Notes on the \(\ell^p\)-Toeplitz algebra on \(\ell^p(\mathbb{N} )\)
There has recently been interest in using the ideas and intuition gained from operator algebra theory to study closed algebras of operators, not on Hilbert spaces, but on \(L^p\) spaces, for \(p\not=2\). This paper is concerned with the \(\ell^p\)-space analogues of the Toeplitz algebra. The classical Toeplitz algebra is the \(C^*\)-algebra acting on \(\ell^2(\mathbb N)\) generated by the unilateral shift. Here we replace \(\ell^2(\mathbb N)\) by \(\ell^p(\mathbb N)\), and study the closed algebra generated by both the unilateral shift, say \(S\), and its ``reverse'', the backward shift, say \(T\) (this being, in the \(C^*\)-case, the adjoint of \(S\)). Then \(TS=1\) but \(1-ST=p_1\), the rank-one projection onto the 1st coordinate. From this, as classically, we find that the Toeplitz algebra \(\mathcal{T}_p\) contains the ideal of compact operators. The paper considers the quotient \(\mathcal{T}_p / K(\ell^p)\) which is a commutative Banach algebra: a major result shown is that the character space is the unit circle \(S^1\). Phillips conjectured that the quotient should actually be (isometrically isomorphic to) the \(\ell^p\) algebra of the circle [\textit{N. C. Phillips}, ``Open problems related to operator algebras on $L^p$ spaces'', Preprint (2014), \url{https://marktomforde.com/graph-algebra-problems/Phillips-Lp-Problems.pdf}]. By Fourier theory, and analogy with the \(C^*\)-algebra case, this means we should consider \(F^p_\lambda(\mathbb Z)\) which is the closed algebra on \(\ell^p(\mathbb Z)\) generated by the action of \(\mathbb Z\) (equivalently, generated by the bilateral shift). This paper proves this conjecture by establishing an explicit isomorphism \(\mathcal{T}_p / K(\ell^p) \rightarrow F^p_\lambda(\mathbb Z)\). The isomorphism maps the convolution operator in \(F^p_\lambda(\mathbb Z)\) to the (image in the quotient of the) partially defined convolution in \(\mathcal{T}_p / K(\ell^p)\). In the final section, some interesting remarks about \(K\)-theory are made. The paper is pleasingly self-contained and easy to read.
scientific article; zbMATH DE number 7456843