## Invariant means and finite representation theory of C∗-algebras (2006)

Citations: | 26 - 8 self |

### BibTeX

@MISC{Brown06invariantmeans,

author = {Nathanial P. Brown},

title = {Invariant means and finite representation theory of C∗-algebras},

year = {2006}

}

### OpenURL

### Abstract

### Citations

437 |
C ∗ -algebras and their automorphism groups
- Pedersen
- 1979
(Show Context)
Citation Context ...So, assume F ⊂ A and ǫ > 0 are given and choose B ⊂ A and φ : B → Mn(C) such that d(F, Bφ) < ǫ and ‖τ|B − trn ◦ φ‖B∗ < ǫ. Let Pτ ∈ A ∗∗ (resp. Ptrn◦φ ∈ B ∗∗ φ ⊂ B∗∗ ⊂ A ∗∗ ) be the central cover (cf. =-=[61]-=-) of the GNS representation of τ (resp. the GNS representation of trn ◦ φ|Bφ ). Let Q = PτPtrn◦φ ∈ A ∗∗ . Then Q commutes with Bφ (⊂ A ∗∗ ) and QBφ is a finite dimensional subalgebra of QA ∗∗ Q ⊂ PτA ... |

343 | Fundamentals of the theory of operator algebras - Kadison, Ringrose - 1983 |

156 |
Completely bounded maps and dilations
- Paulsen
- 2003
(Show Context)
Citation Context ...by πτ ⊙πop τ (a⊗b) = πτ(a)πop τ (b). It is an important fact, essentially due to Murray and von Neumann, that Jπτ(A) ′′ J = πτ(A) ′ and (hence) πτ(A) ′′ = Jπop τ (A)′ J. Completely positive maps (cf. =-=[60]-=-) will play an important role in these notes. We will use the abbreviations c.p. and u.c.p. for ‘completely positive’ and ‘unital representation’ (i.e. a ∗-homomorphism π op τ : A op → B(L 2 (A, τ))) ... |

96 |
The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized
- Rosenberg, Schochet
- 1987
(Show Context)
Citation Context ... and we have assumed a unique tracial state. Hence we only have to know that A satisfies the Universal Coefficient Theorem (UCT) of Rosenberg-Schochet – which it does by results of Rosenberg-Schochet =-=[73]-=- – and that the unique trace is uniform locally finite dimensional for then the previous theorem will imply that (C(M) ⋊h Z) ⊗ U is a nuclear, tracially AF algebra satisfying the UCT and hence it must... |

93 |
C*-algebras of real rank zero
- Brown, Pedersen
- 1991
(Show Context)
Citation Context ...-algebra is QD, since quasidiagonality passes to subalgebras and ASH algebras are QD. Proposition 2.4.2. Elliott’s conjecture predicts that if A is simple, stably finite, nuclear, real rank zero (cf. =-=[15]-=-), has weakly unperforated invariant (see the proof of the previous proposition for this definition) and K0(A) has the Riesz interpolation property (cf. [25, Section IV.6]) then A is tracially AF in t... |

74 |
On non-semisplit extensions, tensor products and exactness of group
- Kirchberg
- 1993
(Show Context)
Citation Context ...ds, τ is an amenable trace if and only if it is the restriction of an invariant mean on B(H). 1 This notion is well known (under different names) and has already shown its importance (cf. [23], [24], =-=[45]-=-, [47], [8], [67], [6]). Indeed, in Connes’ remarkable paper [23] he showed (using the terminology ‘hypertrace’) that a II1-factor is isomorphic to the hyperfinite II1-factor if and only if its unique... |

66 |
Classification of injective factors. Cases II1, II∞, IIIλ, λ ̸= 1
- Connes
- 1976
(Show Context)
Citation Context ... of faithful representation and hence being an invariant mean is an abstract property of a (tracial) state. This notion is well known (under different names) and has already shown its importance (cf. =-=[21]-=-, [22], [40], [42], [7], [60], [5]). Indeed, in Connes’ remarkable paper [21] he showed (using the terminology ‘hypertrace’) that a II1-factor is isomorphic to the hyperfinite II1factor if and only if... |

62 | Analytic K-homology - Higson, Roe - 2000 |

37 |
On the structure of simple C ∗ -algebras tensored with a UHFalgebra
- Rørdam
- 1992
(Show Context)
Citation Context ...onvenient to summarize (special cases of) some known facts. 3 See the section on tracially AF algebras for the definitions of these things.6.1. ELLIOTT’S CLASSIFICATION PROGRAM 69 Theorem 6.1.1 (cf. =-=[70]-=-, [71], [12]). 4 Assume A is simple, nuclear and has a unique tracial state. Then for any UHF algebra U the (simple, nuclear) tensor product algebra A⊗U has a unique trace, real rank zero, stable rank... |

35 |
Generalized inductive limits of finite-dimensional C
- Blackadar, Kirchberg
- 1997
(Show Context)
Citation Context ...hether every invariant mean is quasidiagonal. Though we doubt that this is the case, at the moment we have no good way of distinguishing the two sets. (Compare with Blackadar and Kirchberg’s question =-=[12]-=-: Is every nuclear, stably finite C ∗ -algebra necessarily QD?) If one starts with a QD C ∗ -algebra, however, then it turns out that T(A)QD is precisely the set of traces which can be encoded in the ... |

26 |
Amenable unitary representations of locally compact groups
- Bekka
- 1990
(Show Context)
Citation Context ... .,bn ∈ A, n∑ |τ( i=1 → 0 and τ(a) = lim n→∞ aib ∗ i )| ≤ ‖ n∑ i=1 (6) For all a1, . . ., an, b1, . . .,bn ∈ A, n∑ ai ⊗ bi)‖ ≤ ‖ ‖πτ ⊙ π op τ ( ai ⊗ b ∗ i ‖A⊗Aop. < aPn, Pn >HS , < Pn, Pn >HS i=1 i=1 =-=(7)-=- πτ ⊙ πop τ extends to a representation of A ⊗ Aop . (8) There exists a u.c.p. map Φ : B(H) → πτ(A) ′′ such that Φ(a) = πτ(a), for all a ∈ A. n∑ ai ⊗ bi‖A⊗A op. Proof. The equivalence of (2) and (3) i... |

25 |
Covering dimension and quasidiagonality
- Kirchberg, Winter
(Show Context)
Citation Context ... of the tracial space one immediately gets classification results for such algebras. We should also mention that if A has real rank zero and finite decomposition rank in the sense of Kirchberg-Winter =-=[48]-=- then it can be shown that T(A) = UAT(A)LFD. Hence this class of algebras also fits into the tracial approximation picture. However, for the purposes of classification there is currently little reason... |

24 | Cancellation and stable rank for direct limits of recursive subhomogeneous algebras, preprint
- Phillips
(Show Context)
Citation Context ...0]. Proof. By [54], such crossed products are inductive limits of ‘recursive subhomogeneous’ algebras (which, in particular, are type I), they have stable rank one and weakly unperforated K-theory by =-=[62]-=- and, finally, real rank zero by [63]. □ Remark 6.1.11. Note that the first two results above hold true under the weaker assumption of norm separability of the tracial space. However, Corollary 6.1.10... |

22 |
C ∗ -algebras by example, Fields Institute Monographs 6
- Davidson
- 1996
(Show Context)
Citation Context ...TION, DEFINITIONS AND USEFUL FACTS 11 Proof. That there exist isometries Vk : Cn → H such that ‖φ(a) −V ∗ k aVk‖ → 0 is the first step in proving the usual version of Voiculescu’s Theorem (cf. [3] or =-=[27]-=-). The commutator estimates, which are the important part for us, follow from the previous lemma. □ We will also need the following technical version of Voiculescu’s Theorem. A proof can be found in [... |

22 |
Notes on extensions of C
- Arveson
- 1977
(Show Context)
Citation Context ...riety of II1-factor representations. This general theory is related to various other problems as well. Applications include: (1) A characterization of R ω -embeddable factors in terms of Lance’s WEP. =-=(2)-=- A classification theorem for certain simple, nuclear C ∗ -algebras with unique trace. (3) For a self-adjoint operator, there always exists a filtration such that the finite section method (from numer... |

22 | Inner quasidiagonality and strong NF algebras - Blackadar, Kirchberg, et al. |

22 |
Classification of simple C ∗ -algebras with tracial topological rank zero
- Lin
(Show Context)
Citation Context ...also not enough (see Theorem 7.23). It is interesting to note, however, that the case of an exact Popa algebra with unique trace remains open. On the other hand, inspired by recent work of Huaxin Lin =-=[45]-=-, we will show that approximation properties of traces provide an abstract hypothesis which does ensure that Popa algebras are tracially AF. That is, under certain technical assumptions, we will show ... |

21 | The Novikov conjecture for linear groups
- Guentner, Higson, et al.
- 2005
(Show Context)
Citation Context ...es uniformly embeddable” has been downgraded to a conjecture. Recently Guentner, Higson and Weinberger showed that every linear group is exact and, hence, uniformly embeddable into Hilbert space (cf. =-=[32]-=-). Is there an extremely clever choice of microstates together with an ultraproduct argument which will work here? (9) Is the canonical trace on C ∗ (Γ) (giving the left regular representation) always... |

21 |
Bilinear forms on exact operator spaces and
- Junge, Pisier
(Show Context)
Citation Context ...lgebras. (For example, ‘most’ finite dimensional operator spaces will generate C ∗ -algebras without the local lifting property since the space of n-dimensional operator spaces is not separable – cf. =-=[43]-=- – while every finite dimensional subspace of a C ∗ -algebra with the local lifting property can be identified with a subspace of C ∗ (F∞) and the space of n-dimensional operator subspaces of any sepa... |

19 |
On nuclear C ∗ -algebras
- Lance
- 1973
(Show Context)
Citation Context ...ompare with [6, Corollary 2.11].) For a discrete group Γ the following are equivalent: (1) Γ is amenable. (2) T(C ∗ r (Γ)) = AT(C∗ r (Γ)). (3) AT(C ∗ r (Γ)) ̸= ∅. Proof. (1) =⇒ (2). A result of Lance =-=[49]-=- states that C∗ r(Γ) is nuclear whenever Γ is amenable. Hence for every C∗-algebra B there is a unique C∗-norm on the algebraic tensor product C∗ r(Γ) ⊙ B. In particular, C∗ r(Γ) ⊗ B enjoys the univer... |

18 |
SQ-universality of hyperbolic groups
- Olshanskii
- 1995
(Show Context)
Citation Context ...R AND WEP C ∗ -ALGEBRAS 39 Since the factorization property is equivalent to knowing that τ is an amenable trace on C ∗ (Γ) we can combine Kirchberg’s result quoted above with some work of Olshanskii =-=[56]-=- to get the following connection between amenable traces and one of the major open problems in geometric group theory. Proposition 4.1.13. Every hyperbolic group is residually finite if and only if τ ... |

17 |
C ∗ -Algebras and Numerical Analysis
- Hagen, Roch, et al.
- 2001
(Show Context)
Citation Context ...n order to stay reasonably self-contained we will recall the definitions and (statements of) results that we need, however we highly recommend looking at the recent book of Hagen, Roch and Silbermann =-=[40]-=- (see also Böttcher’s survey article [15]) for more on this subject. Let T ∈ B(H) be given. There are two important questions that one may hope to (approximately) solve via a combination of operator t... |

15 |
Spectral decomposition of all operators in a II1-factor which is embeddable
- Haagerup
- 2001
(Show Context)
Citation Context ...article [54] for precise statements of these equivalences – with a number of simplified proofs – as well as a comprehensive bibliography on the subject.) Moreover, thanks to the deep work of Haagerup =-=[29]-=-, a positive answer to Connes’ problem would nearly complete the relative invariant subspace problem for II1-factors while a negative answer would imply a negative answer to Voiculescu’s ‘unification ... |

14 |
The Effros-Maréchal topology in the space of von Neumann algebras
- Haagerup, Winsløw
(Show Context)
Citation Context ...ely related to a surprising variety of other problems and, as such, is a very important open question. II1-factors which are embeddable into R ω already admit a number of characterizations (see [45], =-=[38]-=-, [58]) but the results of this section show that the difference between embeddability and hyperfiniteness is rather delicate.6.3. CONNES’ EMBEDDING PROBLEM 83 Theorem 6.3.1. If M ⊂ B(L 2 (M)) is a I... |

14 |
On the independence of K-theory and stable rank for simple
- Toms
(Show Context)
Citation Context ... his examples were not stably finite and one could still hope that the Elliott invariant may be complete for simple, stably finite, nuclear C ∗ -algebras. Unfortunately, some recent work of Toms (cf. =-=[75]-=-) has shown that this is also not possible as he has constructed examples of non-isomorphic simple ASH algebras with the same Elliott invariant. On the other hand, some experts have suggested that the... |

12 |
Classification of simple C ∗ -algebras of tracial topological rank zero
- Lin
(Show Context)
Citation Context ...ially AF algebra is that the finite dimensional algebra B from Definition 1.2 is required (for tracially AF algebras) to be ‘large in trace’ (see [51] for the1. INTRODUCTION 5 precise definition and =-=[50]-=- for the classification theorem). Lin’s classification result is so exciting because Popa proved in [67] that every simple, unital, quasidiagonal C ∗ -algebra of real rank zero is a Popa algebra and h... |

12 |
C∗ –algebras associated with groups with Kazhdan’s property T
- Wassermann
- 1991
(Show Context)
Citation Context ...es can be used to solve some K-homological questions and a natural question regarding the passage of quasidiagonality to the Calkin algebra. The main results of this section are taken from [2], [24], =-=[79]-=- and [80]. However, the invariant mean point of view will turn these “technical” results into very natural and intuitively clear theorems. The three questions which are easily solved using amenable tr... |

11 |
Effros, Separable nuclear C ∗ -algebras and injectivity
- Choi, G
- 1976
(Show Context)
Citation Context ... observed the (not very deep) fact that every trace on a nuclear C ∗ -algebra is an invariant mean. Since R is the only II1-factor which can arise from a representation of a nuclear C ∗ -algebra (cf. =-=[19]-=-, [21]) it follows that T(A)u-IM contains all the extreme points of T(A) (when A is nuclear) and thus we arrive at the following stronger (and much deeper) statement. Corollary 3.8. If A is a nuclear ... |

10 |
Lifting problems and local reflexivity for
- Effros, Haagerup
- 1985
(Show Context)
Citation Context ...s to the interested reader. Finally we recall that the Effros-Haagerup lifting theorem asserts that local liftability is equivalent to knowing that for every C ∗ -algebra D the sequence is exact (cf. =-=[28]-=-). 0 → I ⊗ D → A ⊗ D → B ⊗ D → 0 Though we will try our hardest to avoid non-unital C∗-algebras they are occasionally forced upon us. Hence a few words about unitizations and amenable traces on ideals... |

10 | Parrott: Operators commuting with a von Neumann algebra modulo the set of compact operators - Johnson, K - 1972 |

10 |
On quasidiagonal C ∗ -algebras
- Brown
(Show Context)
Citation Context ...n. Since, ι is approximately unitarily equivalent to ι ⊕ π, it is clear how to complete the proof. □ We will also need the following technical version of Voiculescu’s Theorem. A proof can be found in =-=[16]-=- or [17]. Proposition 2.2.2. Let A ⊂ B(H) be in general position and Φ : A → B(K) be a u.c.p. map which is a faithful ∗-homomorphism modulo the compacts (i.e. composing with the quotient map to the Ca... |

10 |
Nonnuclear subalgebras of AF algebras
- Dădărlat
(Show Context)
Citation Context ... be closely related to nuclearity. For example, in [60, pg. 157] Popa asked whether every Popa algebra with unique trace is necessarily nuclear. (Counterexamples were first constructed by Dadarlat in =-=[23]-=-.) More generally, he asked in [60, Remark 3.4.2] whether the hyperfinite II1-factor R was the only II1-factor which could arise from a GNS representation of a Popa algebra. Support for a positive ans... |

9 |
A new proof of the equivalence of injectivity and hyperfiniteness for factors on a separable Hilbert space
- Haagerup
- 1985
(Show Context)
Citation Context ...every C ∗ -algebra B.40 4. EXAMPLES AND SPECIAL CASES Proof. This is an immediate consequence of Theorem 3.2.2 and the celebrated fact that an injective von Neumann algebra is hyperfinite (cf. [23], =-=[33]-=-, [65]) since it is not hard to show that every representation of a nuclear C∗-algebra yields an injective von Neumann algebra (use Lance’s trick from the proof of Theorem 3.1.6 to show the commutant ... |

9 |
A C ∗ -algebra for which Ext(A) is not a group
- Anderson
- 1978
(Show Context)
Citation Context ...algebras are actually traces [32]): (1) For two projections p and q in (matrices over) A, p is equivalent to a proper subprojection of q if and only if τ(p) < τ(q) for all τ ∈ T(A); (2) If f : T(A) → =-=[0, 1]-=- is any continuous, affine function and ǫ > 0 is given then there exists a projection p ∈ A such that |f(τ)−τ(p)| < ǫ for all τ ∈ T(A) (this is essentially a special case of the fact that K0(A) is den... |

8 |
Random matrices and K-theory for exact C
- Haagerup, Thorbjørnsen
(Show Context)
Citation Context ...gebra. 11 The proof typically quoted depends on the uniqueness of the trace on C ∗ r(F2), Haagerup’s remarkable result that every unital, stably finite, exact C ∗ - algebra has a tracial state ([34], =-=[36]-=-) together with Connes’ theorem that injective implies hyperfinite and hence is highly non-trivial. On the other hand, amenable 10 This violates our standing separability assumption, but otherwise cau... |

8 |
A noncommutative moment problem
- Hadwin
- 2001
(Show Context)
Citation Context ...coincide in this case. Proposition 4.1.14. For a free group we have AT(C ∗ (Fn)) = AT(C ∗ (Fn))QD = AT(C ∗ (Fn))LFD. Proof. This follows from a simple observation of Kirchberg (see [45, Lemma 4.5] or =-=[39]-=-). Namely, since R ω is a von Neumann algebra every unitary can be lifted to a unitary in l ∞ (R). A standard approximation argument yields the following improvement: If π : C ∗ (Fn) → R ω is a ∗-homo... |

7 | of unitaries in UHF-algebras and functorial properties of exactness - Commutants - 1994 |

7 |
Property (T ) for II1 factors and unitary representations of Kazhdan groups
- Robertson
- 1993
(Show Context)
Citation Context ...A)LFD.) See Proposition 6.1.23 for related questions. (7) Can an infinite, simple, discrete group with Kazhdan’s property T be embed into the unitary group of an R ω -embeddable factor? (Compare with =-=[69]-=- where it is shown that no such embedding exists into the unitary group of L(Fn) or, more generally, L(Γ) for any a-T-menable discrete group Γ.) (8) Let Γ be a discrete group such that the group von N... |

6 |
Perforated ordered K0-groups
- Elliott, Villadsen
(Show Context)
Citation Context ...ra with stable rank one and weakly unperforated K-theory. But Elliott has shown that every possible weakly unperforated Elliott invariant arises from an ASH algebra (see, for example, the appendix of =-=[31]-=-) and hence we can find a simple ASH algebra B whose Elliott invariant is isomorphic to that of A ⊗ M2∞. If B does not have stable rank one then we can replace B with B ⊗ M2∞ to get a simple ASH algeb... |

6 |
A short proof of “ injectivity implies hyperfiniteness ” for finite von Neumann algebras
- Popa
- 1986
(Show Context)
Citation Context ...he hyperfinite II1-factor R was the only II1-factor which could arise from a GNS representation of a Popa algebra. Support for a positive answer was provided by the following observation of Popa (cf. =-=[65]-=-, [66]): Theorem. Let M be a separable II1-factor. Then M ∼ = R if and only if for every finite subset F ⊂ M and ε > 0 there exists a nonzero finite dimensional C ∗ - subalgebra B ⊂ M with unit e such... |

6 | C ∗ -algebras in numerical analysis
- Böttcher
(Show Context)
Citation Context ...finitions and (statements of) results50 NATHANIAL P. BROWN that we need, however we highly recommend looking at the recent book of Hagen, Roch and Silbermann [35] (see also Böttcher’s survey article =-=[14]-=-) for more on this subject. Let T ∈ B(H) be given. There are two important questions that one may hope to (approximately) solve via a combination of operator theoretic and numerical analytic technique... |

5 |
algebras and numerical linear algebra
- unknown authors
- 1994
(Show Context)
Citation Context ... recover τ (since Q is fixed and finite dimensional, its contribution becomes negligible as n → ∞). □ Remark 10.5. In most cases, part (3) above is a much stronger statement than part (2). Indeed, in =-=[4]-=- Arveson observes how statement (3), which he regards as an analogue of Szegö’s Limit Theorem, can be used to recover the essential spectrum of a self-adjoint operator. Note that we only used exactnes... |

3 |
C.: The structure of C ∗ -algebras of minimal diffeomorphisms
- Lin, Phillips
(Show Context)
Citation Context ...orphic to one of the AH algebras considered in [30] by Lin’s classification theorem [50]. However, the unique trace is locally finite dimensional since the remarkable work of Q. Lin and Phillips (cf. =-=[54]-=-) implies that C(M) ⋊h Z is an inductive limit of type I C∗-algebras (actually very special ‘recursive’ subhomogeneous algebras) and hence Corollary 4.4.4 applies. □ Note that a similar result can be ... |

3 |
Quasidiagonality of direct sums of weighted shifts
- Narayan
- 1992
(Show Context)
Citation Context ...or the classes of essentially normal and (finite direct sums of) weighted shift operators there are nice characterizations of quasidiagonality as well (cf. [27, Corollary IX.7.4, Section IX.8], [74], =-=[55]-=-).6.4. AMENABLE TRACES AND NUMERICAL ANALYSIS 91 Moreover, we should also point out that if a weighted shift satisfies the appropriate hypotheses (cf. [74]) then it is possible to actually construct ... |

3 |
Quasidiagonal weighted shifts
- Smucker
- 1982
(Show Context)
Citation Context ...ever for the classes of essentially normal and (finite direct sums of) weighted shift operators there are nice characterizations of quasidiagonality as well (cf. [27, Corollary IX.7.4, Section IX.8], =-=[74]-=-, [55]).6.4. AMENABLE TRACES AND NUMERICAL ANALYSIS 91 Moreover, we should also point out that if a weighted shift satisfies the appropriate hypotheses (cf. [74]) then it is possible to actually cons... |

3 |
On the existence of quasicentral approximate units relative to normed ideals, Part 1
- Voiculescu
- 1990
(Show Context)
Citation Context ...ly deduce the following result. Corollary 6.5.2. (cf. [24, Theorem 19]) Let Γ be any discrete, non-amenable group. Then C ∗ r(Γ) has no unbounded, finitely summable Fredholm modules. Remark 6.5.3. In =-=[76]-=- Voiculescu defined a notion of ‘subexponential’, unbounded Fredholm module and generalized Connes’ result to this setting (i.e. the 9 See [27] for a nice introduction to Ext and BDF theory. In partic... |

3 |
Notes on hypertraces and C
- Bédos
- 1995
(Show Context)
Citation Context ...ce being an invariant mean is an abstract property of a (tracial) state. This notion is well known (under different names) and has already shown its importance (cf. [21], [22], [40], [42], [7], [60], =-=[5]-=-). Indeed, in Connes’ remarkable paper [21] he showed (using the terminology ‘hypertrace’) that a II1-factor is isomorphic to the hyperfinite II1factor if and only if it has an invariant mean. Continu... |

3 |
Weak expectations and injectivity in operator algebras
- Blackadar
- 1978
(Show Context)
Citation Context ...it ball of M. Since ΠNR has the WEP it follows from [40, Proposition 1.3, Theorem 1.4] that M is embeddable into Rω . □ Remark 9.3. In [44] Lance introduced the WEP as a property of C ∗ -algebras. In =-=[8]-=- Blackadar changed the point of view to make this a W ∗ -notion as follows: A von Neumann algebra M ⊂ B(H) has the WEP relative to a weakly dense C ∗ -subalgebra A ⊂ M if there exists a u.c.p. map Φ :... |

3 | Popa algebras in free group factors
- Brown, Dykema
(Show Context)
Citation Context .... Based on the proof of this result, in joint work with Ken Dykema, we have been able to show that all the free group factors L(Fn) (n < ∞) contain finitely generated, weakly dense Popa algebras (see =-=[18]-=-). Hence it is natural to ask how free entropy dimension behaves on such algebras. For example, is it true that for Popa algebras the free entropy dimension with respect to any trace and any set of ge... |

3 |
algebras and their Automorphism Groups
- unknown authors
- 1979
(Show Context)
Citation Context ...t. So, assume F ⊂ A and ǫ > 0 are given and choose B ⊂ A and φ : B → Mn(C) such that d(F, Bφ) < ǫ and ‖τ|B − trn ◦ φ‖B∗ < ǫ. Let Pτ ∈ A∗∗ (resp. Ptrn◦φ ∈ B∗∗ φ ⊂ B∗∗ ⊂ A∗∗ ) be the central cover (cf. =-=[56]-=-) of the GNS representation of τ (resp. the GNS representation of trn ◦ φ|Bφ ). Let Q = PτPtrn◦φ ∈ A∗∗ . Then Q commutes with Bφ (⊂ A∗∗ ) and QBφ is a finite dimensional subalgebra of QA∗∗Q ⊂ PτA∗∗ ∼ ... |