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Ubiquitin and ubiquitin-like modifiers modulate NK cell-mediated recognition and killing of damaged cells

  • Received: 28 September 2017 Accepted: 21 November 2017 Published: 23 November 2017
  • Efficient elimination of transformed and virus-infected cells by natural killer (NK) cells mainly depends on the recognition of “induced self” ligands by activating receptors, including NKG2D and DNAM1. The surface expression of these ligands in stressed or diseased cells results from the integration of transcriptional, post-transcriptional and post-translational mechanisms. Among post-translational mechanisms, recent findings indicate that ubiquitin and ubiquitin-like modifications, namely ubiquitination and SUMOylation, contribute to a very rapid negative regulation of NKG2D and DNAM1 ligand surface expression promoting either ligand degradation or ligand intracellular retention. On the other hand, accumulating evidences demonstrate that NKG2D receptor expression is down-regulated by ubiquitin-dependent endocytosis upon ligand stimulation. In this scenario, the overall consequence of the post-translational modifications of activating NK cell receptors and of their ligands on target cells is to impair effector cell-mediated recognition of damaged cells. Our review summarizes recent findings on the role of post-translational modifications in the modulation of target cell susceptibility to NK cell-mediated killing.

    Citation: Rosa Molfetta, Beatrice Zitti, Angela Santoni, Rossella Paolini. Ubiquitin and ubiquitin-like modifiers modulate NK cell-mediated recognition and killing of damaged cells[J]. AIMS Allergy and Immunology, 2017, 1(4): 164-180. doi: 10.3934/Allergy.2017.4.164

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  • Efficient elimination of transformed and virus-infected cells by natural killer (NK) cells mainly depends on the recognition of “induced self” ligands by activating receptors, including NKG2D and DNAM1. The surface expression of these ligands in stressed or diseased cells results from the integration of transcriptional, post-transcriptional and post-translational mechanisms. Among post-translational mechanisms, recent findings indicate that ubiquitin and ubiquitin-like modifications, namely ubiquitination and SUMOylation, contribute to a very rapid negative regulation of NKG2D and DNAM1 ligand surface expression promoting either ligand degradation or ligand intracellular retention. On the other hand, accumulating evidences demonstrate that NKG2D receptor expression is down-regulated by ubiquitin-dependent endocytosis upon ligand stimulation. In this scenario, the overall consequence of the post-translational modifications of activating NK cell receptors and of their ligands on target cells is to impair effector cell-mediated recognition of damaged cells. Our review summarizes recent findings on the role of post-translational modifications in the modulation of target cell susceptibility to NK cell-mediated killing.


    In [22] and [23], D. Ruelle has introduced the notion of Smale space. A Smale space is a hyperbolic dynamical system with local product structure. He has constructed groupoids and its operator algebras from the Smale spaces. After the Ruelle's initial study, I. Putnam in [14] (cf. [9], [15], [16], [17], [26], etc.) constructed various groupoids from Smale spaces and studied their C-algebras. The class of Smale spaces contain two important subclasses of topological dynamical systems as its typical examples. One is the class of shifts of finite type, which are sometimes called topological Markov shifts. The other one is the class of hyperbolic toral automorphisms. The study of the former class from the view point of C-algebras is closely related to the study of Cuntz-Krieger algebras as in [7], [8], [10], [11], [12], etc. That of the latter class is closely related to the study of the crossed product C-algebras of the homeomorphisms of the hyperbolic automorphisms on the torus.

    In this paper, we will focus on the study of the latter class, the hyperbolic toral automorphisms from the view points of C-algebras constructed from the associated groupoids as Smale spaces. Let A=[abcd]GL(2,Z) be a hyperbolic matrix. Let q:R2R2/Z2 be the natural quotient map. We denote by R2/Z2 the two-dimensional torus T2 with metric d defined by

    d(x,y)=inf{zw:q(z)=x,q(w)=y,z,wR2} for x,yT2

    where is the Euclid norm on R2. Then the matrix A defines a homeomorphism on T2 which is called a hyperbolic toral automorphism. It is a specific example of an Anosov diffeomorphism on a compact Riemannian manifold (see [4], [25], etc.). Let λu,λs be the eigenvalues of A such that |λu|>1>|λs|. They are both real numbers. Let vu=(u1,u2),vs=(s1,s2) be the normalized eigenvectors for λu,λs, respectively. The direction along vu expands by A, whereas the direction of vs expands by A1. These directions determine local product structure which makes T2 a Smale space. The groupoid GaA introduced by D. Ruelle [22] of the asymptotic equivalence relation is defined by

    GaA={(x,z)T2×T2limnd(Anx,Anz)=limnd(Anx,Anz)=0} (1)

    with its unit space

    (GaA)(0)={(x,x)T2×T2}=T2. (2)

    The multiplication and the inverse operation on GaA are defined by

    (x,z)(z,w)=(x,w),(x,z)1=(z,x) for (x,z),(z,w)GaA.

    As in [14], the groupoid GaA has a natural topology defined by inductive limit topology, which makes GaA étale. The étale groupoid GaA is called the asymptotic groupoid for the hyperbolic toral automorphism (T2,A). We will first see that the groupoid GaA is realized as a transformation groupoid T2×αAZ2 by a certain action αA:Z2Homeo(T2) associated to GaA, so that the C-algebra C(GaA) of the groupoid GaA is isomorphic to the C-algebra of the crossed product C(T2)×αAZ2 by the induced action αA:Z2Aut(C(T2)). As the action αA:Z2Homeo(T2) is free and minimal having a unique invariant ergodic measure, a general theory of C-crossed product ensures that C(T2)×αAZ2 is a simple AT-algebra having a unique tracial state (cf. [13], [14], [17]).

    Let A=[abcd]GL(2,Z) be a hyperbolic matrix which satisfies det(A)=±1, We denote by Δ(A)=(a+d)24(adbc) the discriminant of the characteristic polynomial of the matrix A, which is positive. We will show the following result.

    Theorem 1.1 (Theorem 2.10 and Proposition 3.1). The C-algebra C(GaA) of the étale groupoid GaA for a hyperbolic matrix A=[abcd] is a simple AT-algebra with unique tracial state τ that is isomorphic to the four-dimensional non-commutative torus generated by four unitaries U1,U2,V1,V2 satisfying the following relations:

    U1U2=U2U1,V1V2=V2V1,V1U1=e2πiθ1U1V1,V1U2=e2πiθ2U2V1,V2U1=e2πiθ3U1V2,V2U2=e2πiθ4U2V2,

    where

    θ1=12(1+adΔ(A)),θ2=cΔ(A),θ3=bΔ(A),θ4=12(1adΔ(A)).

    The range τ(K0(C(GaA))) of the tracial state τ of the K0-group K0(C(GaA)) of the C-algebra C(GaA) is

    τ(K0(C(GaA)))=Z+Zθ1+Zθ2+Zθ3inR. (3)

    We note that the slopes θi,i=1,2,3,4 are determined by the formulas (10), (11) for the slopes of the eigenvectors vu=(u1,u2),vs=(s1,s2).

    Since the étale groupoid GaA is a flip conjugacy invariant and the C-algebra C(GaA) has a unique tracial state written τ, we know that the trace value τ(K0(C(GaA))) is a flip conjugacy invariant of the hyperbolic toral automorphism (T2,A).

    As commuting matrices have common eigenvectors, we know that if two matrices A,BGL(2,Z) commute with each other, then the C-algebras C(GaA) and C(GaB) are canonically isomorphic. Hence two matrices [1110] and [2111] have the isomorphic C-algebras. On the other hand, as the range τ(K0(C(GaA))) of the tracial state of the K0-group K0(C(GaA)) is invariant under isomorphism class of the algebra C(GaA), the C-algebra C(GA1) is not isomorphic to C(GA2) for the matrices A1=[1110] and A2=[3121] (Proposition 4.2).

    For a vector (m,n)R2, we write the vector (m,n)t as [mn] and sometimes identify (m,n) with [mn]. A matrix A=[abcd]GL(2,Z) with det(A)=±1 is said to be hyperbolic if A does not have eigenvalues of modulus 1. Let λu,λs be the eigenvalues of A such that |λu|>1>|λs|. They are eigenvalues for unstable direction, stable direction, respectively. We note that b0,c0 because of the conditions adbc=±1 and |λu|>1>|λs|. Take nonzero eigenvectors vu,vs for the eigenvalues λu,λs such that vu=vs=1. We set vu=(u1,u2),vs=(s1,s2)T2 as vectors. The numbers λu,λs,u1,u2,s1,s2 are all real numbers because of the hyperbolicity of the matrix A. It is easy to see that the slopes u1u2,s1s2 are irrational. We set

    rA:=vu|vs.

    Define two vectors

    v1:=vurAvs,v2:=rAvuvs.

    Lemma 2.1. For two vectors x,zT2, the following three conditions are equivalent.

    (ⅰ) (x,z)GaA.

    (ⅱ) z=x+11r2A(m,n)|v1vu for some m,nZ.

    (ⅲ) z=x+11r2A(m,n)|v2vs for some m,nZ.

    Proof. For two vectors x,zT2 regarding them as elements of R2 modulo Z2, we have (x,z)GaA if and only if

    zx+tvux+svs(modZ2) for some t,sR. (4)

    In this case, we see that tvusvs=(m,n) for some m,nZ so that

    tvusvs|vu=(m,n)|vu, (5)
    tvusvs|vs=(m,n)|vs (6)

    and we have

    t=11r2A(m,n)|v1,s=11r2A(m,n)|v2. (7)

    This shows the implications (ⅰ) (ⅱ) and (ⅲ).

    Assume that (ⅱ) holds. By putting s=11r2A(m,n)|v2, we have the equalities both (5) and (6), so that tvusvs=(m,n). Hence the equality (4) holds and we see that (x,z) belongs to the groupoid GaA. This shows that the implication (ⅱ) (ⅰ) holds, and similarly (ⅲ) (ⅰ) holds.

    Let us define an action αA:Z2Homeo(T2) in the following way. We set

    αA(m,n)(x):=x+11r2A(m,n)|v1vu,(m,n)Z2,xT2.

    For a fixed (m,n)Z2, the map xT2αA(m,n)(x)T2 is the parallel transformation along the vector 11r2A(m,n)|v1vu. Hence αA(m,n) defines a homeomorphism on the torus T2. It is clear to see that αA(m,n)αA(k,l)=αA(m+k,n+l) for (m,n),(k,l)Z2.

    Lemma 2.2. Keep the above notation.

    (ⅰ) If αA(m,n)(x)=x for some xT2, then (m,n)=(0,0). Hence the action αA:Z2Homeo(T2) is free.

    (ⅱ) For xT2, the set {αA(m,n)(x)|(m,n)Z2} is dense in T2. Hence the action αA:Z2Homeo(T2) is minimal.

    Proof. (ⅰ) Suppose that αA(m,n)(x)=x for some xT2, so that 11r2A(m,n)|v1vu=(k,l) for some (k,l)Z2. As the slope of the vector vu is irrational, we have (k,l)=(0,0) and hence (m,n)=(0,0).

    (ⅱ) Let v1=(γ1,γ2). As the slope of vs is irrational and vs|v1=0, the slope γ1γ2 of v1 is irrational, so that the set {mγ1+nγ2|m,nZ} is dense in R. Since (m,n)|v1vu=(mγ1+nγ2)vu and the set {x+tvuT2|tR} is dense in T2, we see that the set

    {x+11r2A(m,n)|v1vu(m,n)Z2}

    is dense in T2.

    The action αA:Z2Homeo(T2) induces an action of Z2 to the automorphism group Aut(C(T2)) of C(T2) by fC(T2)fαA(m,n)C(T2). We write it still αA without confusing.

    If a discrete group Γ acts freely on a compact Hausdorff space X by an action α:ΓHomeo(X), the set {(x,αγ(x))X×X|xX,γΓ} has a groupoid structure in a natural way (cf. [2], [18], [19]). The groupoid is called a transformation groupoid written X×αΓ.

    Proposition 2.3. The étale groupoid GaA is isomorphic to the transformation groupoid

    T2×αAZ2={(x,αA(m,n)(x))T2×T2|(m,n)Z2}

    defined by the action αA:Z2Homeo(T2). Hence the C-algebra C(GaA) of the groupoid GaA is isomorphic to the crossed product C(T2)×αAZ2 of C(T2) by the action αA of Z2.

    Proof. By the preceding discussions, a pair (x,z)T2 belongs to the groupoid GaA if and only if z=αA(m,n)(x) for some (m,n)Z2. Since the action αA:Z2Homeo(T2) is free, the groupoid GaA is identified with the transformation groupoid T2×αAZ2 in a natural way. By a general theory of the C-algebras of groupoids ([2], [18]), the C-algebra C(T2×αAZ2) of the groupoid T2×αAZ2 is isomorphic to the crossed product C(T2)×αAZ2.

    Remark 2.4. Define a map αA:Z2T2 by

    αA(m,n):=11r2A(m,n)|v1vu,(m,n)Z2. (8)

    It is easy to see that the étale groupoid GaA may be written

    GaA=T2×αA(Z2) (9)

    as a transformation groupoid.

    We set

    θ1:=u1s2u1s2u2s1,θ2:=u2s2u1s2u2s1, (10)
    θ3:=u1s1u1s2u2s1,θ4:=u2s1u1s2u2s1. (11)

    Lemma 2.5. The real numbers θi,i=1,2,3,4 satisfy

    θ2θ1=θ4θ3=u2u1,θ1θ3=θ2θ4=s2s1, (12)
    θ1+θ4=1. (13)

    Conversely, if real numbers ζi,i=1,2,3,4 satisfy

    ζ2ζ1=ζ4ζ3=u2u1,ζ1ζ3=ζ2ζ4=s2s1, (14)
    ζ1+ζ4=1, (15)

    then we have ζi=θi,i=1,2,3,4.

    Proof. The identities (12) and (13) are immediate. Conversely, suppose that real numbers ζi,i=1,2,3,4 satisfy (14) and (15). As ζ1=u2u1ζ2=u2u1(s2s1)ζ4, the equality (15) implies

    {u2u1(s2s1)+1}ζ4=1,

    so that

    ζ4=u2s1u1s2u2s1

    and hence

    ζ1=u1s2u1s2u2s1,ζ2=u2s2u1s2u2s1,ζ3=u1s1u1s2u2s1.

    Proposition 2.6. For x=(x1,x2)T2, we have

    αA(1,0)(x1,x2)=(x1+θ1,x2+θ2),αA(0,1)(x1,x2)=(x1+θ3,x2+θ4),

    and hence

    αA(m,n)(x1,x2)=(x1+mθ1+nθ3,x2+mθ2+nθ4)for(m,n)Z2.

    Proof. We have

    αA(m,n)(x1,x2)=(x1,x2)+11r2A(m,n)|vurAvsvu=(x1,x2)+11r2A(m,n)|(u1rAs1,u2rAs2)(u1,u2).

    In particular, for (m,n)=(1,0),(0,1), we have

    αA(1,0)(x1,x2)=(x1+11r2A(u1rAs1)u1,x2+11r2A(u1rAs1)u2),αA(0,1)(x1,x2)=(x1+11r2A(u2rAs2)u1,x2+11r2A(u2rAs2)u2).

    We put ξi=11r2A(uirAsi) for i=1,2 so that

    αA(1,0)(x1,x2)=(x1+ξ1u1,x2+ξ1u2), (16)
    αA(0,1)(x1,x2)=(x1+ξ2u1,x2+ξ2u2). (17)

    We then have

    ξ1=11r2A{u1(u1s1+u2s2)s1}=11r2A{u1(1s21)u2s2s1}=11r2A(u1s2u2s1)s2

    and similarly

    ξ2=11r2A{u2(u1s1+u2s2)s2}=11r2A{u2(1s22)u1s1s2}=11r2A(u2s1u1s2)s1.

    Hence we have ξ1ξ2=s2s1. We also have

    ξ1u1+ξ2u2=11r2A{(u1rAs1)u1+(u2rAs2)u2}=11r2A{u21+u22rA(u1s1+u2s2)}=11r2A(1r2A)=1.

    By Lemma 2.5, we have ξ1u1=θ1,ξ1u2=θ2,ξ2u1=θ3,ξ2u2=θ4, proving the desired assertion from the identities (16) and (17).

    We will next express θi,i=1,2,3,4 in terms of the matrix elements a,b,c,d of A.

    Lemma 2.7. The following identities hold.

    (ⅰ)

    aθ1+bθ2=λuθ1,aθ3+bθ4=λuθ3,cθ1+dθ2=λuθ2,cθ3+dθ4=λuθ4,

    and hence

    aθ1+bθ2+cθ3+dθ4=λu. (18)

    (ⅱ)

    aθ3bθ1=λsθ3,aθ4bθ2=λsθ4,cθ3dθ1=λsθ1,cθ4dθ2=λsθ2,

    and hence

    aθ4bθ2cθ3+dθ1=λs.

    Proof. By the identities

    [θ1θ2]=s2u1s2u2s1[u1u2],[θ3θ4]=s1u1s2u2s1[u1u2],[θ3θ1]=u1u1s2u2s1[s1s2],[θ4θ2]=u2u1s2u2s1[s1s2],

    with θ1+θ4=1, we see the desired assertions.

    Lemma 2.8.

    (ⅰ) (aθ1+bθ2)θ4=(cθ3+dθ4)θ1.

    (ⅱ) (aθ3bθ1)θ2=(cθ4+dθ2)θ3.

    Hence we have

    bθ2=cθ3.

    Proof. (ⅰ) By the first and the fourth identities in Lemma 2.7 (ⅰ), we know the identity (ⅰ). The identities of (ⅱ) is similarly shown to those of (ⅰ). By (ⅰ) and (ⅱ) with the identity θ1θ4=θ2θ3, we get bθ2=cθ3.

    Recall that Δ(A) denotes the discriminant (a+d)24(adbc) of the characteristic polynomial of the matrix A. The real number Δ(A) is positive because of the hyperbolicity of A. By elementary calculations, we see the following lemma.

    Lemma 2.9. The identities

    θ1θ4=θ2θ3,θ1+θ4=1,(aθ1+bθ2)θ4=(cθ3+dθ4)θ1,(aθ3bθ1)θ2=(cθ4+dθ2)θ3

    imply

    (θ1,θ2,θ3,θ4) (19)
    ={(12(1+|ad|Δ(A)),|ad|adcΔ(A),|ad|adbΔ(A),12(1|ad|Δ(A)))or(12(1|ad|Δ(A)),|ad|adcΔ(A),|ad|adbΔ(A),12(1+|ad|Δ(A)))ifad,(12,12cb,12bc,12)or(12,12cb,12bc,12)ifa=d. (20)

    We thus have the following theorem.

    Theorem 2.10. The C-algebra C(GaA) of the groupoid GaA for a hyperbolic matrix A=[abcd] is isomorphic to the simple C-algebra generated by four unitaries U1,U2,V1,V2 satisfying the following relations:

    U1U2=U2U1,V1V2=V2V1,V1U1=e2πiθ1U1V1,V1U2=e2πiθ2U2V1,V2U1=e2πiθ3U1V2,V2U2=e2πiθ4U2V2,

    where

    θ1=12(1+adΔ(A)),θ2=cΔ(A),θ3=bΔ(A),θ4=12(1adΔ(A)). (21)

    Hence the C-algebra C(GaA) is isomorphic to the four-dimensional non-commutative torus.

    Proof. As in Lemma 2.2, the action αA:Z2Homeo(T2) is free and minimal, hence the C-crossed product C(T2)×αAZ2 is simple. The C-crossed product is canonically identified with the C-crossed product ((C(T)C(T))×αA(1,0)Z)×αA(0,1)Z. Let U1,U2 be the unitaries in C(T)C(T) defined by U1(t,s)=e2πit,U2(t,s)=e2πis. Let V1,V2 be the implementing unitaries corresponding to the automorphisms αA(1,0),αA(0,1), respectively. By Proposition 2.6, we know the commutation relations among the unitaries U1,U2,V1,V2 for the slopes θ1,θ2,θ3,θ4 satisfying (20). The second values of (20) go to the first of (20) by substituting V1,U1 with V2,U2, respectively. The forth values of (20) go to the third of (20) by substituting V1,U1 with V1,U1, respectively. When a=d, we have Δ(A)=4bc>0 so that ±cb=cΔ(A),±bc=bΔ(A). Hence the first two of (20) include the second two of (20), so that we may unify (20) into (21).

    Since the C-algebra C(GaA) is isomorphic to a four-dimensional non-commutative torus, we know the following proposition by Slawny [24] (see also Putnam[14]).

    Proposition 2.11 (Slawny [24], Putnam [14]). The C-algebra C(GaA) has a unique tracial state.

    Remark 2.12. (ⅰ) We note that the simplicity of the algebra C(GaA) comes from a general theory of Smale space C-algebras as in [14], [17] as well as a unique existence of tracial state on it. It also follows from a general theory of crossed product C-algebras because the action αA of Z2 to Homeo(T2) is free and minimal. It has been shown that a simple higher dimensional non-commutative torus is an AT-algebra by Phillips [13].

    (ⅱ) Suppose that two hyperbolic matrices A,BGL(2,Z) commute each other. By (8) and (9), the equality αA(Z2)=αB(Z2) holds for the commuting matrices A and B, because they have the same eigenvectors. Hence we know that GaA=GaB, so that the C-algebras C(GaA) and GaB) are isomorphic.

    In this section, we will describe the trace values τ(K0(C(GaA))) of the K0-group of the C-algebra C(GaA) in terms of the hyperbolic matrix A.

    In [20], M. A. Rieffel studied K-theory for irrational rotation C-algebras Aθ with irrational numbers θ, which are called two-dimensional non-commutative tori, and proved that τ(K0(Aθ))=Z+Zθ in R, where τ is the unique tracial state on Aθ. In [6], G. A. Elliott (cf. [3], [13], [21], [24], etc.) initiated to study higher-dimensional non-commutative tori. It is well-known the K-groups of the four-dimensional non-commutative torus as in [6] which says

    K0(C(T2)×αAZ2)K1(C(T2)×αAZ2)Z8

    ([6], cf. [24]). For g=(a1,b1,a2,b2),h=(c1,d1,c2,d2)Z4, we define a wedge product ghZ4 by

    (a1,b1,a2,b2)(c1,d1,c2,d2)=(|a1c1b1d1|,|a1c1b2d2|,|a2c2b1d1|,|a2c2b2d2|)

    where |xyzw|=xwyz. Let Θ=[θjk]4j,k=1 be a 4×4 skew symmetric matrix over R. We regard the matrix Θ as a linear map from Z4Z4 to R by Θ(xy)=Θxy. Then ΘΘ:(Z4Z4)(Z4Z4)=4Z4R is defined by

    (ΘΘ)(x1x2)(x3x4)=12!2!σS4sgn(σ)Θ(xσ(1)xσ(2))Θ(xσ(3)xσ(4))

    for x1,x2,x3,x4Z4. Although we may generally define nΘ:2nZ4R, the wedge product 2nZ4=0 for n>3, so that

    exp(Θ)=1Θ12(ΘΘ)16(ΘΘΘ):evenZ4R

    becomes

    exp(Θ)=1Θ12(ΘΘ).

    Let AΘ be the C-algebra generated by four unitaries uj,j=1,2,3,4 satisfying the commutation relations ujuk=e2πiθjkukuj,j,k=1,2,3,4. The C-algebra AΘ is called the four-dimensional non-commutative torus ([6]). If Θ is non-degenerate, the algebra AΘ has a unique tracial state written τ. By Elliott's result in [6], there exists an isomorphism h:K0(AΘ)evenZ4 such that exp(Θ)h=τ, so that we have

    exp(Θ)(evenZ4)=τ(K0(AΘ)). (22)

    Proposition 3.1. Let τ be the unique tracial state on C(GaA). Then we have

    τ(K0(C(GaA)))=Z+Zθ1+Zθ2+Zθ3inR. (23)

    Proof. Take the unitaries U1,U2,V1,V2 and the real numbers θ1,θ2,θ3,θ4 as in Theorem 2.10. We set the real numbers θjk,j,k=1,2,3,4 such as θjj=θ12=θ21=θ34=θ43=0 for j=1,2,3,4 and θ13=θ4,θ14=θ3,θ23=θ2,θ24=θ1. Let u1=V2,u2=V1,u3=U2,u4=U1 so that we have the commutation relations

    ujuk=e2πiθjkukuj,j,k=1,2,3,4.

    As θ1θ4=θ2θ3, we have

    θ12θ34θ13θ24+θ14θ23=0.

    By (22) or [6] (cf. [3,2.21], [13,Theorem 3.9]), we have

    τ(K0(C(GaA)))=Z+Z(θ12θ34θ13θ24+θ14θ23)+1j<k4Zθjk=Z+Zθ1+Zθ2+Zθ3.

    Remark 3.2. (ⅰ) It is straightforward to see that the skew symmetric matrix Θ=[θjk]4j,k=1 in our setting above is non-degenerate.

    (ⅱ) Suppose that two hyperbolic toral automorphisms (T2,A) and (T2,B) are topologically conjugate. We then know that both the C-algebras C(GaA) and C(GaB) are isomorphic. Since they have unique tracial states τA and τB respectively, we see that

    τA(K0(C(GaA)))=τB(K0(C(GaB))).

    We may also find a matrix MGL(2,Z) such that AM=MB by [1]. We then directly see that the ranges τA(K0(C(GaA))) and τB(K0(C(GaB))) coincide by using the formula (23). Similarly we may directly show that the equality τA(K0(C(GaA)))=τA1(K0(C(GaA1))) by the formula (23).

    In this section, we will present some examples.

    1. A=[1110]. Since a=b=c=1,d=0, we have by Theorem 2.10,

    (θ1,θ2,θ3,θ4)=(12(1+15),15,15,12(515). (24)

    By the formula (23), we have

    τ(K0(C(GaA)))=Z+5+510Z.

    Proposition 4.1. Let A be the matrix [1110]. Put θ=12(1+15). Then the C-algebra C(GaA) is isomorphic to the tensor product AθA5θ between the irrational rotation C-algebras Aθ and A5θ with its rotation angles θ and 5θ respectively.

    Proof. Let U1,U2,V1,V2 be the generating unitaries in Theorem 2.10. Since

    (θ1,θ2,θ3,θ4)=(θ,2θ1,2θ1,1θ)

    by (24), we have

    U1U2=U2U1,V1V2=V2V1,V1U1=e2πiθU1V1,V1U2=e2πi2θU2V1,V2U1=e2πi2θU1V2,V2U2=e2πiθU2V2,

    We set

    u1=U1U22,u2=U2,v1=V1V22,v2=V2.

    It is straightforward to see that the following equalities hold

    u1u2=u2u1,v1v2=v2v1,v1u1=e2πi5θu1v1,v1u2=u2v1,v2u1=u1v2,v2u2=e2πiθu2v2.

    Since the C-algebra C(u1,u2,v1,v2) generated by u1,u2,v1,v2 coincides with C(GaA), we have

    C(GaA)C(u1,v1)C(u2,v2)A5θAθ.

    2. A=[3121]. Since a=3,b=d=1,d=2, we have by Theorem 2.10,

    (θ1,θ2,θ3,θ4)=(3+36,33,36,336)

    and

    λu=aθ1+bθ2+cθ3+dθ4=2+3,λs=aθ4bθ2cθ3+dθ1=23.

    Since θ4=1θ1, θ2=2θ3, θ1=12+θ3, the formula (23) tells us

    τ(K0(C(GaA)))=Z+Zθ1+Zθ2+Zθ3=12Z+36Z.

    Proposition 4.2. Let A1=[1110] and A2=[3121]. Then the C-algebra C(GaA1) is not isomorphic to C(GaA2).

    Proof. Since the C-algebra C(GaA) has the unique tracial state τ, the range τ(K0(C(GaA))) of τ of the K0-group K0(C(GaA)) is invariant under isomorphism class of the C-algebra. As

    τ(K0(C(GaA1)))=Z+5+510Z,τ(K0(C(GaA1)))=12Z+36Z,

    we see that τ(K0(C(GaA1)))τ(K0(C(GaA2))), so that the C-algebra C(GA1) is not isomorphic to C(GA2).

    The author would like to deeply thank the referee for careful reading and lots of helpful advices in the presentation of the paper. This work was supported by JSPS KAKENHI Grant Numbers 15K04896, 19K03537.

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