States of Toeplitz–cuntz Algebras

  • Published 1999


We give a spatial method of analyzing the state space of a Toeplitz– Cuntz algebra T On; our method applies equally well to the Cuntz algebra O∞. We study the singular–essential decomposition of a state, and give a new characterization of essentiality. We apply our results to the problem of extending a pure essential product state of Fn, the fixed–point algebra of the action of the gauge group, to T On. Introduction Toeplitz–Cuntz algebras have played an integral rôle in the study of normal –endomorphisms of B(H) ([6],[7],[11],[12]), much as the spectral C–algebras of Arveson’s continuous tensor product systems have been basic to the study of one– parameter semigroups of normal –endomorphisms ([2], [3], [4], [5]). Motivated by a view of the Toeplitz–Cuntz algebra T On as the spectral C –algebra of a product system over the natural numbers N, in §1 we develop a spatial method of analyzing its state space. Our method is a discrete version of the one given by Arveson in [5] for constructing states of his spectral C–algebras. A feature of our method is that it applies for any n ∈ { 1, 2, . . . ,∞}, and hence gives a unified approach for studying both Toeplitz–Cuntz algebras and the Cuntz algebra O∞, which in this paper is thought of as T O∞. In §2 we study the singular–essential decomposition of a state of T On, among other things giving a useful new characterization of essentiality. In §3 we study pure essential product states of Fn, the fixed–point algebra of the action of the gauge group on T On. In particular, we use the results of §1 to give a procedure for extending any such state to T On, and deduce from our characterization of essentiality that every state extension can be thus obtained. The author would like to thank M. Laca for the many helpful discussions during the preparation of this paper, and W. Arveson for the motivating guidance in advance of the work. 1. States of T On From T On to K and back. Fix n ∈ { 1, 2, . . . ,∞} for the duration of this paper, and let T On be the corresponding Toeplitz-Cuntz algebra ([8]). Specifically, T On is a unital C–algebra generated by distinguished elements { vi } n i=1 which satisfy v j vi = δij1, and is universal for these relations in the sense that if U is a unital C–algebra generated by elements { ui } n i=1 which satisfy u ∗ jui = δij1, then vi 7→ ui extends to a morphism of T On onto U . For each multi–index μ = { i1, . . . , ik } we write |μ| = k and set vμ = vi1 · · · vik ; of course v∅ = 1. The set of all multi–indices will be denoted W , and we set Wk = {μ ∈ W : |μ| = k }. With this notation, T On = span{ vμv ∗ ν : μ, ν ∈ W }. 1991 Mathematics Subject Classification. Primary 46L30, 46A22, 47D25.

Cite this paper

@inproceedings{FOWLER1999StatesOT, title={States of Toeplitz–cuntz Algebras}, author={NEAL J. FOWLER}, year={1999} }