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Publications and source records attributed to Katsunori Kawamura.
Bosons and fermions are described by using canonical generators of Cuntz algebras on any permutative representation. We show a fermionization of bosons which universally holds on any permutative representation of the Cuntz algebra ${\cal O}_{2}$. As examples, we show fermionizations on the Fock space and the infinite wedge.
A measurable map $T$ on a measure space induces a representation $Π_{T}$ of a Cuntz algebra ${\cal O}_{N}$ when $T$ satisfies a certain condition. For such two maps $τ$ and $σ$ and representations $Π_τ$ and $Π_σ$ associated with them, we show that $Π_τ$ is the restriction of $Π_σ$ when $τ$ is a jump transformation of $σ$. Especially, the Gauss map $τ_1$ and the Farey map $σ_1$ induce representations $Π_{τ_1}$ of ${\cal O}_{\infty}$ and that $Π_{σ_1}$ of ${\cal O}_{2}$, respectively, and $Π_{τ_1}=Π_{σ_1}|_{{\cal O}_{\infty}}$ with respect to a certain embedding of ${\cal O}_{\infty}$ into ${\cal O}_{2}$.
Bosons and fermions are described by using canonical generators of Cuntz algebras on any permutative representation. According to branching laws associated with these descriptions, a certain representation of the Cuntz algebra $\co{2}$ induces Fock representations ${\cal H}_{B}$ and ${\cal H}_{F}$ of bosons and fermions simultaneously. From this, a unitary operator $U$ from ${\cal H}_{B}$ to ${\cal H}_{F}$ is obtained. We show the explicit formula of the action of $U$ on the standard basis of ${\cal H}_{B}$. It is shown that $U$ preserves the particle number of ${\cal H}_{B}$ and ${\cal H}_{F}$.
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 {\rm III} factor representations of Cuntz-Krieger algebras which is different from results of the tensor product of factors of type {\rm III}.
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.
It is often inevitable to introduce an indefinite-metric space in quantum field theory, for example, which is explained for the sake of the manifestly covariant quantization of the electromagnetic field. We show two more evident mathematical reasons why such indefinite metric appears. The first idea is the replacement of involution on an algebra. For an algebra ${\cal A}$ with an involution $\sdag$ such that a representation of the involutive algebra $({\cal A},\sdag)$ brings an indefinite-metric space, we replace the involution $\sdag$ with a new one * on ${\cal A}$ such that $({\cal A},*)$ is a well-known involutive algebra acting on a representation space with positive definite metric. This explains that non-isomorphic two involutive algebras are transformed each other by the replacement of involution. The second is that a covariant (Hilbert space) representation $({\cal H},π,U)$ of an involutive dynamical system $(({\cal A},*),{\bf Z}_{2},α)$ brings a Krein space representation of the algebra ${\cal A}$ with the replaced involution. For example, we show representations of abnormal CCRs, CARs and pseudo-Cuntz algebras arising from those of standard CCRs, CARs and Cuntz algebras.
Cirelli, Manià and Pizzocchero generalized quantum mechanics by Kähler geometry. Furthermore they proved that any unital C$^{*}$-algebra is represented as a function algebra on the set of pure states with a noncommutative $*$-product as an application. The ordinary quantum mechanics is regarded as a dynamical system of the projective Hilbert space ${\cal P}({\cal H})$ of a Hilbert space ${\cal H}$. The space ${\cal P}({\cal H})$ is an infinite dimensional Kähler manifold of positive constant holomorphic sectional curvature. In general, such dynamical system is constructed for a general Kähler manifold of nonzero constant holomorphic sectional curvature $c$. The Hilbert ball $B_{\cal H}$ is defined by the open unit ball in ${\cal H}$ and it is a Kähler manifold with $c<0$. We introduce the quantum mechanics on $B_{\cal H}$. As an application, we show the structure of the noncommutative function algebra on $B_{\cal H}$.
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 from restrictions of representations of ${\cal O}_{\infty}$ on ${\cal B}$. For example, it is shown that the Fock representation of ${\cal B}$ is given as the restriction of the standard representation of ${\cal O}_{\infty}$ on ${\cal B}$.
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.
We introduce a nonsymmetric, associative tensor product among representations of Cuntz algebras by using embeddings. We show the decomposition formulae of tensor products for permutative representations explicitly We apply decomposition formulae to determine properties of endomorphisms.
It is often inevitable to introduce an indefinite-metric space in quantum field theory. There is a problem to determine the metric structure of a given representation space of field operators. We show the systematic method to determine such indefinite-metric explicitly. At first, we choose a new involution $*$ of field operators instead of the original involution $\sdag$ such that there is a Hilbert space $({\cal H},<\cdot|\cdot>)$ with the positive-definite metric $<\cdot|\cdot>$ which is consistent with $*$. Next we find another hermitian form $(\cdot|\cdot)$ on ${\cal H}$ such that $({\cal H},(\cdot|\cdot))$ is a Krein space and $(\cdot|\cdot)$ is consistent with $\sdag$. We apply this method to various models and show that our results coincide with known results.
We generalize the Serre-Swan theorem to non-commutative C$^{*}$-algebras. For a Hilbert C$^{*}$-module $X$ over a C$^{*}$-algebra ${\cal A}$, we introduce a hermitian vector bundle $\exx$ associated to $X$. We show that there is a linear subspace $Γ_{X}$ of the space of all holomorphic sections of ${\cal E}_{X}$ and a flat connection $D$ on ${\cal E}_{X}$ with the following properties: (i) $Γ_{X}$ is a Hilbert ${\cal A}$-module with the action of ${\cal A}$ defined by $D$, (ii) the C$^{*}$-inner product of $Γ_{X}$ is induced by the hermitian metric of ${\cal E}_{X}$, (iii) ${\cal E}_{X}$ is isomorphic to an associated bundle of an infinite dimensional Hopf bundle, (iv) $Γ_{X}$ is isomorphic to $X$.
Previously, we have shown that the CAR algebra for fermions is embedded in the Cuntz algebra ${\cal O}_{2}$ in such a way that the generators are expressed in terms of polynomials in the canonical generators of the latter, and it coincides with the U(1)-fixed point subalgebra ${\cal A}\equiv {\cal O}_{2}^{U(1)}$ of ${\cal O}_{2}$ for the canonical gauge action. Based on this embedding formula, some properties of ${\cal A}$ are studied in detail by restricting those of ${\cal O}_{2}$. Various endomorphisms of ${\cal O}_{2}$, which are defined by polynomials in the canonical generators, are explicitly constructed, and transcribed into those of ${\cal A}$. Especially, we investigate branching laws for a certain family of such endomorphisms with respect to four important representations, i.e., the Fock representation, the infinite wedge representation and their duals. These endomorphisms are completely classified by their branching laws. As an application, we show that the reinterpretation of the Fock vacuum as the Dirac vacuum is described in representation theory through a mixture of fermions.
We introduce representations of the Cuntz algebra $\con$ which are parameterized by sequences in the set of unit vectors in ${\bf C}^{N}$. These representations are natural generalizations of permutative representations by Bratteli-Jorgensen and Davidson-Pitts. We show their existence, irreducibility, equivalence, uniqueness of irreducible decomposition and decomposition formulae by using parameters of representations.
In our previous articles, we have presented a class of endomorphisms of the Cuntz algebras which are defined by polynomials of canonical generators and their conjugates. We showed the classification of some case under unitary equivalence by help of branching laws of permutative representations. In this article, we construct an automaton which is called the Mealy machine associated with the endomorphism in order to compute its branching law. We show that the branching law is obtained as outputs from the machine for the input of information of a given representation.
We generalize permutative representations of the Cuntz algebras for the \cka\ $\coa$ for any $A$. We characterize cyclic permutative representations by notions of cycle and chain, and show their existence and uniqueness. We show necessary and sufficient conditions for their irreducibility and equivalence. In consequence, we have a complete classification of permutative representations of $\coa$. Furthermore we show decomposition formulae and that the uniqueness of irreducible decomposition holds for permutative representation.
A unitary equivalence class of endomorphisms of a unital C$^{*}$-algebra ${\cal A}$ is called a {\it sector} of ${\cal A}$. We introduced permutative endomorphisms of the Cuntz algebra ${\cal O}_N$ in the previous work. Branching laws of permutative representations of ${\cal O}_N$ by them are computed by directed regular graphs. In this article, we classify sectors associated with permutative endomorphisms of ${\cal O}_N$ by their graph invariants concretely.
Based on an embedding formula of the CAR algebra into the Cuntz algebra ${\mathcal O}_{2^p}$, properties of the CAR algebra are studied in detail by restricting those of the Cuntz algebra. Various $\ast$-endomorphisms of the Cuntz algebra are explicitly constructed, and transcribed into those of the CAR algebra. In particular, a set of $\ast$-endomorphisms of the CAR algebra into its even subalgebra are constructed. According to branching formulae, which are obtained by composing representations and $\ast$-endomorphisms, it is shown that a KMS state of the CAR algebra is obtained through the above even-CAR endomorphisms from the Fock representation. A $U(2^p)$ action on ${\mathcal O}_{2^p}$ induces $\ast$-automorphisms of the CAR algebra, which are given by nonlinear transformations expressed in terms of polynomials in generators. It is shown that, among such $\ast$-automorphisms of the CAR algebra, there exists a family of one-parameter groups of $\ast$-automorphisms describing time evolutions of fermions, in which the particle number of the system changes by time while the Fock vacuum is kept invariant.