1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

4.3. Amenable actions 125 (3) lls *a Ti -7iJl2---* 0 for alls Er, wheres E Cc(r, A) is the function which sends s 1---+ lA and a ...

126 4. Constructions Define X = L a;^1 (T(p)) ® ep,p pEF and note that X = X*. Hence compression by X is a c.p. map and a comput ...

4.3. Amenable actions 127 Definition 4.3.5. An action of r on a compact space Xis called (topolog- ically) amenable (or, equival ...

128 4. Constructions Using the inequality (a - b)^2 :::; Ja^2 - b^2 J for all positive numbers a, b, we then get lls *a Ti -Till ...

4.4. x )<I r -algebras 129 The proof of this lemma shows that if X is an amenable r-space, then we can assume each map mi: X ...

130 4. Constructions Definition 4.4.1. We say a function h: Xx r ~ <C is of positive type if for any finite sequence s1, ... ...

4.4. x )<l r -algebras 131 where equality holds ifs= e. In particular, (T, T) = E(g* *a g) = 1. Note also that (s *a T, s *a ...

132 4. Constructions Remark 4.4.4. This theorem provides natural examples of subalgebras of nuclear C*-algebras which are not nu ...

4.5. Compact group actions and graph C*-algebras 133 Exercise 4.4.3. Prove that if Y is a compact r-space with a r-equivariant c ...

134 4. Constructions Proof. If a EA+ and 7r(a) = 0, then 7r(Ea(a)) = E13(7r(a)) = 0. D Theorem 4.5.2. Let A be a C*-algebra and ...

4.5. Compact group actions and graph C*-algebras 135 and r(e) is the vertex at which it ends). An operator algebraist likes to t ...

136 4. Constructions algebras. Can you find the finite graph which yields 02? Hint: It has a single vertex!) Though the details ...

4.6. Ountz-Pimsner algebras 137 Let A be a C*-algebra and 'H be a linear space which is a right A-module. An A-valued inner prod ...

138 4. Constructions and note that Hean is canonically isomorphic to .e; Q9 1-i equipped with the A-valued inner product where t ...

4.6. Cuntz-Pimsner algebras 139 semi-inner product defined by (e &>r/,e0rJ) = (rJ',7rd(e,e))rJ). We denote by 1i®BK the H ...

140 4. Constructions where T~ = [T6 · · · T~nl E JIB(AEEin, 1-i) and likewise for Try. Note that for operators x and y, llxyll = ...

4.6. Ountz-Pimsner algebras 141 Theorem 4.6.4. Let A be a C*-algebra and X be a Banach A-module. Then, the subset AX = {ax : a E ...

142 4. Constructions for every a E ('., e,'f] E 1i and a,b EA. Let Oj:: JIB(H)-+ JIB(F(H)) be the *-representation given by cr;: ...

4.6. Cuntz-Pimsner algebras 143 Since F(1i) =span T(1i)1i^1590 , it coincides with the GNS Hilbert A-module for (T(1i), EH) by u ...

144 4. Constructions (3) There is an action I (called the gauge action) of 1l on 0(1i) such that /z(a) =a and /z(Be) = zSe for e ...