Representations of Cuntz algebras associated to quasi-stationary Markov measures

Abstract

In this paper, we answer the question of equivalence, or singularity, of two given quasi-stationary Markov measures on one-sided infinite words, as well as the corresponding question of equivalence of associated Cuntz algebra O N -representations. We do this by associating certain monic representations of O N to quasi-stationary Markov measures and then proving that equivalence for a pair of measures is decided by unitary equivalence of the corresponding pair of representations.

Representations of Cuntz-Krieger relations, dynamics on Bratteli diagrams, and path-space measures

“…A Markov measure satisfying (2.17) is called a quasi-stationary measure. We remark that condition (2.17) appeared first in [DJ14b] in a different context.…”

“…In this case X B is a Cantor group. The case of O N has been studied in recent papers [DJ14a,DJ14b]. In both cases, Cuntz, and Cuntz-Krieger, by a representation we mean an assignment of isometries in a Hilbert space H, assigning to every letter from the finite alphabet an isometry, and assigned in such a way that distinct isometries have orthogonal ranges, adding up to the identity operator in H, in the case of O N .…”

Minimal Representations of Cuntz Algebras

“…As is well known, it is a core component for the Cuntz algebra to study states on the Cuntz algebra. There are a large number of the extensions of Cuntz states on Cuntz algebras that have been discussed by many authors [10,14], One of the extension is an f -sub-Cuntz state. Proof Suppose that (H ω , π ω , Ω ω ) is the GNS representation of O n , then…”

“…Later, Jeong [20] studied the irreducible representations of the Cuntz algebra; Fowler and Laca [17] generalized the extensions of pure states on O n and Bergmann and Conti [3] proposed the induced representation of extended Cuntz algebra. Furthermore, Kawamura et al [24,25] studied the permutation representations of the Cuntz algebra O ∞ , the classification of sub-Cuntz states and pure states on the Cuntz algebra arising from geometric progressions; Dutkay et al [13][14][15] introduced atomic representations, monic representations of the Cuntz algebra and representations of the Cuntz algebras associated to Lemma 2.5 ([31]) Suppose that (H 1 , π 1 , Ω 1 ) and (H 2 , π 2 , Ω 2 ) are representations of a C *algebra A. There is a unitary u : H 1 → H 2 such that Ω 2 = u(Ω 1 ) and π 2 (x) = uπ 1 (x)u * for all x ∈ A if and only if (π 1 (x)Ω 1 , Ω 1 ) = (π 2 (x)Ω 2 , Ω 2 ) for all x ∈ A.…”

Separable representations, KMS states, and wavelets for higher-rank graphs

“…We note that Λ-semibranching function systems also provide a template for establishing the existence of faithful representations of C * (Λ) on other Hilbert spaces. Indeed, examining recent work of Bezuglyi and Jorgensen [3], and Jorgensen and Dutkay [9,8], we conjecture that the Perron-Frobenius measure of Definition 2.5, although a useful and canonical example of a probability measure, is potentially just one of many measures that could give faithful representations of Cuntz-Krieger C * -algebras associated to strongly connected finite k-graphs.…”