Notes on the ℓ ^p-Toeplitz algebra on ℓ ^p(ℕ)

For p ∈ (1, ∞), especially p ≠ 2, the ℓ^p-Toeplitz algebra Τ_p on ℓ^p(ℕ) is the Banach subalgebra of L(ℓ^p(ℕ)) generated by the unilateral shift S and its reverse, the backwards shift T, which contains the algebra K(ℓ^p(ℕ)) of all compact operators as an ideal. In this note, we show that the maximal ideal space of Τ_p/K(ℓ^p(ℕ)) is homeomorphic to the unit circle S¹. Furthermore, the quotient algebra Τ_p/K(ℓ^p(ℕ)) is isometrically isomorphic to the closed subalgebra of L(ℓ^p(ℤ)) generated by the bilateral shift and its inverse, namely, the reduced group ℓ^p-algebra F_λ^p(ℤ). This solves an open problem raised by N. C. Phillips. As an application, we show that the K-theory groups of the ℓ^p-Toeplitz algebras Τ_p do not depend on p ∈ (1, ∞).