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Chapter 2 Projections and Unitary Elements

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Chapter 2 Projections and unitary elements The K-theory of a C∗-algebra is constructed from equivalence classes of its projections and from equivalence classes of its unitary elements. For that reason, we shall consider several equivalence relations and look at the relations between them. The K-groups will be defined only in the following chapters. This chapter is mainly based on Chapter 2 of the book [RLL00]. 2.1 Homotopy classes of unitary elements Definition 2.1.1. For any topological space Ω, one says that a; b 2 Ω are homotopic in Ω if there exists a continuous map v : [0; 1] 3 t 7! v(t) 2 Ω with v(0) = a and v(1) = b. In such a case one writes a ∼h b in Ω. Clearly, the relation ∼h defines an equivalence relation on Ω, and one says that v is a continuous path from a to b in Ω. Note that if Ω0 is another topological space with a; b 2 Ω0 as well, then a; b could be homotopic in Ω without being homotopic in Ω0. Thus, mentioning the ambient space Ω is crucial for the definition of the homotopy relation. On the other hand, we shall often just write t 7! v(t) for the continuous path, without specifying t 2 [0; 1]. In the next statement, we consider this equivalence relation in the set U(C) of all unitary elements of a unital C∗-algebra C. Clearly, this set is a group (for the multiplica- tion) but not a vector space. Note also that if u0; u1; v0; v1 2 U(C) satisfy u0 ∼h u1 and v0 ∼h v1, then u0v0 ∼h u1v1. Indeed, if t 7! u(t) and t 7! v(t) denote the corresponding continuous paths, then t 7! u(t)v(t) is a continuous map between u0v0 and u1v1. In the sequel we shall denote by U0(C) the set of elements in U(C) which are homotopic to 1 2 C. Let us also recall from Lemma 1.2.8 that for any unitary element u, one has σ(u) 2 T. Lemma 2.1.2. Let C be a unital C∗-algebra. Then: ia (i) If a 2 C is self-adjoint, then e belongs to U0(C), (ii) If u 2 C is unitary and σ(u) =6 T, then u 2 U0(C), 15 16 CHAPTER 2. PROJECTIONS AND UNITARY ELEMENTS (iii) If u; v 2 C are unitary with ku − vk < 2, then u ∼h v. Proof. i) In the proof of Lemma 1.2.8 it has already been observed that if a is self- adjoint, then eia is unitary. By considering now the map [0; 1] 3 t 7! v(t) := eita 2 U(C), one easily observes that this map is continuous, and that v(0) = 1 and v(1) = eia. As ia ia a consequence, e ∼h 1, or equivalently e 2 U0(C). ii) Since σ(u) =6 ( T, there) exists θ 2 R such that eiθ 62 σ(u). Let us then define v : σ(u) ! R by v ei(θ+t) = θ + t for any t 2 (0; 2π) such that ei(θ+t) 2 σ(u). Since σ(u) is a closed set in T, it follows that v is continuous. In addition, one has that eiv(z) = z for any z 2 σ(u). Thus, if one sets a := v(u), one infers that a is a self-adjoint ia element of C and that u = e . As a consequence of (i), one deduces that u 2 U0(C). iii) If ku − vk < 2, it follows that kv∗u − 1k = kv∗(u − v)k < 2. Then, from the estimates jzj ≤ r(a) ≤ kak valid for any z 2 σ(a) and any a 2 C, one infers that −2 62 σ(v∗u − 1), or equivalently −1 62 σ(v∗u). Since v∗u is a unitary element of ∗ C, one infers then from (ii) that v u ∼h 1. Finally, by multiplying the corresponding continuous path on the left by v (or by using the remark made just before the statement of the lemma), one infers that u ∼h v, as expected. Let us stress that the previous lemma states that for any self-adjoint a 2 C, eia is a unitary element of U0(C). However, not all unitary elements of C are of this form, and the point (ii) has only provided a sufficient condition for being of this form. Later on, we shall construct unitary elements which are not obtained from a self-adjoint element. Let us observe that since unitary elements of Mn(C) have only a finite spectrum, one can directly infer from the previous statement (ii) the following corollary: Corollary 2.1.3. The unitary group in Mn(C) is connected, or in other words ( ) ( ) U0 Mn(C) = U Mn(C) : By considering matrix algebras, the following statement can easily be proved: ∗ Lemma 2.1.4( (Whitehead)) . Let C be a unital C -algebra, and let u; v 2 U(C). Then one has in U M2(C) ( ) ( ) ( ) ( ) u 0 uv 0 vu 0 v 0 ∼ ∼ ∼ : 0 v h 0 1 h 0 1 h 0 u In particular, one infers that ( ) ( ) u 0 1 0 ∼ (2.1) 0 u∗ h 0 1 ( ) in U M2(C) . 2.1. HOMOTOPY CLASSES OF UNITARY ELEMENTS 17 0 1 C Proof. Since ( 1 0 ) is a unitary element of M2( ), one infers from the previous corollary 0 1 ∼ 1 0 that ( 1 0 ) h ( 0 1 ). Then, by observing that ( ) ( )( )( )( ) u 0 u 0 0 1 v 0 0 1 = ; 0 v 0 1 1 0 0 1 1 0 u 0 1 0 v 0 1 0 uv 0 one readily infers that the r.h.s. is homotopic to ( 0 1 )( 0 1 )( 0 1 )( 0 1 ) = ( 0 1 ). The other relations can be proved similarly. Let us add some more information on U0(C). Proposition 2.1.5. Let C be a unital C∗-algebra. Then, ∗ (i) U0(C) is a normal subgroup of U(C), i.e. vuv 2 U0(C) whenever u 2 U0(C) and v 2 U(C), (ii) U0(C) is open and closed relative to U(C), ia1 ia2 ian (iii) An element u 2 C belongs to U0(C) if and only if u = e e : : : e for some self-adjoint elements a1; : : : ; an 2 C. Exercise 2.1.6. Provide the proof of this statement, see also [RLL00, Prop. 2.1.6]. Based on the content of this proposition, the following lemma can be proved: Lemma 2.1.7. Let C, Q be unital C∗-algebras, and let ' : C!Q be a surjective (and hence unit preserving) ∗-homomorphism. Then: ( ) (i) ' U0(C) = U0(Q), ( ) 2 U Q 2 U C u 0 (ii) For each u ( ) there exists v 0 M2( ) such that '(v) = ( 0 u∗ ), 2 U Q 2 U C ∼ (iii) If(u )( ), and if there exists v ( ) such that u h '(v), then u belongs to ' U(C) . Proof. i) Since a unital ∗-homomorphism( is) continuous and maps unitary elements on unitary elements, it follows that ' U0(C) is contained in U0(Q). Conversely, if u ib1 ib2 ibn belongs to U0(Q), then u = e e : : : e for some self-adjoint elements b1; : : : ; bn 2 Q by Proposition 2.1.5.(iii). Since ' is surjective, there exists aj 2 C such that bj = '(aj) 2 f g for any j 1; : : : ; n . Note that aj can be chosen( self-adjoint) since otherwise the ∗ ∗ ∗ element (aj + aj )=2 is self-adjoint and satisfies ' (aj + aj )=2 = (bj + bj )=2 = bj. Then, ia1 ia2 ian by setting v = e e : : : e one gets, again by Proposition 2.1.5, that v 2 U0(C) and that '(v) = u. ( ) 2 U Q u 0 U Q ii) For any u ( ) consider the element( ( 0 u∗)) which( belongs) to 0 M2( ) by (2.1). By applying then the point (i) to U0 M2(C) and U0 M2(Q) instead of U0(C) and U0(Q), one immediately deduces the second statement. ∗ ∗ ∗ iii) If u ∼h '(v), then u'(v) = u'(v ) is homotopic to 1 2 Q, i.e. u'(v ) 2 U0(Q). ∗ 2 U C By (i) it follows that u'(v ) = '(w)( for some) w 0( ). Consequently, one infers that u = '(wv), or in other words u 2 ' U(C) . 18 CHAPTER 2. PROJECTIONS AND UNITARY ELEMENTS Recall that for any unital C∗-algebra C, one denotes by GL(C) the group of its invertible elements. The set of elements of GL(C) which are homotopic to 1 is denoted by GL0(C). Clearly, U(C) is a subgroup of GL(C). The following statement establishes a more precise link between these two groups. Before this, observe that for any a 2 C, the element a∗a is positive, as recalled in Proposition 1.2.7. Thus, one can define jaj := (a∗a)1=2 which is also a positive element of C, and call it the absolute value of a. Proposition 2.1.8. Let C be a unital C∗-algebra. (i) If a belongs to GL(C), then jaj belongs to GL(C) as well, and w(a) := ajaj−1 belongs to U(C). In addition, the equality a = w(a)jaj holds. (ii) The map w : GL(C) 3 a 7! w(a) 2 U(C) is continuous, satisfies w(u) = u for any u 2 U(C), and verifies w(a) ∼h a in GL(C) for any a 2 GL(C), (iii) If v0; v1 2 U(C) satisfies v0 ∼h v1 in GL(C), then v0 ∼h v1 in U(C).
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