Subalgebras of the Cuntz C^*-Algebra

Abstract

In this paper we exploit the fact that a Cuntz C$^*$-algebra is a groupoid C$^*$-algebra to facilitate the study of non-self-adjoint subalgebras of $O_n$. The Cuntz groupoid is not principal and the spectral theorem for bimodules does not apply in full generality. We characterize the bimodules (over a natural masa) which are determined by their spectra in the Cuntz groupoid; these are exactly the ones which are invariant under the guage automorphisms and exactly the ones which are generated by the Cuntz partial isometries which they contain. We investigate analytic subalgebras of $O_n, n$ finite, by studying cocycles on the Cuntz groupoid. In contrast to AF groupoids, there are no cocycles which are integer valued or bounded and vanish precisely on the natural diagonal. $O_n$ contains a canonical UHF subalgebra; each strongly maximal triangular subalgebra of the UHF subalgebra has an extension to a strongly maximal triangular subalgebra of $O_n$ and each trivially analytic subalgebra of the UHF subalgebra has a proper analytic extension. We also study the Volterra subalgebra of $O_n$. We identify the spectrum of the Volterra subalgebra and use this to prove a theorem of Power that the radical is equal to the closed commutator ideal. We also show that the Volterra subalgebra is maximal triangular but not strongly maximal triangular.