We show that a tensor product among representation of certain C$^{*}$-algebras induces a bialgebra. Let $\tilde{\cal O}_{*}$ be the smallest unitization of the direct sum of Cuntz algebras \[{\cal O}_{*}\equiv {\bf C}\oplus {\cal O}_{2}\oplus {\cal O}_{3}\oplus{\cal O}_{4}\oplus ....\] We show that there exists a non-cocommutative comultiplication $Δ$ and a counit $ε$ of $\tilde{\cal O}_{*}$. From $Δ,\vep$ and the standard algebraic structure, $\tilde{\cal O}_{*}$ is a C$^{*}$-bialgebra. Furthermore we show the following: (i) The antipode on $\tilde{\cal O}_{*}$ never exist. (ii) There exists a unique Haar state on $\tilde{\cal O}_{*}$. (iii) For a certain one-parameter bialgebra automorphism group of $\tilde{\cal O}_{*}$, a KMS state on $\tilde{\cal O}_{*}$ exists.
Let ${\sf CK}_{*}$ denote the C$^{*}$-algebra defined by the direct sum of all Cuntz-Krieger algebras. We introduce a comultiplication $Δ_ϕ$ and a counit $ε$ on ${\sf CK}_{*}$ such that $Δ_ϕ$ is a nondegenerate $*$-homomorphism from ${\sf CK}_{*}$ to ${\sf CK}_{*}\otimes {\sf CK}_{*}$ and $ε$ is a $*$-homomorphism from ${\sf CK}_{*}$ to ${\bf C}$. From this, ${\sf CK}_{*}$ is a counital non-commutative non-cocommutative C$^{*}$-bialgebra. Furthermore, C$^{*}$-bialgebra automorphisms, a tensor product of representations and C$^{*}$-subbialgebras of ${\sf CK}_{*}$ are investigated.
The direct sum ${\cal O}_{*}$ of all Cuntz algebras has a non-cocommutative comultiplication $Δ_φ$ such that there exists no antipode of any dense subbialgebra of the C$^{*}$-bialgebra $({\cal O}_{*},Δ_φ)$. From states equations of ${\cal O}_{*}$ with respect to the tensor product, we construct an operator $W$ for $({\cal O}_{*},Δ_φ)$ such that $W^{*}$ is an isometry, $W(x\otimes I)W^{*}=Δ_φ(x)$ for each $x\in {\cal O}_{*}$ and $W$ satisfies the pentagon equation.
If phantom Cuntz-Krieger algebras do not exist, then real rank zero Cuntz-Krieger algebras can be characterized by outer properties. In this survey paper, a summary of the known results on non-existence of phantom Cuntz-Krieger algebras is given.
Let ${\cal O}_{*}$ be the C$^{*}$-algebra defined as the direct sum of all Cuntz algebras. Then ${\cal O}_{*}$ has a non-cocommutative comultiplication $Δ_ϕ$ and a counit $ε$. Let ${\rm BI}({\cal O}_{*})$ denote the set of all closed biideals of the C$^{*}$-bialgebra $({\cal O}_{*},Δ_ϕ,ε)$ and let ${\cal P}({\bf P})$ denote the power set of the set of all prime numbers. We show a one-to-one correspondence between ${\rm BI}({\cal O}_{*})$ and ${\cal P}({\bf P})$. Furthermore, we show that for any ${\cal I}$ in ${\rm BI}({\cal O}_{*})$, there exists a C$^{*}$-subbialgebra ${\cal B}_{\cal I}$ of …
We extend the definition of an extension of a right Hilbert module to the setting of Hilbert bimodules and show that an extension of Hilbert bimodules induces an extension of Cuntz-Pimsner algebras. We also study the Cuntz-Pimsner algebra associated to the multiplier bimodule and show that an extension can be realised as a restricted direct-sum bimodule.
We investigate subalgebras $A$ of the Cuntz algebra ${\mathcal O}_n$ that arise as finite direct sums of corners of the UHF-subalgebra ${\mathcal F}_n$. For such an $A$, we completely determine its normalizer group inside ${\mathcal O}_n$.
Bosons and fermions are often written by elements of other algebras. M. Abe gave a recursive realization of the boson by formal infinite sums of the canonical generators of the Cuntz algebra ${\cal O}_{\infty}$. We show that such formal infinite sum always makes sense on a certain dense subspace of any permutative representation of ${\cal O}_{\infty}$. In this meaning, we can regard as if the algebra ${\cal B}$ of bosons was a unital $*$-subalgebra of ${\cal O}_{\infty}$ on a given permutative representation by keeping their unboundedness. By this relation, we compute branching laws arising fr…
Let $M_{*}({\bf C})$ denote the C$^{*}$-algebra defined as the direct sum of all matrix algebras $\{M_{n}({\bf C}):n\geq 1\}$. It is known that $M_{*}({\bf C})$ has a non-cocommutative comultiplication $Δ_φ$. We show that the C$^{*}$-bialgebra $(M_{*}({\bf C}),Δ_φ)$ has a universal $R$-matrix $R$ such that the quasi-cocommutative C$^{*}$-bialgebra $(M_{*}({\bf C}),Δ_φ,R)$ is triangular.
Let ${\cal A}_{0}(*)$ denote the direct sum of a certain set of UHF algebras and let ${\cal A}(*)\equiv {\bf C}\oplus {\cal A}_{0}(*)$. We introduce a non-cocommutative comultiplication $Δ_ϕ$ on ${\cal A}(*)$, and give an example of comodule-C$^{*}$-algebra of the C$^{*}$-bialgebra $({\cal A}(*),Δ_ϕ)$. With respect to $Δ_ϕ$, we define a non-symmetric tensor product of *-representations of UHF algebras and show tensor product formulas of GNS representations by product states.
Let ${\Bbb F}_{n}$ 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 $Δ_φ$ on $\bigoplus C^{*}({\Bbb F}_{n})$ such that $(\bigoplus C^{*}({\Bbb F}_{n}),\,Δ_φ)$ is a non-cocommutative C$^{*}$-bialgebra. With respect to $Δ_φ$, the tensor product $π\otimes_φπ'$ of any two representations $π$ and $π'$ of free groups is defined. The operation $\ptimes$ is associative and non-commutative. We compute its tensor product formulas of sev…
In a pure C*-algebra (i.e., one having suitable regularity properties in its Cuntz semigroup), any element on which all bounded traces vanish is a sum of 7 commutators.
We produce a variety of odd bounded Fredholm modules and odd spectral triples on Cuntz-Krieger algebras by means of realizing these algebras as "the algebra of functions on a non-commutative space" coming from a sub shift of finite type. We show that any odd $K$-homology class can be represented by such an odd bounded Fredholm module or odd spectral triple. The odd bounded Fredholm modules that are constructed are finitely summable. The spectral triples are $θ$-summable although their bounded transform, when constructed using the sign-function, will already on the level of analytic $K$-cycles …
We characterize, in terms of elementary properties, the abelian monoids which are direct limits of finite direct sums of monoids of the form $(Z/nZ)\sqcup\{0\}$ (where 0 is a new zero element), for positive integers $n$. The key properties are the Riesz refinement property and the requirement that each element $x$ has finite order, that is, $(n+1)x=x$ for some positive integer $n$. Such monoids are necessarily semilattices of abelian groups, and part of our approach yields a characterization of the Riesz refinement property among semilattices of abelian groups. Further, we describe the monoids…
In this paper we extend work of Kawamura, see kawamura, for Cuntz-Krieger algebras O_A for infinite matrices A. We generalize the definition of branching systems, prove their existence for any given matrix A and show how they induce some very concrete representations of O_A. We use these representations to describe the Perron-Frobenius operator, associated to an nonsingular transformation, as an infinite sum and under some hypothesis we find a matrix representation for the operator. We finish the paper with a few examples.
We give a short proof of the nuclearity property of a class of Cuntz-Pimsner algebras associated with a Hilbert A-bimodule M, where A is a separable and nuclear C*-algebra. We assume that the left A-action on the bimodule M is given in terms of compact module operators and that M is direct summand of the standard Hilbert module over A.
A row co-isometry is a family $(V_i)_{i=0}^{N-1}$ of operators on a Hilbert space, subject to the relation $$\sum_{i=0}^{N-1}V_iV_i^*=I.$$ As shown in \cite{BJK00}, row co-isometries appear as compressions of representations of Cuntz algebras. In this paper we will present some general constructions of Parseval frames for Hilbert spaces, obtained by iterating the operators $V_i$ on a finite set of vectors. The constructions are based on random walks on finite graphs. As applications of our constructions we obtain Parseval Fourier bases on self-affine measures and Parseval Walsh bases on the in…
It is shown that the Cuntz semigroup is a complete invariant for the C*-algebras that can be realized as an inductive limit of a sequence of finite direct sums of splitting interval algebras.
Given a quantum graph $\mathcal{G}=(B,ψ,A)$, we define a C*-correspondence $E_\mathcal{G}$ over the noncommutative vertex C*-algebra $B$, called the quantum edge correspondence. For a classical graph $\mathcal{G}$, $E_\mathcal{G}$ is the usual graph correspondence spanned by the edges of $\mathcal{G}$. When the quantum adjacency matrix $A\colon B\to B$ is completely positive, we show that $E_\mathcal{G}$ is faithful if and only if $\ker(A)$ does not contain a central summand of $B$. In this case, we show that the Cuntz-Pimsner algebra $\mathcal{O}_{E_\mathcal{G}}$ is isomorphic to a quotient o…
We extend the usual theory of universal C*-algebras from generators and relations in order to allow some relations to be described using the strong operator topology. In particular, we can allow some infinite sum relations. We prove a universal property for the algebras we define and we show how the Cuntz algebra of infinite isometries as well as the Exel-Laca algebras can be described using infinite sum relations. Finally, we give some sufficient conditions for when a C*-algebra generated by projections and partial isometries is a universal C*-algebra using only norm relations, in case one st…
We characterize the non-commutative Aleksandrov--Clark measures and the minimal realization formulas of contractive and, in particular, isometric non-commutative rational multipliers of the Fock space. Here, the full Fock space over $\mathbb{C} ^d$ is defined as the Hilbert space of square--summable power series in several non-commuting formal variables, and we interpret this space as the non-commutative and multi-variable analogue of the Hardy space of square--summable Taylor series in the complex unit disk. We further obtain analogues of several classical results in Aleksandrov--Clark measur…
We prove a classification theorem for purely infinte simple C*-algebras that is strong enough to show that the tensor products of two different irrational rotation algebras with the same even Cuntz algebra are isomorphic. In more detail, let C be be the class of simple C*-algebras A which are direct limits A = lim A_k, in which each A_k is a finite direct sum of algebras of the form C(X) \otimes M_n \otimes O_m, where m is even, O_m is the Cuntz algebra, X is either a point, a compact interval, or the circle S^1, and each map A_k ---> A is approximately absorbing. ("Approximately absorbing" is…
The Weyl group of the Cuntz algebra O_n, with n finite, is investigated. This is (isomorphic to) the group of polynomial automorphisms of O_n, namely those induced by unitaries that can be written as finite sums of words in the canonical generating isometries and their adjoints. A necessary and sufficient algorithmic combinatorial condition is found for deciding when a polynomial endomorphism restricts to an automorphism of the canonical diagonal MASA. Some steps towards a general criterion for invertibility of such endomorphisms on the whole of O_n are also taken. A condition for verifying in…
The notion of permutative representation is generalized to the $2$-adic ring $C^*$-algebra $\mathcal{Q}_{2}$. Permutative representations of $\mathcal{Q}_2$ are then investigated with a particular focus on the inclusion of the Cuntz algebra $\mathcal{O}_2\subset\mathcal{Q}_2$. Notably, every permutative representation of $\mathcal{O}_2$ is shown to extend automatically to a permutative representation of $\mathcal{Q}_2$ provided that an extension whatever exists. Moreover, all permutative extensions of a given representation of $\mathcal{O}_2$ are proved to be unitarily equivalent to one anothe…
We construct KMS-states from $\mathrm{Li}_1$-summable semifinite spectral triples and show that in several important examples the construction coincides with well-known direct constructions of KMS-states for naturally defined flows. Under further summability assumptions the constructed KMS-state can be computed in terms of Dixmier traces. For closed manifolds, we recover the ordinary Lebesgue integral. For Cuntz-Pimsner algebras with their gauge flow, the construction produces KMS-states from traces on the coefficient algebra and recovers the Laca-Neshveyev correspondence. For a discrete group…
To any periodic, unital and full C*-dynamical system (A, α, R) an invertible operator s acting on the Banach space of trace functionals of the fixed point algebra is canonically associated. KMS states correspond to positive eigenvectors of s. A Perron-Frobenius type theorem asserts the existence of KMS states at inverse temperatures equal the logarithms of the inner and outer spectral radii of s (extremal KMS states). Examples arising from subshifts in symbolic dynamics, self-similar sets in fractal geometry and noncommutative metric spaces are discussed. Certain subshifts are naturally associ…
