Tensor Products of Type III Factor Representations of Cuntz–Krieger Algebras

Kawamura, Katsunori

doi:10.1007/s10468-012-9362-2

Cited by 8 publications

(9 citation statements)

References 25 publications

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“…Such bialgebra structures do not appear before one takes direct sums. With respect to their comultiplications, new tensor products among representations of these C * -algebras and their tensor product formulas were obtained [11,14]. In [12], we gave a general method to construct a C *bialgebra from a given system of C * -algebras and special * -homomorphisms among them.…”

Section: Motivationmentioning

confidence: 99%

Non-Cocommutative C-Bialgebra Defined as the Direct Sum of Free Group C-Algebras

Kawamura¹

2015

Glasgow Math. J.

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Leti ${\Bbb F}$n be the free group of rank n and let $\bigoplus C^{*}({\Bbb F}_{n})$ denote the direct sum of full group C*-algebras $C^{*}({\Bbb F}_{n})$ of ${\Bbb F}_{n} (1\leq n<\infty$). We introduce a new comultiplication Δϕ on $\bigoplus C^{*}({\Bbb F}_{n})$ such that $(\bigoplus C^{*}({\Bbb F}_{n}),\,\Delta_{\varphi})$ is a non-cocommutative C*-bialgebra. With respect to Δϕ, the tensor product π⊗ϕπ′ of any two representations π and π′ of free groups is defined. The operation ×ϕ is associative and non-commutative. We compute its tensor product formulas of several representations.

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Section: Motivationmentioning

confidence: 99%

Non-Cocommutative C-Bialgebra Defined as the Direct Sum of Free Group C-Algebras

Kawamura¹

2015

Glasgow Math. J.

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“…The state ρ a is called the quasi-free state on O n by a [3,19]. It is known that the GNS representation by ρ a is a type III factor representation; ρ a ∼ ρ b if and only if a = b; ρ a ⊗ ϕ ρ b = ρ a⊠b [25,36,39]. Fix a ∈ Λ n and let (H, π, Ω) denote the GNS representation by ρ a .…”

Section: Sub-cuntz States Associated With Permutative Representationsmentioning

confidence: 99%

Classification of Sub-Cuntz States

Kawamura

2014

Algebr Represent Theor

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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.

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“…About properties of (O * , ϕ ) and its representations, see [13,14,16,20]. About generalizations of (O * , ϕ ), see [15,21].…”

Section: * -Bialgebra (O * ϕ )mentioning

confidence: 99%

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

Kawamura

2013

Linear Algebra and its Applications

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A new construction method of R-matrix is given. Let A be a C * -bialgebra with a comultiplication without the assumption of the quasi-cocommutativity. For two states ω and ψ of A which satisfy certain conditions, we construct a unitary R-matrix R(ω, ψ) of the C * -bialgebra (A, ) on the tensor product of GNS representation spaces associated with ω and ψ. The set {R(ω, ψ) : ω, ψ} satisfies a kind of Yang-Baxter equation. Furthermore, we show a nontrivial example of such R-matrices for a non-quasi-cocommutative C * -bialgebra.

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Tensor Products of Type III Factor Representations of Cuntz–Krieger Algebras

Cited by 8 publications

References 25 publications

Non-Cocommutative C-Bialgebra Defined as the Direct Sum of Free Group C-Algebras

Non-Cocommutative C-Bialgebra Defined as the Direct Sum of Free Group C-Algebras

Classification of Sub-Cuntz States

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

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Tensor Products of Type III Factor Representations of Cuntz–Krieger Algebras

Cited by 8 publications

References 25 publications

Non-Cocommutative C*-Bialgebra Defined as the Direct Sum of Free Group C*-Algebras

Non-Cocommutative C*-Bialgebra Defined as the Direct Sum of Free Group C*-Algebras

Classification of Sub-Cuntz States

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

Contact Info

Product

Resources

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Non-Cocommutative C-Bialgebra Defined as the Direct Sum of Free Group C-Algebras

Non-Cocommutative C-Bialgebra Defined as the Direct Sum of Free Group C-Algebras