Let p>1, the l^p Toeplitz algebra is a Banach algebra generated by unilateral shift and its reverse on l^p(N). This algebra contains the compact operators on l^p(N) as a closed two‐sided ideal. In this talk, we show that the quotient by this ideal is isometrically isomorphic to the reduced group l^p operator algebra of the integers. This answers a question of Phillips. As an application, we show that the K‐theory of the l^p Toeplitz algebra is independent of p.

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