R-Matrices and the Yang–Baxter equation on GNS representations on C∗-bialgebras

Kawamura, Katsunori

doi:10.1016/j.laa.2012.08.006

Cited by 2 publications

(2 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For Cuntz algebras which are typical examples of separable infinite simple C * -algebras (see § 1.2), representations and states have been studied by many authors [3,4,8,9,11,14,15,19,20,22,24,25,36,41,42,46]. They have various applications, for example, endomorphisms of B(H) [6,10,21], iterated function systems [7], Markov measures [17,18], wavelets [26], continued fractions [37], construction of R-matrices [35], construction of multiplicative isometries [33], invariant measures [28], and string theory [2]. But their classifications have not been finished yet.…”

Section: Motivation 1classification Problem Of Pure States On Cuntz A...mentioning

confidence: 99%

Pure States on Cuntz Algebras Arising from Geometric Progressions

Kawamura

2016

Algebr Represent Theor

Self Cite

View full text Add to dashboard Cite

Let O n denote the Cuntz algebra for n ≥ 2. We introduce an embedding f of O m into O n arising from a geometric progression of Cuntz generaters of O n . By identifying O m with f (O m ), we extend Cuntz states on O m to O n . We show (i) a necessary and sufficient condition of the uniqueness of the extension, (ii) the complete classification of all such extensions up to unitary equivalence of their GNS representations, and (iii) the decomposition formula of a mixing state into a convex hull of pure states. The complete set of invariants of all GNS representations by such pure states is given as a certain set of complex unit vectors.Mathematics Subject Classifications (2010). 46K10.

show abstract

Section: Motivation 1classification Problem Of Pure States On Cuntz A...mentioning

confidence: 99%

Pure States on Cuntz Algebras Arising from Geometric Progressions

Kawamura

2016

Algebr Represent Theor

Self Cite

View full text Add to dashboard Cite

show abstract

“…We illustrate a rough classification of states on O n (2 ≤ n < ∞) as follows: 2). Cuntz states are completely classified pure states with explicit complete invariants, and are used to construct multiplicative isometries ( [37], § 3) and R-matrices ( [38], § 3.2) (see also [30,36]). In this study, we select sub-Cuntz states as a target of complete classification because they are natural generalizations of Cuntz states.…”

Section: States On Cuntz Algebrasmentioning

confidence: 99%

Classification of Sub-Cuntz States

Kawamura

2014

Algebr Represent Theor

Self Cite

View full text Add to dashboard Cite

Let O n denote the Cuntz algebra for 2 ≤ n < ∞. With respect to a homogeneous embedding of O n m into O n , an extension of a Cuntz state on O n m to O n is called a sub-Cuntz state, which was introduced by Bratteli and Jorgensen. We show (i) a necessary and sufficient condition of the uniqueness of the extension, (ii) the complete classification of pure sub-Cuntz states up to unitary equivalence of their GNS representations, and (iii) the decomposition formula of a mixing sub-Cuntz state into a convex hull of pure sub-Cuntz states. Invariants of GNS representations of pure sub-Cuntz states are realized as conjugacy classes of nonperiodic homogeneous unit vectors in a tensor-power vector space. It is shown that this state parameterization satisfies both the U (n)-covariance and the compatibility with a certain tensor product. For proofs of main theorems, matricizations of state parameters and properties of free semigroups are used.Mathematics Subject Classifications (2010). 46K10.

show abstract

R-Matrices and the Yang–Baxter equation on GNS representations on C∗-bialgebras

Cited by 2 publications

References 25 publications

Pure States on Cuntz Algebras Arising from Geometric Progressions

Pure States on Cuntz Algebras Arising from Geometric Progressions

Classification of Sub-Cuntz States

Contact Info

Product

Resources

About