Compact operators and nest representations of limit algebras

Katsoulis, Elias G.; Peters, Justin R.

doi:10.1090/s0002-9947-07-04071-8

Cited by 4 publications

(3 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [25], it is shown that the fundamental relation for strongly maximal TAF algebras can be recovered from the (perhaps infinite dimensional) representation theory of such algebras. Since various TAF algebras (as well as our tensor algebras) are actually tensor algebras of correspondences, we expect that the completion of the classification scheme of Muhly and Solel [37] mentioned in the introduction will profit from a better understanding of their nest representations.…”

Section: Open Problems and Future Directionsmentioning

confidence: 99%

Operator algebras for multivariable dynamics

2011

Self Cite

View full text Add to dashboard Cite

Let X be a locally compact Hausdorff space with n proper continuous self maps σ i : X → X for 1 ≤ i ≤ n. To this we associate two conjugacy operator algebras which emerge as the natural candidates for the universal algebra of the system, the tensor algebra A(X, τ ) and the semicrossed product C 0 (X) × τ F + n . We develop the necessary dilation theory for both models. In particular, we exhibit an explicit family of boundary representations which determine the C*-envelope of the tensor algebra.We introduce a new concept of conjugacy for multidimensional systems, called piecewise conjugacy. We prove that the piecewise conjugacy class of the system can be recovered from the algebraic structure of either A(X, σ) or C 0 (X) × σ F + n . Various classification results follow as a consequence. For example, if n = 2 or 3, or the space X has covering dimension at most 1, then the tensor algebras are algebraically isomorphic (or completely isometrically isomorphic) if and only if the systems are piecewise topologically conjugate.We define a generalized notion of wandering sets and recurrence. Using this, it is shown that A(X, σ) or C 0 (X) × σ F + n is semisimple if and only if there are no generalized wandering sets. In the metrizable case, this is equivalent to each σ i being surjective and v-recurrent points being dense for each v ∈ F + n .

show abstract

Section: Open Problems and Future Directionsmentioning

confidence: 99%

Operator algebras for multivariable dynamics

2011

Self Cite

View full text Add to dashboard Cite

show abstract

“…The representation theory for non-selfadjoint operator algebras, although still in its infancy, is already playing an important role. Given a non-selfadjoint algebra, its nest representations are thought of as the proper generalization of the irreducible ones and a general theory around them is developing [26,6,21]. This paper contributes to the further development of that theory by analyzing such representations for algebras associated to directed graphs.…”

Section: Introductionmentioning

confidence: 99%

Nest Representations of Directed Graph Algebras

Davidson

Katsoulis

2006

Proc. Lond. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. This paper is a comprehensive study of the nest representations for the free semigroupoid algebra L G of countable directed graph G as well as its norm-closed counterpart, the tensor algebra T + (G). We prove that the finite dimensional nest representations separate the points in L G , and a fortiori, in T + (G). The irreducible finite dimensional representations separate the points in L G if and only if G is transitive in components (which is equivalent to being semisimple). Also the upper triangular nest representations separate points if and only if for every vertex x ∈ V(G) supporting a cycle, x also supports at least one loop edge.We also study faithful nest representations. We prove that L G (or T + (G)) admits a faithful irreducible representation if and only if G is strongly transitive as a directed graph. More generally, we obtain a condition on G which is equivalent to the existence of a faithful nest representation. We also give a condition that determines the existence a faithful nest representation for a maximal type N nest.

show abstract

“…Nest representations and their kernels were originally introduced by Lamoureux [29] as a generalization for primitive ideals. The second author, in joint work with Peters [25] and Kribs [22], applied the concept of a nest representation to the classification theory for non-selfadjoint operator algebras, including graph algebras [22] and limit algebras [25]. (See also [45].)…”

Section: Semicrossed Productsmentioning

confidence: 99%