Polynomial Endomorphisms of the Cuntz Algebras Arising from Permutations: I ? General Theory

Kawamura, Katsunori

doi:10.1007/s11005-005-0344-8

Cited by 26 publications

(27 citation statements)

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“…Reducible endomorphisms of O 2 . There are ten reducible endomorphisms of O 2 [18]. In the sequel we follow closely the discussion in [4,Section 6].…”

Section: Resultsmentioning

confidence: 98%

“…2 These values are plotted in the last column of Table 1 (below), modelled on those in [18,19,26], where for multiindices α and β we denote s α s * β by s α,β . 3 Notice that by the results in [18,19] (24) and ρ (14) (23) , which clearly give index 1.…”

Section: Resultsmentioning

confidence: 99%

“…those arising from permutation unitaries in the core UHF-subalgebra. In particular, Kawamura [18,19] analyzed systematically some properties like properness, irreducibility and branching laws of rank-two (or "quadratic") permutation endomorphisms of O 2 . More recently Skalski and Zacharias computed Voiculescu's topological entropy [29] of the same endomorphisms [26], thus extending Choda's computation of the entropy of the canonical shift of O n [2].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Computing the Jones index of quadratic permutation endomorphisms of O2

Conti

Szymański

2009

Journal of Mathematical Physics

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“…Reducible endomorphisms of O 2 . There are ten reducible endomorphisms of O 2 [18]. In the sequel we follow closely the discussion in [4,Section 6].…”

Section: Resultsmentioning

confidence: 98%

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Computing the Jones index of quadratic permutation endomorphisms of O2

Conti

Szymański

2009

Journal of Mathematical Physics

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“…From studies of branching laws of a certain class of representations of Cuntz algebras [20,21,22], we found a non-symmetric tensor product of representations of Cuntz algebras [23]. Subsequently, this tensor product was explained by the C * -bialgebra which is defined by the direct sum of all Cuntz algebras [24].…”

Section: Motivationmentioning

confidence: 99%

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

Kawamura

2012

Algebr Represent Theor

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We introduced a non-symmetric tensor product of any two states or any two representations of Cuntz-Krieger algebras associated with a certain non-cocommutative comultiplication in previous our work. In this paper, we show that a certain set of KMS states is closed with respect to the tensor product. From this, we obtain formulae of tensor product of type III factor representations of Cuntz-Krieger algebras which is different from results of the tensor product of factors of type III. (2000). 46K10, 46L30, 46L35. Mathematics Subject Classifications

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“…Many authors have generalized and further studied these representations. For example, motivated by the work of Kawamura in [17], Lawson studied primitive partial permutation representations of polycyclic monoids and their relations to branching function systems (see [20]).…”

Section: Permutative Representations Of Ultragraph C*-algebrasmentioning

confidence: 99%

Branching systems and general Cuntz–Krieger uniqueness theorem for ultragraph C^*-algebras

Royer

2016

Int. J. Math.

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We give a notion of branching systems on ultragraphs. From this we build concrete representations of ultragraph C*-algebras on the bounded linear operators of Hilbert spaces. To each branching system of an ultragraph we describe the associated Perron-Frobenius operator in terms of the induced representation. We show that every permutative representation of an ultragraph C*-algebra is unitary equivalent to a representation arising from a branching system. We give a sufficient condition on ultragraphs such that a large class of representations of the C*-algebras of these ultragraphs is permutative. To give a sufficient condition on branching systems so that their induced representations are faithful we generalize Szymański's version of the Cuntz-Krieger uniqueness theorem for ultragraph C*-algebras.Ultragraphs are combinatorial objects that generalize directed graphs. Roughly speaking an ultragraph is a graph where the image of the range map does not belong to the set of vertices, but instead to the power set on vertices. The concept was introduced by Mark Tomforde in [24] with an eye towards C*-algebra applications. In particular, Tomforde showed in [24] how to associate a C*-algebra to an ultragraph and proved that there exist ultragraph C*-algebras that are neither Exel-Laca algebras nor graph C*-algebras. So the study of ultragraph C*-algebras is of interest, but furthermore, ultragraph C*-algebras were key in answering the long-standing question of whether an Exel-Laca algebra is Morita equivalent to a graph algebra (see [16]).Recently the scope of ultragraphs has surpassed the realm of C*-algebras, reaching applications to symbolic dynamics. In particular, ultragraphs are fundamental in characterizing one-sided shift spaces over infinite alphabets. Furthermore, questions regarding the dynamics of one-sided shift spaces over infinite alphabets were answered using ultragraphs and their C*-algebras (see [13]).The above evidence leads us to believe that there are still many applications of ultragraphs to be found. Indeed, among the results we present in this paper, we will show a connection (via branching systems) between ultragraphs and the Perron-Frobenius operator from the ergodic theory (see Section 5), therefore generalizing results previously obtained for graph algebras and Cuntz-Krieger C*-algebras (see [10,12]).Branching systems are not just important as a way to connect ultragraphs to the ergodic theory. They have appeared in fields as random walks, symbolic dynamics, wavelet theory and are strongly connected to the representation theory of combinatorial algebras. In particular, Bratteli and Jorgensen have initiated the study of wavelets and representations of the Cuntz algebra via branching systems in [3,4]. After this, many results relating branching systems and representations of generalized Cuntz algebras were obtained, see

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Polynomial Endomorphisms of the Cuntz Algebras Arising from Permutations: I ? General Theory

Abstract: We introduce a class of endomorphisms of the Cuntz algebras which are defined by polynomials of generators. We show their classification under unitary equivalence by help of permutative representations.

Cited by 26 publications

References 8 publications

Computing the Jones index of quadratic permutation endomorphisms of O2

Computing the Jones index of quadratic permutation endomorphisms of O2

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

Branching systems and general Cuntz–Krieger uniqueness theorem for ultragraph C^*-algebras

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Polynomial Endomorphisms of the Cuntz Algebras Arising from Permutations: I ? General Theory

Abstract: We introduce a class of endomorphisms of the Cuntz algebras which are defined by polynomials of generators. We show their classification under unitary equivalence by help of permutative representations.

Cited by 26 publications

References 8 publications

Computing the Jones index of quadratic permutation endomorphisms of O2

Computing the Jones index of quadratic permutation endomorphisms of O2

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

Branching systems and general Cuntz–Krieger uniqueness theorem for ultragraph C*-algebras

Contact Info

Product

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Branching systems and general Cuntz–Krieger uniqueness theorem for ultragraph C^*-algebras