), and define One can now verify Φ is an isomorphism that maps the unilateral shift to V: By property 1 above, Φ is linear. The map Φ is injective because Tf is not compact for any non-zero f ∈ C(T) and thus Tf + K = 0 implies f = 0. Since the range of Φ is a C*-algebra, Φ is surjective by the minimality of C*(V). Property 2 and the continuous functional calculus ensure that Φ preserves the *-operation. Finally, the semicommutator property shows that Φ is multiplicative. Therefore the theorem holds. References Operator theory Invariant subspaces C*-algebras Theorems in functional analysis de:Shiftoperator#Wold-Zerlegung