九州大学学術情報リポジトリ
Kyushu University Institutional Repository
トレース空間が有限次元である作用素環の基本群に ついて
川原, 崇司
https://doi.org/10.15017/1931724
出版情報：Kyushu University, 2017, 博士（数理学）, 課程博士 バージョン：
権利関係：
FUNDAMENTAL GROUP OF OPERATOR ALGEBRAS WITH FINITE DIMENSIONAL TRACE SPACE
TAKASHI KAWAHARA
1
Acknowledgment
First and foremost, I would like to thank my adviser, Yasuo Watatani, for his support, encouragement, guidance, patience and the many hours of his time over the years. Without his help, I could not have written this thesis. I also thank Norio Nawata for his suggestions and guidances.
I would also like to thank many people in Kyushu University. In particular, I would thank my wonderful colleague Kei Hasegawa for many discussions and helpful comments. I would also thank Toshihiko Masuda for his suggestions and guidances.
Finally, I thank my parents and my sisters for their support. Without their support, I could not study mathematics.
2
Contents
Acknowledgment 2
1. Introduction 4
2. Preliminaries 5
2.1. C∗-algebra 5
2.2. Tracial state space 7
2.3. Hilbert modules 9
2.4. K-theory 13
2.5. Inductive limit ofC∗-algebras and groups 16
2.6. Finite von Neumann algebra and trace space 25
2.7. Equivalence bimodule of von Neumann algebras 25
3. Fundamental group ofC∗-algebras 26
3.1. Definition of Fundamental group ofC∗-algebras 26 3.2. Forms of fundamental groups and some example 30 3.3. Exact sequence of Picard group and scaling group 38 4. Fundamental group of finite von Neumann algebra with finite
dimensional normal trace space 40
4.1. Definition in the case of von Neumann algebras 40
4.2. The form of fundamental groups 44
4.3. Relation with the fundamental group ofC∗-algebra 50 5. Realization of fundamental group of AF-algebra 51
References 58
3
1. Introduction
LetMbe a factor of type II1 with a normalized traceτ. Murray and von Neu- mann introduced the fundamental groupF(M) ofMin [16]. They showed that if Mis hyperfinite, thenF(M) =R×+. Since then there has been many works on the computation of the fundamental groups. Voiculescu showed thatF(L(F∞)) of the group factorL(F∞) of the free groupF∞contains the positive rationals in [30] and Radulescu proved thatF(L(F∞)) =R×+in [23]. Connes [4] showed thatF(L(G)) is a countable group ifGis an ICC group with property (T). Popa and Vaes showed that either countable subgroup of R×+ or any uncountable group belonging to a certain ”large” class can be realized as the fundamental group of some factor of type II1 in [21] and in [22].
Nawata and Watatani [19], [20] introduce the fundamental group of simpleC∗- algebras with unique trace whose study is essentially based on the computation of Picard groups by Kodaka [11], [12], [13]. Nawata defined the fundamental group of non-unital C∗-algebras [17] and calculate the Picard group of some projectionless C∗-algebras with strict comparison by the fundamental groups[18].
In this paper, we define the fundamental group ofC∗-algebras with finite dimen- sional trace space and finite von Neumann algebras with finite dimensional normal trace space. This fundamental group is a ”numerical invariant”. This study is based on the work in [19], [20].
First, we will consider the fundamental group ofC∗-algebras with finite dimen- sional trace space. LetAbe a unital C∗-algebra with finite dimensional bounded trace space. The fundamental groupF(A) ofAand the determinant fundamental group Fdet(A) are defined by using self-similarity and the extremal points of the bounded trace space of A. Then the groupsF(A) and Fdet(A) are multiplicative subgroups ofGLn(R) and R×+ respectively. We shall show that K-theoretical ob- struction and positivity restrict the element of the fundamental group and we will have thatA=DU(σ) for some diagonal matrixDand for some permutation unitary U(σ) for anyAinF(A). If the unitalC∗-algebrasAandBwith finite dimensional trace space are Morita equivalent, thenF(A) = (DU(σ))−1F(B)(DU(σ)) for some diagonal matrixDand for some permutation unitaryU(σ). After showing the basic facts on the fundamental groups of unitalC∗-algebras, we computeF(A) of several C∗-algebrasA. Moreover, we shall show that given any group Gin GL2(R) which is isomorphic toZ2and whose elements have the formDU(σ), there exists a simple AF-algebraAsuch thatF(A) =G. Furthermore, we shall show that for anyn∈N there exist uncountably many mutually non-isomorphic simple (non)nuclear unital C∗-algebrasA with n-dimensional trace space such that F(A) ={In}, whereIn
is a unit inMn(C).
Second, we will consider the fundamental group of finite von Neumann alge- bras with finite dimensional normal trace space. As is the same with C∗-algebra, we define the fundamental group F(M) and the determinant fundamental group Fdet(M) by using self-similarity and the extremal points of the tracial state space ofM. ThenF(M) is a multiplicative subgroup ofGLn(R) andFdet(M) is that of R×+. We shall determine the form ofF(M) completely. As in the same with the case of the fundamental group ofC∗-algebras with finite dimensional trace space, if the finite von Neumann algebrasMandN with finite dimensional normal trace space are Morita equivalent, then F(M) = (DU(σ))−1F(N)(DU(σ)) for some diagonal matrixD and for some permutation unitaryU(σ). A result [21, Cor.5.3] by Popa
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implies that for any family of countable groups {Gi}i∈N there exists a family of II1-factors{ Mi}i∈Nsuch thatF(Mi) =Gi and thatMi is not stably isomorphic each other. By using the fact, we will show that there exists a finite von Neumann algebra M with finite dimensional normal trace space such that F(M) = G for any countable groups G which have that form as fundamental group. Let A be a unital C∗-algebra with n-dimensional trace space. We say {φi}ni=1 the set of extreme points of the tracial state spaceT(A). Putφ= 1
n
∑n
i=1φn. Thenπφ(A)w is a finite von Neumann algebra with finite dimensional normal trace space and F(A)⊂F(πφ(A)w).
At last, we will consider the realization of fundamental groups ofAF-algebras.
Leta, bbe non-zero positive real numbers. We suppose thata̸=b. Then we can no- tice that there is no finite von Neumann algebraMwith finite dimensional normal trace space such thatF(M) ={
[ an 0 0 bn
]
:n∈Z} by the form of fundamental group of von Neumann algebras. If a is an algebraic number and b is a rational (or a certain algebraic number), then there is no simpleAF-algebraAin a specific class such that F(A) = {
[ an 0 0 bn
]
:n∈Z}. However, we show this as one of the main theorems at last, ifais transcendal, then there is a simpleAF-algebraA such thatF(A) ={
[ an 0 0 bn
]
:n∈Z}.
2. Preliminaries
2.1. C∗-algebra. We recall some definitions on Banach algebras. A Banach space X is aC-linear space with a norm ∥ · ∥ : A → R, such that X is complete with respect to the norm. ABanach algebraAis an algebra overCwhich has a norm making it into a Banach space and satisfying ∥ab∥ ≤ ∥a∥∥b∥ for any a, b in A . An involution ∗ on a Banach algebra A is an anti-linear map a 7→ a∗ such that (a∗)∗=a, (ab)∗=b∗a∗ for anya, binA.
Definition 2.1. A C∗-algebra A is a Banach algebra with an involution which satisfies the following condition: ∥a∗a∥=∥a∥2for anyain A.
This equivalence is calledC∗-condition. We say that C∗-algebra A is unital if Ahas a multiplicative identity 1A. By the definition of the involution and by this condition, we obtain 1∗A= 1A,∥1A∥= 1 and the following proposition.
Proposition 2.2. The involution of aC∗-algebra is isometric.
Proof. Let A be a C∗-algebra and let a be a non-zero element of A. By C∗- condition,∥a∗a∥=∥a∥2. SinceAis a Banach algebra,∥a∗a∥ ≤ ∥a∗∥∥a∥. Therefore
∥a∥ ≤ ∥a∗∥. Replacingabya∗, we can get∥a∗∥ ≤ ∥a∥. Hence∥a∥=∥a∗∥. □ The following proposition yields information about invertibility. It holds on the case of Banach Algebras with identity.
Proposition 2.3. LetAbe aC∗-algebra and letabe a element ofA. If∥a−1∥<1, thenais invertible.
Proof. Setb =∑∞
n=0(1−a)n. By assumption, the sum is absolutely convergent.
Thereforebis well-defined. Moreoverab=ba= 1. Hence ais invertible. □
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LetAbe aC∗-algebra. We say that a non-empty subsetBofAis aC∗-subalgebra of A if it is closed with respect to the norm ofA and closed under the algebraic operations of A: addition, multiplication, adjoint, scalar multiplication. Let B a C∗-algebra. A ∗-homomorphism φ : A −→ B is a ∗-algebraic homomorphism:
φ(a+b) =φ(a)+φ(b),φ(ab) =φ(a)φ(b),φ(λa) =λφ(a),φ(a)∗=φ(a∗) for anya, b inAand for anyλinC. IfAandBare unital andφis a∗-homomorphism fromA intoBsuch thatφ(1A) = 1B,φis called to be a unit-preserving∗-homomorphism.
We denote byφ= 0 the∗-homomorphismφ:A → Bsuch thatφ(a) = 0 for anya∈ A. We supposeAandBare unital. We say thatAandBare∗-isomorphic if there exists a one to one unit preserving∗-homomorphismφfromAontoB. IfAandB are non-unital, the ∗-isomorphismφ:A −→ B is a bijective∗-homomorphism. A subsetI of Ais called an ideal if I is algebraically two-sided ideal of Aand I is closed with respect to the norm ofA. Every ideal ofC∗-algebras is automatically closed under the involution. The quotientA/I is aC∗-algebra with a norm∥·∥A/I
defined by∥[a]I∥A/I =inf{ ∥a+x∥:x∈ I }, where [a]I is an equivalent class ofa inA/I.
Definition 2.4. LetAbe a unitalC∗-algebra and letabe an element ofA. The spectrum ofais the set of all complex numbersλsuch thata−λ1Ais not invertible inA. We denote it by sp(a).
The set sp(a) is a non-empty compact subset of C(See [6]). If ais invertible in Aand is in aC∗-subalgebraB ofA, thenais invertible in B(See [6]). We denote by C∗(a, 1A) the smallest sub-C∗-algebra of A generated by a and 1A. Since C∗(a, 1A) is aC∗-subalgebra of any C∗-algebra which has the element a, sp(a) is independent ofC∗-algebras.
Here we recall some definitions of elements in a C∗-algebra. Let A be a C∗- algebra. An element a in Ais called normal ifaa∗ =a∗a. An element ain A is calledself-adjointifa=a∗. An elementainAis calledpositiveifais self-adjoint andsp(a)⊂R+, whereR+ is the set of all real numbersαsuch thatα≥0.
LetX be a compact set. We denote byC(X) the continuous function algebra on X. ThenC(X) is a commutative C∗-algebra. Moreover f ∈ C(X) is self-adjoint if and only iff is aR-valued function andf is positive if and only iff(x)≥0 for anyxinX.
Theorem 2.5. (Functional calculus) Let Abe a unitalC∗-algebra and letabe a normal element ofA. There is a∗-isomorphism Φ :C∗(a, 1A)−→C(sp(a)) such that Φ(a) =z, where zis the functionz:sp(a)−→Csuch thatz(λ) =λ.
Letf be a continuous function onsp(a). We can define f(a) inA by Φ−1(f).
Since both self-adjoint elements and positive elements are normal, we can apply the previous theorem to these elements. We show the following proposition by the previous theorem. Let a, b self-adjoint elements of a C∗-algebra. We denote by a≥bifa−bis positive.
Proposition 2.6. Let a be a positive element of a unital C∗-algebra. If there existsδ >0 such thata≥δ·1A, thenais invertible.
Proof. We showsp(a) does not have 0. By 2.5, we can consideraas the functionz ofC(sp(a)). Conversely, suppose 0∈sp(a). Then z(0) = 0< δ. This leads to the
contradiction. □
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Proposition 2.7. Let A be a unitalC∗-algebra, let a be a self-adjoint element of A, let b be an element of A and letf be a continuous function on sp(a). If b commutes witha,b commutates withf(a).
Proof. It follows from the facts thatf(a)∈C∗(a, 1A) and that bcommutes with
any element ofC∗(a, 1A). □
In general, polar decomposition cannnot be done in a C∗-algebra. However, it can be for invertible elements. An elementuin a unitalC∗-algebra is calledunitary ifuu∗=u∗u= 1A.
Proposition 2.8. LetAbe a unitalC∗-algebra and letabe an invertible element ofA. Then there exists a unitaryusuch thata=u(a∗a)12.
Proof. Sinceais invertible,a∗anda∗aare so. Therefore (a∗a)−12 is inC∗(a∗a, 1A).
Putu=a(a∗a)−12. Thenuu∗=u∗u= 1. □
Next proposition is a representation by unitary elements.
Proposition 2.9. Every element in a unitalC∗-algebra is a linear combination of four unitaries.
Proof. LetAbe a unitalC∗-algebra and letabe an element ofAsuch that∥a∥= 1.
Seta1= 12(a+a∗) anda2= 2i1(a−a∗). Thena1anda2are self-adjoint,a=a1+ia2 and∥ai∥ ≤1. Therefore it is sufficient to show the case thata is self-adjoint. Let u=a+i(1−a2)12. Thenuis unitary anda= 12(u+u∗). □ 2.2. Tracial state space. Let A be a C∗-algebra. A bounded trace on A is a bounded linear mapτ:A →Cwith the trace property: τ(ab) =τ(ba) for anya, b in A. A trace τ isself −adjointifτ(a)∈R for every self-adjoint elementa in A, and a traceτ ispositiveifτ(a)≥0 for every positive elementainA. Ifτ is a positive bounded trace with ||τ|| = 1, thenτ is called atracial state. We denote byT(A) the set of tracial states onAand by linCT(A) the C-linear span ofT(A).
We consider the relation between extreme points ofT(A) and basis of linCT(A).
Lemma 2.10. LetA be a unital C∗-algebra, and letτ be a linear functional on A. The followings are equivalent.
(1)τ satisfies the trace property.
(2)τ(a) =τ(u∗au) for any unitaryuinA, for anyain A
Proof. We suppose (1). Let u be a unitary in A, and let a be an element in A. Then
τ(u∗au) = τ(auu∗)
= τ(a).
We suppose (2). Let a, b be elements in A. By 2.9, we can put b =∑4 i=1ciui, whereci is a complex number andui is a unitary inA. Therefore,
τ(ab) = τ(a(c1u1+c2u2+c3u3+c4u4))
= c1τ(au1) +c2τ(au2) +c3τ(au3) +c4τ(au4)
= c1τ(u1a) +c2τ(u2a) +c3τ(u3a) +c4τ(u4a)
= τ((c1u1+c2u2+c3u3+c4u4)a)
= τ(ba)
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□ Proposition 2.11. Let A be a unital C∗-algebra, then linCT(A) is the set of bounded traces onA.
Proof. Letτ be a bounded trace onA. We defineτ :A →Cbyτ(a) =τ(a∗) and define τ1, τ2 :A → Cby τ1(a) = 1
2(τ(a) +τ(a)) and byτ2(a) = 1
2i(τ(a)−τ(a)).
Then τ is also a bounded trace on A, so τ1, τ2 are bounded self-adjoint traces, and τ =τ1+iτ2. Therefore it is sufficient to show the case τ is self-adjoint. By Hahn decomposition, We can obtain the unique pair of positive linear functionals (τ+, τ−) satisfying τ = τ+−τ− and that ||τ|| =||τ+||+||τ−||. We show that τ+ andτ− satisfy the trace property. Let ube a unitary in A. We defineαu:A → A byαu(a) =u∗au, thenαu is an automorphism,τ±◦αuis a linear functional onA, and ||τ±|| =||τ±◦αu||. Then||τ||=||τ+◦αu||+||τ−◦αu||. However,τ satisfies the trace property, so
τ+(a)−τ−(a) = τ(a)
= τ(u∗au)
= τ+(u∗au)−τ−(u∗au)
= τ+◦αu(a)−τ−◦αu(a)
Therefore, τ =τ+◦αu−τ−◦αu and ||τ|| =||τ+◦αu||+||τ−◦αu||. Because of the uniqueness of Hahn decomposition, τ± =τ±◦αu Therefore, τ± has the trace property andτ± is bounded positive trace. By adjusting the value of norm,τ is a
linear combination of tracial states. □
We consider the case thatAis unital and thatdimlinCT(A)<∞. ThenT(A) is compact in the weak∗- topology so the set of extreme points is nonempty. Now, we show the relation between extreme points of T(A) and basis of linCT(A). We denote by∂eT(A) the set of extreme points ofT(A).
Lemma 2.12. Let A be a unital C∗-algebra and let τ1, τ2,· · · , τl be in ∂eT(A).
Suppose if i ̸= j, then τi ̸= τj. Then τ1, τ2,· · ·, τl are linearly independent in linCT(A).
Proof. We assume ∑l
j=1αjτj = 0, whereαj is a complex number. Then, we can get αj1, αj2 ∈ R satisfying αj = αj1+iαj2. By transposing the real part and imaginary part, it is enough to show if∑l
j=1βjτj = 0 forβj ∈R, thenβj = 0 for any j. We suppose there exists j such thatβj ̸= 0. By transposing the negative part, ∑l
j=1βj1τj =∑l
j=1βj2τj, where 0 =βj1 ≤ βj2 or 0 =βj2 ≤βj1. Because T(A) is simplex(See [29]), βj1 = βj2 = 0. This contradicts our assumption, so βj = 0 for allj. Therefore,τ1, τ2,· · · , τlare linearly independent in linCT(A). □ The next proposition is important for the expression of fundamental group we define at the next section.
Proposition 2.13. Let A be a unital C∗-algebra. If dimlinCT(A) < ∞, then
∂eT(A) is a basis of linCT(A).
Proof. By Klein-Milman‘s Theorem and 2.11,∂eT(A) generates linCT(A). In ad- dition, by 2.12, all elements of∂eT(A) are linearly independent. □
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2.3. Hilbert modules. In this section, we review some standard facts on Hilbert modules.
Definition 2.14. Let Abe a C∗-algebra and let E be a right A-module. An A- valued pre-inner product onE is a map⟨·,·⟩:E × E −→ A such that⟨ξ, η+ζ⟩=
⟨ξ, η⟩+⟨ξ, ζ⟩, ⟨ξ, λη⟩ = λ⟨ξ, η⟩, ⟨ξ, ηa⟩ = ⟨ξ, η⟩a, ⟨ξ, η⟩ = ⟨η, ξ⟩∗ and ⟨ξ, ξ⟩ is a positive element of A for any ξ, η, ζ in E, for any λ in C and for any ain A. If
⟨ξ, ξ⟩= 0 impliesξ= 0, then ⟨·,·⟩is called anA-valued inner product.
We will denote by∥ · ∥ the norm ofA. We define∥ · ∥A:E −→R+ by∥ξ∥A=
∥⟨ξ, ξ⟩∥12.
Proposition 2.15. LetAbe aC∗-algebra and letEbe a rightA-module with anA- valued pre-inner product⟨·,·⟩A. Then⟨ξ, η⟩∗A⟨ξ, η⟩A≤ ∥⟨ξ, ξ⟩∥A⟨η, η⟩A,∥ξ+η∥A≤
∥ξ∥A+∥η∥Aand∥ξb∥A≤ ∥ξ∥A∥b∥ for anyξ, η inE and for anyb inA. This can be found in [14]. Therefore∥ · ∥Ais a norm onE.
Definition 2.16. Let Abe a C∗-algebra. A right Hilbert A-moduleE is a right A-module such that it has anA-valued inner product and is complete with respect to the norm induced by the inner product.
Similarly, we can define left Hilbert modules.
We will use the term ”Hilbert module” without the words ”lef t”or ”right” when no confusion can arise.
Let E be a Hilbert A-module. The additive span of {⟨ξ, η⟩ : ξ, η ∈ E} is an algebraic ideal ofA. We denote by JE the closure of it. The Hilbert moduleE is calledf ull ifJE =A. We show the following proposition related to the fullness of Hilbert modules.
Proposition 2.17. LetAbe aC∗-algebra, letE be a full right HilbertA-module andabe an element ofA. Ifξa= 0 for any ξinE, then a= 0.
Proof. Letηandξbe elements ofE. By hypothesis, 0 =⟨η, ξa⟩=⟨η, ξ⟩a. Therefore
ba= 0 for any bin JE. SinceJE =A,a= 0. □
We will consider the analogs of bounded operators for Hilbert modules.
Definition 2.18. Let A be aC∗-algebra and E, F are right Hilbert A-modules.
A map T : E −→ F is adjointable if there exists a map T∗ : E −→ F such that
⟨T ξ, η⟩=⟨ξ, T∗η⟩for anyξinE and for anyη inF. The operatorT∗ is called the adjoint operator of T. We denote by L(E,F) the set of all adjointable operators fromE intoF and setL(E) =L(E,E).
Adjointable operators are C-linear, A-linear and bounded(See [14]). The set L(E,F) is a Banach space and L(E) is aC∗-algebra. Even ifAis not unital,L(E) is unital.
Definition 2.19. Let Abe a C∗-algebra and letE be a right Hilbert A-module.
Fixξ∈ F and η∈ E. We define the mapθξ,η :E → F byθξ,η(ζ) =ξ⟨η, ζ⟩. Since θ∗ξ,η =θη,ξ,θξ,η is an adjointable operator. We denote byK(E,F) the closed linear span of the set {θξ,η :ξ∈ F, η∈ E}. We will writeK(E) =K(E,E) as in the same case ofL(E).
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Let θξ,η be in K(E). Since T θξ,η = θT ξ,η and θξ,ηT = θξ,T∗η where T is an adjoint operator onE,K(E) is an ideal ofL(E). If K(E) have the identity operator I onE, thenK(E) =L(E).
We next introduce the basis of Hilbert modules. This is necessary to define the fundamental group ofC∗-algebras.
Definition 2.20. LetEbe a right HilbertA-module. A finite basis{ξi}ki=1 ofE is a family ofE such thatη =∑k
i=1ξi⟨ξi, η⟩for anyη inE.
IfEis a left HilbertA-module, then the equation is replaced byη=∑k
i=1⟨η, ξi⟩ξi. In general, there doesn’t always exist finite bases as in the case of infinite dimen- sional Hilbert spaces. If a right Hilbert module E has a finite basis {ξi}ki=1, then the operator ∑k
i=1θξi,ξi in K(E) is identity on E. ThereforeK(E) =L(E). From now on, we discuss imprimitivity bimodules, which have finite basis.
Definition 2.21. Let A and B be C∗-algebras. A C-module E is said to be an A-Bimprimitivity bimodule ifE is simultaneously a full left HilbertA-module with anA-valued left inner productA⟨·,·⟩ and a full right HilbertB-module with a B- valued right inner product⟨·,·⟩B which satisfies the condition : A⟨ξ, η⟩ζ=ξ⟨η, ζ⟩B for any ξ, η, ζ in E. Let E and F be A-B imprimitivity bimodule. We call E is isomorphic toF if there exists a C-module homomorphism Φ :E → F such that Φ(aξb) =aΦ(ξ)b,A⟨Φ(ξ),Φ(η)⟩=A⟨ξ, η⟩,⟨Φ(ξ),Φ(η)⟩B =⟨ξ, η⟩B for anyξ, η in E, for anyainAand for anyb inB.
Example 2.22. LetA be a unital C∗-algebra. We define anA-valued left inner product onAA⟨·,·⟩and anA-valued right inner product onA ⟨·,·⟩A byA⟨a, b⟩= ab∗ and⟨a, b⟩A=a∗bfor anya, binA. ThenAis anA-Aimprimitivity bimodule.
Example 2.23. LetAbe a unitalC∗-algebra and letpbe a projection ofMn(A) such that the linear span of {a∗pb : a, b ∈ An} is dense in A. Then pAn is a pMn(A)p-Aimprimitivity bimodule.
LetE be anA-Bimprimitivity bimodule. Then⟨aξ, η⟩B=⟨ξ, a∗η⟩B and
A⟨ξb, η⟩=A⟨ξ, ηb∗⟩for any ξ, η inE, for anyain Aand for anyb inB. We write
∥ · ∥A and∥ · ∥B for the norms induced by the A-valued inner product and by the B-valued inner product respectively. Then∥ξ∥A=∥ξ∥Bfor anyξinE. These were proved in [1]. We will denote byEB the imprimitivity bimoduleE if we considerE as a right HilbertB-module.
Lemma 2.24. Let A and B be aC∗-algebra and letE be an A-B imprimitivity bimodule. Then sup{∥aξ∥B:ξ∈ E, ∥ξ∥B= 1}=∥a∥
Proof. We define a map Φ :A → L(EB) by Φ(a)(ξ) =aξ. By 2.17, the map is an injective∗-homomorphism. Therefore Φ is isometry (See [6]). □ Proposition 2.25. Let A and B are unital C∗-algebras and let E be an A-B imprimitivity bimodule. ThenE has a finite basis as a right HilbertB module.
Proof. Letθξ,η be an element ofK(EB). By the condition of imprimitivity bimod- ules, θξ,η(ζ) = A⟨ξ, η⟩ζ. Now, we can define a map Ψ : K(EB) → A such that Ψ(θξ,η) = A⟨ξ, η⟩. By 2.24 and by the fullness, Ψ is an ∗-isomorphism. Since A is unital,K(EB) =L(EB). By definition ofK(EB), there existξi, ηi in E such that
∥∑k
i=1θξi,ηi−I∥<1. Therefore there exists an adjoint operatorT inL(EB) such
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that ∑k
i=1θT ξi,ηi =I by 2.3. Replacing T ξi by ξi, we can obtain ξi, ηi in E such that∑k
i=1θξi,ηi =I. Put K=∑k
i=1θξi+ηi,ξi+ηi. SinceK≥2I, Kis positive and invertible. Then ∑k
i=1θ
K−12(ξi+ηi),K−12(ξi+ηi)=I. Hence {K−12(ξi+ηi)}ki=1 is a
basis ofEB. □
Let A be a unital C∗-algebra. We call that a projection p in Mk(A) is self- similar full projection if the linear span of{a∗pb a, b∈ Ak}is dense inAand there exists an∗-isomorphism fromAontopMk(A)p. We consider the relation between A-Aimprimitivity bimodules and self-similar full projections. We denote byeithe element ofAk whosej-th element isδij, whereδij is Kronecker’s delta.
Proposition 2.26. Let A be a unital C∗-algebra and let p is a self-similar full projection inMk(A). ThenpAkis anA-Aimprimitivity bimodule with finite basis {pei}ki=1.
Proof. It follows by 2.23. □
Proposition 2.27. LetAbe a unitalC∗-algebra and letEbe anA-Aimprimitivity bimodule with a finite right basis{ξi}ki=1. Putp= (⟨ξi, ξj⟩A)ij, which is a projec- tion in Mk(A). Then there exists an isomorphism Φ :E →pAk as a right Hilbert A-module such that Φ(ξi) =pei, the isomorphism induces an∗-isomorphism from A ontopMk(A)p, pis a self-similar full projection, andpAk becomes isomorphic toE as an A-Aimprimitivity bimodule.
Proof. We set a map Φ : E → pAk by Φ(η)j1 =⟨ξj, η⟩. Then Φ is isomorphism as a right Hilbert A-module. Since EA is full, pAk is so. Therefore the linear span of {a∗pb a, b ∈ Ak} is dense in A. Moreover, as in the same proof of 2.25, A ∼=K(EA) =L(EA). SinceL(EA)=∼L(pAk),A ∼=L(pAk) =pMk(A)p. Put Ψ the
∗-isomorphism fromAontopMk(A)p. Then Ψ(a)ij=⟨ξi, aξj⟩A. □ Here we discuss the internal tensor product of Hilbert modules. Our purpose is to construct a tensor product of two imprimitivity bimodule,A-Bimprimitivity bimoduleE andB-C imprimitivity bimoduleF. For a treatment of a more general case we refer the reader to [1]. We denote by E ⊙BF the quotient of the tensor product vector spaceE ⊙ F overCby the subspace which is a linear span of the set {ξb⊙η−ξ⊙bη:ξ∈ E, η∈ F, b∈ B}. We define an operation (E ⊙BF)×C → E ⊙BF by (ξ⊗η, c)7→ξ⊗ηc. ThenE ⊙BF is a rightC-module by the operation. We set aC-valued inner product onE ⊙ F by⟨ξ1⊗η1, ξ2⊗η2⟩C =⟨η1,⟨ξ1, ξ2⟩Bη2⟩C.
This inner product is well-defined(See [14]). The completion ofE⊙Fwith respect to the norm induced by theC-valued inner product is the internal tensor product of E andF. We denote it byE ⊗ F. SinceF is a full rightC-Hilbert module,E ⊗ F is also full. We can also see thatE ⊗ F is a leftA-module bya(ξ⊗η) =aξ⊗η. In the same manner, we can set aA-valued inner product onE ⊙F byA⟨ξ1⊗η1, ξ2⊗η2⟩=
A⟨ξ1⟨η1, η2⟩B, ξ2⟩. ThenA⟨ξ1⊗η1, ξ2⊗η2⟩ξ3⊗η3=ξ1⊗η1⟨ξ2⊗η2, ξ3⊗η3⟩C. Thus, we can extend theA-valued inner product onE ⊗ F. SimilarlyE ⊗ F is a full left HilbertA-module. HenceE ⊗ F is anA-Cimprimitivity bimodule.
Example 2.28. LetAbe a unitalC∗-algebra and letE be anA-Aimprimitivity bimodule. Then E ⊗ A and A ⊗ E are isomorphic to E. Let B and C be unital C∗-algebras, letE1and E2 be A-B imprimitivity bimodules. and letF1 andF2 be B-C imprimitivity bimodules. IfE1 is isomorphic toE2 andF1is isomorphic toF2, thenE1⊗ F1is isomorphic toE2⊗ F2.
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Next proposition determines the basis ofE ⊗ F.
Proposition 2.29. LetA,BandCbe unitalC∗-algebras, letEbe aA-Bimprim- itivity bimodule and let F be a B-C imprimitivity bimodule. If we consider the tensor productE ⊗ F as a right HilbertC-module, then{ξi⊗ηj}ki,j=11,k2 is a basis of E ⊗ F, where{ξi}ki=11 and{ηj}kj=12 are finite bases ofEB andFC respectively.
Proof. SinceE ⊙ F is dense in E ⊗ F, it is sufficient to show ξ⊗η =∑k1,k2 i,j=1ξi⊗ ηj⟨ξi⊗ηj, ξ⊗η⟩C for anyξ⊗η in E ⊙ F. In fact,
k∑1,k2 i,j=1
ξi⊗ηj⟨ξi⊗ηj, ξ⊗η⟩C =
k∑1,k2 i,j=1
ξi⊗ηj⟨ηj,⟨ξi⊗ξ⟩Bη⟩C
=
k1
∑
i=1
ξi⊗ ⟨ξi, ξ⟩Bη
=
k1
∑
i=1
ξi⟨ξi, ξ⟩B⊗η
= ξ⊗η
Hence{ξi⊗ηj}i,j=1k1,k2 is a basis ofE ⊗ F. □ Remark 2.30. LetEbe a right HilbertA-module and letχbe aA-Aimprimitivity bimodule. Then we can also define a right HilbertA-moduleE ⊗χas in the same manner. IfE and χhave a right bases {ξi}i=1k1 , {ηj}j=1k2 , then{ξi⊗ηj}ki,j=11,k2 is a basis ofE ⊗χ.
Our purpose is to define a Picard group on aC∗-algebra. The internal tensor is the multiplication on the group. The following definition of ”dual module” is the inverse on the group.
Definition 2.31. LetAandBbeC∗-algebras and letE be anA-Bimprimitivity bimodule. Put the setE∗={ξ∗:ξ∈ E}. We define an addition, a leftBaction and a rightAaction onE∗ byξ∗+η∗= (ξ+η)∗, bξ∗a= (a∗ξb∗)∗forξ∗, η∗inE∗,binB andain A. ThenE∗ is a leftB-module and a rightA-module. We set aB-valued inner productB⟨·,·⟩and anA-valued inner product⟨·,·⟩AbyB⟨ξ∗, η∗⟩=⟨ξ, η⟩Band by⟨ξ∗, η∗⟩A=A⟨ξ, η⟩forξ∗, η∗ in E∗. SinceE is anA-B imprimitivity bimodule, E∗ is a B-Aimprimitivity bimodule. We callE∗ the dual module ofE.
Example 2.32. LetAbe aC∗-algebra andE be anA-Aimprimitivity bimodule.
ThenE∗⊗ E andE ⊗ E∗ are isomorphic toA.
Now we define the Picard group on a unitalC∗-algebra.
Definition 2.33. Let A be a unital C∗-algebra. The Picard group on A is the set of isomorphic classes [E] ofA-Aimprimitivity bimodules E with the operation induced by the internal tensor product.
The unit element of this group is [A]. By 2.22, 2.28 and 2.32, this definition is well-defined. From now on, we consider the relation between automorphisms and Picard group.
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Definition 2.34. LetAbe a unitalC∗-algebra. An automorphismαofAis a unit- preserving, injective and surgective∗-homomorphism from Aonto A. We denote by Aut(A) the set of automorphisms ofA, which forms a group under the product of composition. An automorphism αis called inner if there exists a unitary u in A such thatα(a) =uau∗ = adu(a) for any a in A. We denote by Int(A) the set of inner automorphisms. Then Int(A) is a normal subgroup of Aut(A). We denote by Out(A) the quotient group Aut(A)/Int(A).
Definition 2.35. Let A be a unital C∗-algebra and let αbe an automorphism.
We consider the setAas aC-module, define a left A-action onAbya·ξ=aξ for a, ξ∈ Aand define a right A-action onAbyξ·a=ξα(a) fora, ξ∈ A. Moreover, we define a leftA-valued inner product A⟨·,·⟩byA⟨ξ, η⟩=ξη∗ and define a right A-valued inner product⟨·,·⟩Aby⟨ξ, η⟩A=α−1(ξ∗η). Then the C-moduleAwith these actions and inner products is anA-Aimprimitive bimodule. We denote it by Eα.
LetAbe a unitalC∗-algebra and letαandβ be automorphisms ofA. We define a map Φ :Eα◦β → Eα⊗ Eβby Φ(a) =a⊗1. Then Φ is an isomorphism as anA − A imprimitivity bimodule.
Proposition 2.36. LetAbe a unitalC∗-algebra and letαandβbe automorphisms of A. As an A − A imprimitivity bimodule Eα is isomorphic toEβ if and only if there exists a unitaryuin Asuch thatα= adu◦β.
Proof. Let Φ :Eα→ Eβ be an isomorphism. Put Φ(1) =u. SinceA⟨Φ(1),Φ(1)⟩=
A⟨1,1⟩and⟨Φ(1),Φ(1)⟩A=⟨1,1⟩A,uis a unitary. Moreover, au = aΦ(1)
= Φ(a)
= Φ(1·α−1(a))
= u·α−1(a)
= uβ◦α−1(a)
Therefore α(a) = adu◦β(a). Conversely, we suppose α= adu◦β. We define a map Φ :Eα→ Eβ by Φ(a) =au. Then Φ is an isomorphism. □
Therefore the following proposition is established.
Proposition 2.37. LetAbe a unitalC∗-algebra. We define a mapρA: Out(A)→ P ic(A) by ρA([α]) = [Eα] for any [α] in Out(A). Then this map is well-defined.
Moreover, the map is an injective group homomorphism.
2.4. K-theory. In this section, we recall some definitions and facts of the functor K0(See [28]). When we discuss the concepts in this section, we consider the case of unital C∗-algebras. These are important tools to compute some examples of fundamental groups we introduce in this paper.
Definition 2.38. A monoidSis a set with an associative operation·if there exists a unitesuch thatx·e=e·x=xfor anyxinS. We call a monoid S is abelian if x·y=y·xfor anyx, y in S.
Definition 2.39. LetAbe aC∗-algebra. We write byPn(A) the set of projections in Mn(A) and write byP∞(A) the disjoint union ofPn(A) for allnin N. We set
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the relation∼0 onP∞(A). We denote byp∼0q forp∈Pn(A) and forq∈Pm(A) if there existsv inMm,n(A) such thatp=v∗v andq=vv∗.
This relation is an equivalence relation. We consider the projections with this relation as a same element in theK0-group. PutD(A) =P∞(A)/∼0. We would like to form a group by the setD(A). We make it into a monoid and obtain a group by the Grothendieck construction. Usually, the construction makes a semigroup (which has no additive unit) into a group. In this paper, we consider the case of monoids for some properties of the construction. We define an operation ⊕ on P∞(A) by p⊕q = diag(p, q). This operation is associative. The following proposition shows well-definedness of this addition onD(A).
Proposition 2.40. LetAbe aC∗-algebra. We define an addition + onD(A) by [p]D+ [q]D= [p⊕q]Dfor equivalence classes [p]D,[q]Din D(A), wherepandqare elements ofP∞(A). Then this addition is well-defined andD(A) forms an abelian monoid with a unit [0A]D, where 0Ais a zero element of A.
Proof. We first show ifp1∼0p2and q1∼0q2forpi, qi in P∞(A), thenp1⊕q1∼0
p2⊕q2. Suppose p1 ∈ Mn1(A), p2 ∈ Mn2(A), q1 ∈ Mk1(A), and q2 ∈ Mk2(A).
By the definition of the relation ∼0, there exist an element v of Mn1,n2(A) and an element w of Mk1,k2(A) such that p1 = vv∗, p2 = v∗v, q1 = ww∗, and q2 = w∗w. Put u1 =
[ v 0n1,k2
0k1,n2 w ]
, where 0n,m is a zero-element of Mn,m(A).
Then p1⊕q1 = u1u∗1 and p2⊕q2 = u∗1u1. Therefore the addition + is well- defined and associative since the operation⊕is associative. We shall showp⊕q∼0
q⊕p. Let p be a projection of Mn(A) and q be a projection of Mm(A) Put u2 =
[ 0n,m p q 0m,n
]
. Then p⊕q = u2u∗2 and q⊕p = u∗2u2. Therefore the addition + is commutative. Finally, we shall showp∼0p⊕0 for any projectionp in P∞(A). Putu3 = (p0). Thenp=u3u∗3 andp⊕0 =u∗3u3. Therefore [0]D is a
unit. □
LetS be an abelian monoid with an associative operation + and with a unit 0.
We define a relation ∼on S×S. We denote by (x0, y0)∼(x1, y1) for any xi, yi in S if there existsz in S such thatx0+y1+z=x1+y0+z. This relation is an equivalence relation. We writeG(S) for the quotientS×S/∼.
Proposition 2.41. LetS be an abelian monoid with an associative operation + and with a unit 0. We define an operation + on G(S) by (x0, y0) + (x1, y1) = (x0+x1, y0+y1). ThenG(S) is an abelian group by this operation +.
Proof. We show + onG(S) is well-defined. Suppose (x0, y0)∼(x′0, y0′) and (x1, y1)∼(x′1, y1′). Then there existz0, z1 such that x0+y′0+z0 =x′0+y0+z0 and that x1+y1′ +z1 = x′1+y1+z1. Adding the both sides respectively, x0+ x1+y′0+y1′ +z0+z1=x′0+x′1+y0+y1+z0+z1. Therefore the operation + is well-defined. Since 0 is a unit onS, (0,0) is a unit. Let (x, y) be onG(S). Then (y, x) is an inverse element of (x, y) because (x, x)∼(0,0) for anyxin S. Since +
onS is commutative, + onG(S) is so. □
We call the groupG(S) isGrothendieck groupofS. We define a mapγ:S→ G(S) byγ(x) = (x,0). Then the mapγis additive. We call it Grothendieck map ofS. From now on, we introduce some properties ofG(S) with respect to the map
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