2911:
2320:
3144:
3604:
700:
1189:
865:
4504:
3326:
416:
1630:
3718:
975:
It is clear that the extension of a finite-dimensional C*-algebra by another finite-dimensional C*-algebra is again finite-dimensional. More generally, the extension of an AF algebra by another AF algebra is again AF.
577:
whose objects are isomorphism classes of finite-dimensional C*-algebras and whose morphisms are *-homomorphisms modulo unitary equivalence. By the above discussion, the objects can be viewed as vectors with entries in
4708:
Lawrence G. Brown. Extensions of AF Algebras: The
Projection Lifting Problem. Operator Algebras and Applications, Proceedings of symposia in pure mathematics, vol. 38, Part 1, pp. 175-176, American Mathematical Soc.,
3423:
2144:
2947:. The Effros-Handelman-Shen theorem says the converse is true. Every Riesz group, with a given scale, arises as the dimension group of some AF algebra. This specifies the range of the classifying functor
2016:
4771:
604:
790:
4648:
4000:
3111:
is given entrywise order. The two properties of Riesz groups are preserved by direct limits, assuming the order structure on the direct limit comes from those in the inductive system. So (
1509:
513:
304:
1080:
215:
4093:
3920:
4676:
is one with a separable predual and contains a weakly dense AF C*-algebra. Murray and von
Neumann showed that, up to isomorphism, there exists a unique hyperfinite type II
4570:
826:
4642:
4606:
4543:
4338:
2338:
Two preliminary facts are needed before one can sketch a proof of
Elliott's theorem. The first one summarizes the above discussion on finite-dimensional C*-algebras.
4692:
exhibited a family of non-isomorphic type III hyperfinite factors with cardinality of the continuum. Today we have a complete classification of hyperfinite factors.
1327:
102:. For unital AF algebras, both existence and uniqueness follow from the fact the Murray-von Neumann semigroup of projections in an AF algebra is cancellative.
76:
3221:
331:
1520:
3615:
3599:{\displaystyle (G,G^{+})=\varinjlim (H_{k},H_{k}^{+}),\quad {\mbox{where}}\quad (H,H_{k}^{+})=(\mathbb {Z} ^{n_{k}},\mathbb {Z} _{+}^{n_{k}}).}
730:
is unital. The inductive system specifying an AF algebra is not unique. One can always drop to a subsequence. Suppressing the connecting maps,
4689:
2205:
if it is scale-preserving. Two dimension group are said to be isomorphic if there exists a contractive group isomorphism between them.
4271:, in a sense appropriate for ordered groups. Thus UHF algebras are simple C*-algebras. In general, the groups which are not dense in
2048:
937:
Due to the presence of matrix units in the inductive sequence, AF algebras have the following local characterization: a C*-algebra
2676:
We sketch the proof for the non-trivial part of the theorem, corresponding to the sequence of commutative diagrams on the right.
4733:
2665:. Given an isomorphism between dimension groups, one constructs a diagram of commuting triangles between the direct systems of
816:, with the nodes connected by appropriate downward arrows, is the Bratteli diagram of an AF algebra. A Bratteli diagram of the
4324:
It was proposed by
Elliott that other classes of C*-algebras may be classifiable by K-theoretic invariants. For a C*-algebra
1929:
4654:
In the literature, one can find several conjectures of this kind with corresponding modified/refined
Elliott invariants.
695:{\displaystyle A=\varinjlim \cdots \rightarrow A_{i}\,{\stackrel {\alpha _{i}}{\rightarrow }}A_{i+1}\rightarrow \cdots ,}
3136:'s, each with the canonical order structure. This hinges on the following technical lemma, sometimes referred to as the
4664:
4784:
740:
3938:
1450:
451:
98:: the lift must be unique up to approximate unitary equivalence. Classification then follows from what is known as
820:
is given on the right. The two arrows between nodes means each connecting map is an embedding of multiplicity 2.
3132:
A key step towards the Effros-Handelman-Shen theorem is the fact that every Riesz group is the direct limit of
1184:{\displaystyle M_{\infty }(A)=\varinjlim \cdots \rightarrow M_{n}(A)\rightarrow M_{n+1}(A)\rightarrow \cdots .}
325:
is said to be the multiplicity of Φ. In general, a unital homomorphism between finite-dimensional C*-algebras
258:
4831:
4157:
173:
4011:
4826:
3876:
1262:). It is clear that ~ is an equivalence relation. Define a binary operation + on the set of equivalences
1195:
2327:
Elliott's theorem says that the dimension group is a complete invariant of AF algebras: two AF algebras
40:
4651:
by
Elliott stated that the Elliott invariant classifies simple unital separable amenable C*-algebras.
4294:
3331:
where all the connecting homomorphisms in the directed system on the right hand side are positive.
860:{\displaystyle 1\rightrightarrows 2\rightrightarrows 4\rightrightarrows 8\rightrightarrows \dots }
43:
C*-algebras. Approximate finite-dimensionality was first defined and described combinatorially by
4499:{\displaystyle {\mbox{Ell}}(A)=((K_{0}(A),K_{0}(A)^{+},\Gamma (A)),K_{1}(A),T^{+}(A),\rho _{A}),}
2852:
By induction, we have a diagram of commuting triangles as indicated in the last diagram. The map
125:
121:
94:: a homomorphism between invariants must lift to a *-homomorphism of algebras. Second, one shows
4821:
4772:
On the
Classification of Inductive Limits of Sequences of Semisimple Finite-Dimensional Algebras
4548:
4290:
574:
66:
4611:
4575:
4512:
2849:
with a unitary conjugation if needed, we have a commutative triangle on the level of algebras.
2423:
can be "moved back", on the level of algebras, to some finite stage in the inductive system.
1365:
1333:
4848:
4755:
4298:
1004:
965:
144:
8:
4673:
1216:
106:
1276:
2825:
further down if necessary, we obtain diagram 4, a commutative triangle on the level of
1380:
69:
for AF-algebras serves as a prototype for classification results for larger classes of
4785:
Regularity properties in the classification program for separable amenable C-algebras
3321:{\displaystyle (G,G^{+})=\varinjlim (\mathbb {Z} ^{n_{k}},\mathbb {Z} _{+}^{n_{k}}),}
3086:) is an ordered group that is unperforated and has the Riesz decomposition property.
993:
518:
In the non-unital case, the equality is replaced by ≤. Graphically, Φ, equivalently (
48:
1245:
797:
528:
79:
stably finite C*-algebras. Its proof divides into two parts. The invariant here is
411:{\displaystyle \Phi :\oplus _{1}^{s}M_{n_{k}}\rightarrow \oplus _{1}^{t}M_{m_{l}}}
4253:
4101:
A similar argument applies in general. Observe that the scale is by definition a
2246:
are both finite-dimensional, corresponding to each partial multiplicities matrix
1074:) canonically, into the "upper left corner". Consider the algebraic direct limit
813:
70:
32:
1625:{\displaystyle (K_{0}(A),K_{0}(A)^{+})=(\mathbb {Z} ^{m},\mathbb {Z} _{+}^{m}).}
1344:. While we could choose matrices of arbitrarily large dimension to stand in for
532:
73:
4842:
3713:{\displaystyle \Gamma (H_{1})=\{v\in H_{1}^{+}|\phi _{1}(v)\in \Gamma (G)\},}
2967:
2944:
2835:
For finite-dimensional algebras, two *-homomorphisms induces the same map on
1896:
A finer invariant is needed to detect isomorphism classes. For an AF algebra
1427:
1007:
and serves as the range of a kind of "dimension function." For an AF algebra
114:
59:
4681:
4268:
4102:
595:
110:
44:
4235:
817:
20:
2910:
4545:
is the tracial positive linear functionals in the weak-* topology, and
4310:
147:
28:
2319:
2335:
are isomorphic if and only if their dimension groups are isomorphic.
2231:
between AF algebras in fact induces a contractive group homomorphism
1919:), to be the subset whose elements are represented by projections in
1361:
36:
1661:
87:
949:
is "almost contained" in some finite-dimensional C*-subalgebra.
4160:
are AF and unital. Their dimension groups are the subgroups of
2538:
The proof of the lemma is based on the simple observation that
2250:, there is a unique, up to unitary equivalence, *-homomorphism
2139:{\displaystyle \Gamma (A)=\{x\in K_{0}(A)\,|\,0\leq x\leq \}.}
2842:
if and only if they are unitary equivalent. So, by composing
1720:
can be identified with the partial multiplicities matrix of
238:
Up to unitary equivalence, a unital *-homomorphism Φ :
109:
world are the hyperfinite factors, which were classified by
2201:
A group homomorphism between dimension group is said to be
714:
is a finite-dimensional C*-algebra and the connecting maps
3783:'s (with the number of copies possible less than that in
2208:
The dimension group retains the essential properties of
51:
gave a complete classification of AF algebras using the
3143:
2011:{\displaystyle \Gamma (A)=\{\,|\,p^{*}=p^{2}=p\in A\}.}
1411:
is Murray-von
Neumann equivalent to a subprojection of
1352:, our result will be equivalent regardless. This makes
128:, AF C*-algebras are noncommutative generalizations of
4343:
3504:
3193:, as indicated in the adjacent diagram, such that ker(
2606:
are isomorphic if and only if their dimension groups (
1003:
is an invariant of C*-algebras. It has its origins in
4614:
4578:
4551:
4515:
4341:
4289:
Commutative C*-algebras, which were characterized by
4014:
3941:
3879:
3618:
3426:
3224:
3185:) be a positive homomorphism. Then there exists maps
2051:
1932:
1523:
1453:
1279:
1083:
918:
is itself an AF algebra. Given a
Bratteli diagram of
829:
743:
607:
582:
and morphisms are the partial multiplicity matrices.
454:
334:
261:
176:
2938:
1336:
of two finite-dimensional matrices corresponding to
4319:
1407:) carries a natural order structure: we say ≤ if
4734:Inductive limits of finite dimensional C*-algebras
4636:
4600:
4564:
4537:
4498:
4087:
3994:
3914:
3712:
3598:
3320:
2954:for AF-algebras and completes the classification.
2818:Using the property of the direct limit and moving
2805:Φ shows that the second diagram commutes for some
2795:such that the first diagram on the right commutes.
2138:
2010:
1624:
1503:
1321:
1183:
930:gives inductive system that specifies an ideal of
859:
784:
694:
507:
410:
298:
209:
153:
4760:Dimension groups and their affine representations
4148:. The same argument as above proves the theorem.
1676:between AF algebras induces a group homomorphism
1444:For example, for a finite-dimensional C*-algebra
598:of a sequence of finite-dimensional C*-algebras:
158:
4840:
105:The counterpart of simple AF C*-algebras in the
25:approximately finite-dimensional (AF) C*-algebra
4185:) is the group of rational numbers of the form
785:{\displaystyle A={\overline {\cup _{n}A_{n}}}.}
3790:). So this gives a finite-dimensional algebra
1383:construction gives an abelian group, which is
1198:(self-adjoint idempotents) in this algebra by
723:are *-homomorphisms. We will assume that each
421:is specified, up to unitary equivalence, by a
3995:{\displaystyle \varinjlim (G_{k},G_{k}^{+}),}
1504:{\displaystyle A=\oplus _{k=1}^{m}M_{n_{k}},}
508:{\displaystyle \sum _{k}r_{lk}n_{k}=m_{l}.\;}
167:takes the following form, up to isomorphism:
3704:
3641:
3609:Dropping to a subsequence if necessary, let
2408:The lemma can be extended to the case where
2130:
2067:
2002:
1948:
86:with its natural order structure; this is a
2924:, the adjacent diagram commutates for each
2815:Comparing diagrams 1 and 2 gives diagram 3.
2323:Commutative diagrams for Elliott's theorem.
934:. In fact, every ideal arises in this way.
163:An arbitrary finite-dimensional C*-algebra
3350:) is a countable Riesz group with scale Γ(
2943:The dimension group of an AF algebra is a
2661:The crux of the proof has become known as
2471:. Then for any a contractive homomorphism
1861:are not isomorphic but stably isomorphic;
504:
4806:, Encyclopaedia of Mathematical Sciences
4758:, Handelman, D.E., and Shen C.L. (1980),
3852:. There is a corresponding homomorphism
3568:
3546:
3290:
3268:
2928:. By uniqueness of direct limit of maps,
2101:
2095:
1966:
1960:
1601:
1586:
643:
4684:obtained the analogous result for the II
3845:determines a finite-dimensional algebra
3406:Consider first the special case where Γ(
3142:
2909:
2318:
812:} in the obvious way. For instance, the
299:{\displaystyle \Phi (a)=a\otimes I_{r},}
62:with sufficiently nice order structure.
4657:
2767:). By the previous lemma, there exists
2531:is unique up to unitary equivalence in
2344:For two finite-dimensional C*-algebras
1843:can only distinguish AF algebras up to
1043:matrices whose entries are elements of
869:(A Bratteli diagram of the CAR algebra)
805:is formed by the Bratteli diagrams of {
4841:
4164:. For example, for the 2 × 2 matrices
1802:
1635:Two essential features of the mapping
945:is separable and any finite subset of
926:of nodes, the subdiagram generated by
4804:Classification of Nuclear C*-Algebras
4782:Elliott, G.A. and Toms, A.S. (2008),
4267:) ∩ , with 1 = . All such groups are
3215:) can be expressed as a direct limit
3125:) is a Riesz group for an AF algebra
3009:The following property is called the
2883:is the direct limit of the sequence {
2864:is the direct limit of the sequence {
2731:)) be a dimension group isomorphism.
2405:is unique up to unitary equivalence.
2038:element is the maximal element of Γ(
210:{\displaystyle \oplus _{k}M_{n_{k}},}
4797:A User's Guide for Operator Algebras
4088:{\displaystyle \cup _{k}\phi _{k}=.}
3870:. Induction gives a directed system
2314:
3915:{\displaystyle A=\varinjlim A_{k},}
3772:has the canonical order structure,
3354:), then there exists an AF algebra
2735:Consider the composition of maps Φ
2549:) is finitely generated and, since
227:denotes the full matrix algebra of
13:
4418:
3692:
3619:
2966:with a partial order is called an
2052:
1933:
1089:
335:
262:
14:
4860:
4814:
2939:The Effros-Handelman-Shen theorem
2497:), there exists a *-homomorphism
2460:be the canonical homomorphism of
2378:), there exists a *-homomorphism
2352:, and a contractive homomorphism
1022:) can be defined as follows. Let
979:
535:with nodes corresponding to each
4752:, American Mathematical Society.
4665:approximately finite-dimensional
4320:Elliott's classification program
2898:are mutual inverses. Therefore,
2303:is the direct limit of those of
58:functor whose range consists of
4572:is the natural pairing between
3751:. Now consider the order ideal
3510:
3502:
2957:
2663:Elliott's intertwining argument
2299:), then the dimension group of
154:Definition and basic properties
4712:
4702:
4631:
4625:
4595:
4589:
4532:
4526:
4490:
4474:
4468:
4452:
4446:
4430:
4427:
4421:
4406:
4399:
4383:
4377:
4364:
4361:
4355:
4349:
4158:uniformly hyperfinite algebras
4098:This proves the special case.
4079:
4060:
4054:
4035:
3986:
3955:
3701:
3695:
3686:
3680:
3666:
3635:
3622:
3590:
3541:
3535:
3511:
3496:
3465:
3446:
3427:
3334:
3312:
3263:
3244:
3225:
2974:of elements ≥ 0 is called the
2673:by applying the second lemma.
2238:on the dimension groups. When
2127:
2114:
2097:
2092:
2086:
2061:
2055:
1962:
1957:
1951:
1942:
1936:
1616:
1581:
1575:
1566:
1559:
1543:
1537:
1524:
1368:. We denote this semigroup by
1316:
1304:
1298:
1292:
1286:
1280:
1172:
1169:
1163:
1144:
1141:
1135:
1122:
1100:
1094:
851:
845:
839:
833:
683:
648:
630:
585:
549:and the number of arrows from
373:
271:
265:
159:Finite-dimensional C*-algebras
1:
4748:, Field Institute Monographs
4725:
2917:Furthermore, on the level of
1713:are both finite-dimensional,
1217:Murray-von Neumann equivalent
527:), can be represented by its
4293:, are AF precisely when the
4275:are the dimension groups of
3011:Riesz decomposition property
774:
563:is the partial multiplicity
531:. The Bratteli diagram is a
7:
4827:Encyclopedia of Mathematics
4301:. The continuous functions
4151:
1733:respects direct limits. If
252:is necessarily of the form
10:
4865:
4737:, Trans. Amer. Math. Soc.
3797:whose dimension group is (
3410:) = with maximal element
3392:) = with maximal element
3107:) is a Riesz group, where
2433:be finite-dimensional and
2186:, its dimension group is (
4788:, Bull. Amer. Math. Soc.
4662:In a related context, an
4565:{\displaystyle \rho _{A}}
2798:Same argument applied to
100:the intertwining argument
4695:
4637:{\displaystyle K_{0}(A)}
4601:{\displaystyle T^{+}(A)}
4538:{\displaystyle T^{+}(A)}
3388:)). In particular, if Γ(
3093:is finite-dimensional, (
2556:respects direct limits,
4744:Davidson, K.R. (1996),
4731:Bratteli, Ola. (1972),
1767:) is the direct limit ∪
1705:). In particular, when
1430:whose positive cone is
1060:) can be embedded into
1035:) be the C*-algebra of
734:can also be written as
122:noncommutative geometry
4795:Fillmore, P.A.(1996),
4769:Elliott, G.A. (1976),
4746:C*-algebras by Example
4638:
4602:
4566:
4539:
4500:
4089:
3996:
3916:
3714:
3600:
3322:
3148:
2914:
2324:
2140:
2012:
1626:
1505:
1323:
1185:
861:
786:
696:
509:
442:) satisfying, for all
431:partial multiplicities
412:
300:
211:
67:classification theorem
60:ordered abelian groups
4799:, Wiley-Interscience.
4718:Davidson 1996, p. 77.
4639:
4603:
4567:
4540:
4501:
4316:is one such example.
4090:
3997:
3917:
3715:
3601:
3323:
3146:
2913:
2774:and a *-homomorphism
2322:
2141:
2025:is unital with unit 1
2013:
1825:)) is isomorphic to
1627:
1506:
1366:cancellation property
1334:orthogonal direct sum
1324:
1186:
984:
941:is AF if and only if
862:
787:
697:
510:
413:
301:
212:
4658:Von Neumann algebras
4612:
4576:
4549:
4513:
4339:
4299:totally disconnected
4012:
3939:
3877:
3817:forward by defining
3616:
3424:
3403:is unital with = .
3222:
3162:) be a Riesz group,
3089:It is clear that if
3035:, then there exists
2890:}. It is clear that
2049:
1930:
1664:. A *-homomorphism
1521:
1451:
1277:
1081:
1005:topological K-theory
952:The projections in ∪
827:
741:
605:
452:
332:
259:
174:
145:totally disconnected
90:. First, one proves
4802:Rørdam, M. (2002),
4674:von Neumann algebra
3985:
3779:is a direct sum of
3664:
3589:
3534:
3495:
3311:
3207:Every Riesz group (
3147:The Shen criterion.
3140:in the literature.
2986:is unperforated if
2658:)) are isomorphic.
1803:The dimension group
1615:
1480:
1332:where ⊕ yields the
390:
355:
107:von Neumann algebra
4810:, Springer-Verlag.
4634:
4598:
4562:
4535:
4496:
4347:
4252:) is the group of
4116:}, one can choose
4085:
3992:
3971:
3950:
3912:
3894:
3710:
3650:
3596:
3566:
3520:
3508:
3481:
3460:
3318:
3288:
3258:
3149:
2915:
2325:
2219:A *-homomorphism
2136:
2008:
1845:stable isomorphism
1622:
1599:
1501:
1460:
1381:Grothendieck group
1379:). Performing the
1322:{\displaystyle +=}
1319:
1181:
1114:
914:). In particular,
857:
782:
692:
622:
505:
464:
408:
376:
341:
296:
207:
120:In the context of
41:finite-dimensional
4762:, Amer. J. Math.
4346:
4332:is defined to be
4330:Elliott invariant
4211:. The scale is Γ(
3943:
3887:
3744:for some element
3507:
3453:
3251:
2596:Theorem (Elliott)
2315:Elliott's theorem
2167:)) is called the
1660:is a (covariant)
1107:
897:takes the form ∪
889:), then an ideal
873:If an AF algebra
777:
664:
615:
455:
49:George A. Elliott
4856:
4835:
4719:
4716:
4710:
4706:
4643:
4641:
4640:
4635:
4624:
4623:
4607:
4605:
4604:
4599:
4588:
4587:
4571:
4569:
4568:
4563:
4561:
4560:
4544:
4542:
4541:
4536:
4525:
4524:
4505:
4503:
4502:
4497:
4489:
4488:
4467:
4466:
4445:
4444:
4414:
4413:
4398:
4397:
4376:
4375:
4348:
4344:
4254:dyadic rationals
4233:
4231:
4230:
4227:
4224:
4202:
4200:
4199:
4196:
4193:
4094:
4092:
4091:
4086:
4078:
4077:
4053:
4052:
4034:
4033:
4024:
4023:
4001:
3999:
3998:
3993:
3984:
3979:
3967:
3966:
3951:
3921:
3919:
3918:
3913:
3908:
3907:
3895:
3719:
3717:
3716:
3711:
3679:
3678:
3669:
3663:
3658:
3634:
3633:
3605:
3603:
3602:
3597:
3588:
3587:
3586:
3576:
3571:
3562:
3561:
3560:
3559:
3549:
3533:
3528:
3509:
3505:
3494:
3489:
3477:
3476:
3461:
3445:
3444:
3327:
3325:
3324:
3319:
3310:
3309:
3308:
3298:
3293:
3284:
3283:
3282:
3281:
3271:
3259:
3243:
3242:
2982:. One says that
2598:Two AF algebras
2416:on the level of
2145:
2143:
2142:
2137:
2126:
2125:
2100:
2085:
2084:
2017:
2015:
2014:
2009:
1989:
1988:
1976:
1975:
1965:
1915:), denoted by Γ(
1900:, we define the
1631:
1629:
1628:
1623:
1614:
1609:
1604:
1595:
1594:
1589:
1574:
1573:
1558:
1557:
1536:
1535:
1510:
1508:
1507:
1502:
1497:
1496:
1495:
1494:
1479:
1474:
1328:
1326:
1325:
1320:
1246:partial isometry
1206:). Two elements
1190:
1188:
1187:
1182:
1162:
1161:
1134:
1133:
1115:
1093:
1092:
966:approximate unit
964:in fact form an
922:and some subset
866:
864:
863:
858:
798:Bratteli diagram
791:
789:
788:
783:
778:
773:
772:
771:
762:
761:
751:
701:
699:
698:
693:
682:
681:
666:
665:
663:
662:
661:
651:
646:
642:
641:
623:
590:A C*-algebra is
529:Bratteli diagram
514:
512:
511:
506:
500:
499:
487:
486:
477:
476:
463:
417:
415:
414:
409:
407:
406:
405:
404:
389:
384:
372:
371:
370:
369:
354:
349:
305:
303:
302:
297:
292:
291:
216:
214:
213:
208:
203:
202:
201:
200:
186:
185:
4864:
4863:
4859:
4858:
4857:
4855:
4854:
4853:
4839:
4838:
4820:
4817:
4728:
4723:
4722:
4717:
4713:
4707:
4703:
4698:
4687:
4679:
4660:
4619:
4615:
4613:
4610:
4609:
4583:
4579:
4577:
4574:
4573:
4556:
4552:
4550:
4547:
4546:
4520:
4516:
4514:
4511:
4510:
4484:
4480:
4462:
4458:
4440:
4436:
4409:
4405:
4393:
4389:
4371:
4367:
4342:
4340:
4337:
4336:
4322:
4280:
4262:
4247:
4228:
4225:
4222:
4221:
4219:
4217:
4197:
4194:
4189:
4188:
4186:
4184:
4177:
4170:
4156:By definition,
4154:
4146:
4140:
4132:
4121:
4114:
4073:
4069:
4048:
4044:
4029:
4025:
4019:
4015:
4013:
4010:
4009:
3980:
3975:
3962:
3958:
3942:
3940:
3937:
3936:
3931:
3903:
3899:
3886:
3878:
3875:
3874:
3869:
3865:
3858:
3851:
3844:
3837:
3830:
3823:
3816:
3810:, ). Next move
3809:
3803:
3796:
3789:
3778:
3771:
3765:. Because each
3764:
3757:
3750:
3742:
3736:
3729:
3674:
3670:
3665:
3659:
3654:
3629:
3625:
3617:
3614:
3613:
3582:
3578:
3577:
3572:
3567:
3555:
3551:
3550:
3545:
3544:
3529:
3524:
3503:
3490:
3485:
3472:
3468:
3452:
3440:
3436:
3425:
3422:
3421:
3415:
3397:
3371:
3364:
3337:
3304:
3300:
3299:
3294:
3289:
3277:
3273:
3272:
3267:
3266:
3250:
3238:
3234:
3223:
3220:
3219:
3176:
3124:
3117:
3106:
3099:
3065:
3058:
3051:
3040:
3033:
3022:
2960:
2953:
2941:
2934:
2923:
2906:are isomorphic.
2888:
2869:
2848:
2841:
2831:
2824:
2811:
2804:
2794:
2787:
2780:
2773:
2762:
2755:
2748:
2741:
2722:
2711:
2696:
2685:
2649:
2638:
2623:
2612:
2590:
2584:
2577:
2572:
2562:
2555:
2544:
2522:
2517:
2510:
2492:
2481:
2465:
2458:
2451:
2446:
2422:
2396:
2373:
2362:
2317:
2308:
2297:
2290:
2285:
2268:
2237:
2214:
2184:
2169:dimension group
2162:
2155:
2121:
2117:
2096:
2080:
2076:
2050:
2047:
2046:
2042:) and in fact,
2037:
2030:
1984:
1980:
1971:
1967:
1961:
1931:
1928:
1927:
1910:
1888:
1881:
1874:
1867:
1860:
1853:
1847:. For example,
1842:
1831:
1820:
1813:
1805:
1797:
1788:
1781:
1772:
1762:
1754:
1747:
1742:
1732:
1719:
1700:
1689:
1682:
1659:
1645:
1610:
1605:
1600:
1590:
1585:
1584:
1569:
1565:
1553:
1549:
1531:
1527:
1522:
1519:
1518:
1490:
1486:
1485:
1481:
1475:
1464:
1452:
1449:
1448:
1435:
1420:
1401:
1388:
1373:
1278:
1275:
1274:
1257:
1214:are said to be
1151:
1147:
1129:
1125:
1106:
1088:
1084:
1082:
1079:
1078:
1069:
1055:
1030:
1017:
1001:
990:
988:
982:
962:
957:
912:
902:
887:
882:
828:
825:
824:
814:Pascal triangle
810:
767:
763:
757:
753:
752:
750:
742:
739:
738:
728:
722:
713:
671:
667:
657:
653:
652:
647:
645:
644:
637:
633:
614:
606:
603:
602:
588:
568:
561:
554:
547:
540:
526:
495:
491:
482:
478:
469:
465:
459:
453:
450:
449:
441:
400:
396:
395:
391:
385:
380:
365:
361:
360:
356:
350:
345:
333:
330:
329:
287:
283:
260:
257:
256:
250:
243:
225:
196:
192:
191:
187:
181:
177:
175:
172:
171:
161:
156:
134:
85:
57:
33:inductive limit
17:
12:
11:
5:
4862:
4852:
4851:
4837:
4836:
4816:
4815:External links
4813:
4812:
4811:
4800:
4793:
4780:
4767:
4753:
4742:
4727:
4724:
4721:
4720:
4711:
4700:
4699:
4697:
4694:
4685:
4677:
4659:
4656:
4633:
4630:
4627:
4622:
4618:
4597:
4594:
4591:
4586:
4582:
4559:
4555:
4534:
4531:
4528:
4523:
4519:
4507:
4506:
4495:
4492:
4487:
4483:
4479:
4476:
4473:
4470:
4465:
4461:
4457:
4454:
4451:
4448:
4443:
4439:
4435:
4432:
4429:
4426:
4423:
4420:
4417:
4412:
4408:
4404:
4401:
4396:
4392:
4388:
4385:
4382:
4379:
4374:
4370:
4366:
4363:
4360:
4357:
4354:
4351:
4321:
4318:
4278:
4260:
4245:
4234:, 1}. For the
4215:
4182:
4175:
4168:
4153:
4150:
4144:
4138:
4130:
4119:
4112:
4096:
4095:
4084:
4081:
4076:
4072:
4068:
4065:
4062:
4059:
4056:
4051:
4047:
4043:
4040:
4037:
4032:
4028:
4022:
4018:
4003:
4002:
3991:
3988:
3983:
3978:
3974:
3970:
3965:
3961:
3957:
3954:
3949:
3946:
3929:
3923:
3922:
3911:
3906:
3902:
3898:
3893:
3890:
3885:
3882:
3867:
3863:
3856:
3849:
3842:
3835:
3828:
3821:
3814:
3807:
3801:
3794:
3787:
3776:
3769:
3762:
3755:
3748:
3740:
3734:
3727:
3721:
3720:
3709:
3706:
3703:
3700:
3697:
3694:
3691:
3688:
3685:
3682:
3677:
3673:
3668:
3662:
3657:
3653:
3649:
3646:
3643:
3640:
3637:
3632:
3628:
3624:
3621:
3607:
3606:
3595:
3592:
3585:
3581:
3575:
3570:
3565:
3558:
3554:
3548:
3543:
3540:
3537:
3532:
3527:
3523:
3519:
3516:
3513:
3501:
3498:
3493:
3488:
3484:
3480:
3475:
3471:
3467:
3464:
3459:
3456:
3451:
3448:
3443:
3439:
3435:
3432:
3429:
3413:
3395:
3369:
3362:
3336:
3333:
3329:
3328:
3317:
3314:
3307:
3303:
3297:
3292:
3287:
3280:
3276:
3270:
3265:
3262:
3257:
3254:
3249:
3246:
3241:
3237:
3233:
3230:
3227:
3174:
3138:Shen criterion
3122:
3115:
3104:
3097:
3063:
3056:
3049:
3042:≥ 0 such that
3038:
3031:
3020:
2959:
2956:
2951:
2940:
2937:
2932:
2921:
2908:
2907:
2886:
2867:
2850:
2846:
2839:
2833:
2829:
2822:
2816:
2813:
2809:
2802:
2796:
2792:
2785:
2778:
2771:
2760:
2753:
2746:
2739:
2720:
2709:
2694:
2683:
2647:
2636:
2621:
2610:
2588:
2582:
2575:
2568:
2560:
2553:
2542:
2520:
2515:
2508:
2490:
2479:
2463:
2456:
2449:
2442:
2420:
2394:
2371:
2360:
2316:
2313:
2312:
2311:
2306:
2295:
2288:
2281:
2274:
2266:
2235:
2212:
2182:
2160:
2153:
2147:
2146:
2135:
2132:
2129:
2124:
2120:
2116:
2113:
2110:
2107:
2104:
2099:
2094:
2091:
2088:
2083:
2079:
2075:
2072:
2069:
2066:
2063:
2060:
2057:
2054:
2035:
2026:
2019:
2018:
2007:
2004:
2001:
1998:
1995:
1992:
1987:
1983:
1979:
1974:
1970:
1964:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1908:
1886:
1879:
1872:
1865:
1858:
1851:
1840:
1829:
1818:
1811:
1804:
1801:
1800:
1799:
1793:
1786:
1776:
1768:
1760:
1752:
1745:
1738:
1730:
1725:
1717:
1698:
1687:
1680:
1657:
1643:
1633:
1632:
1621:
1618:
1613:
1608:
1603:
1598:
1593:
1588:
1583:
1580:
1577:
1572:
1568:
1564:
1561:
1556:
1552:
1548:
1545:
1542:
1539:
1534:
1530:
1526:
1512:
1511:
1500:
1493:
1489:
1484:
1478:
1473:
1470:
1467:
1463:
1459:
1456:
1433:
1418:
1399:
1386:
1371:
1330:
1329:
1318:
1315:
1312:
1309:
1306:
1303:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1255:
1192:
1191:
1180:
1177:
1174:
1171:
1168:
1165:
1160:
1157:
1154:
1150:
1146:
1143:
1140:
1137:
1132:
1128:
1124:
1121:
1118:
1113:
1110:
1105:
1102:
1099:
1096:
1091:
1087:
1064:
1051:
1026:
1015:
999:
989:
986:
983:
981:
980:Classification
978:
960:
953:
910:
898:
885:
878:
871:
870:
867:
856:
853:
850:
847:
844:
841:
838:
835:
832:
808:
793:
792:
781:
776:
770:
766:
760:
756:
749:
746:
726:
718:
709:
703:
702:
691:
688:
685:
680:
677:
674:
670:
660:
656:
650:
640:
636:
632:
629:
626:
621:
618:
613:
610:
587:
584:
566:
559:
552:
545:
538:
533:directed graph
522:
516:
515:
503:
498:
494:
490:
485:
481:
475:
472:
468:
462:
458:
437:
419:
418:
403:
399:
394:
388:
383:
379:
375:
368:
364:
359:
353:
348:
344:
340:
337:
307:
306:
295:
290:
286:
282:
279:
276:
273:
270:
267:
264:
248:
241:
223:
218:
217:
206:
199:
195:
190:
184:
180:
160:
157:
155:
152:
132:
83:
55:
15:
9:
6:
4:
3:
2:
4861:
4850:
4847:
4846:
4844:
4833:
4829:
4828:
4823:
4819:
4818:
4809:
4805:
4801:
4798:
4794:
4791:
4787:
4786:
4781:
4778:
4775:, J. Algebra
4774:
4773:
4768:
4765:
4761:
4757:
4754:
4751:
4747:
4743:
4740:
4736:
4735:
4730:
4729:
4715:
4705:
4701:
4693:
4691:
4683:
4675:
4671:
4667:
4666:
4655:
4652:
4650:
4647:The original
4645:
4628:
4620:
4616:
4592:
4584:
4580:
4557:
4553:
4529:
4521:
4517:
4493:
4485:
4481:
4477:
4471:
4463:
4459:
4455:
4449:
4441:
4437:
4433:
4424:
4415:
4410:
4402:
4394:
4390:
4386:
4380:
4372:
4368:
4358:
4352:
4335:
4334:
4333:
4331:
4327:
4317:
4315:
4312:
4308:
4304:
4300:
4296:
4292:
4287:
4285:
4281:
4274:
4270:
4266:
4259:
4255:
4251:
4244:
4240:
4237:
4214:
4210:
4206:
4192:
4181:
4174:
4167:
4163:
4159:
4149:
4147:
4137:
4133:
4126:
4122:
4115:
4108:
4104:
4099:
4082:
4074:
4070:
4066:
4063:
4057:
4049:
4045:
4041:
4038:
4030:
4026:
4020:
4016:
4008:
4007:
4006:
3989:
3981:
3976:
3972:
3968:
3963:
3959:
3952:
3947:
3944:
3935:
3934:
3933:
3928:
3909:
3904:
3900:
3896:
3891:
3888:
3883:
3880:
3873:
3872:
3871:
3862:
3855:
3848:
3841:
3834:
3827:
3820:
3813:
3806:
3800:
3793:
3786:
3782:
3775:
3768:
3761:
3758:generated by
3754:
3747:
3743:
3733:
3726:
3707:
3698:
3689:
3683:
3675:
3671:
3660:
3655:
3651:
3647:
3644:
3638:
3630:
3626:
3612:
3611:
3610:
3593:
3583:
3579:
3573:
3563:
3556:
3552:
3538:
3530:
3525:
3521:
3517:
3514:
3499:
3491:
3486:
3482:
3478:
3473:
3469:
3462:
3457:
3454:
3449:
3441:
3437:
3433:
3430:
3420:
3419:
3418:
3416:
3409:
3404:
3402:
3398:
3391:
3387:
3383:
3379:
3375:
3368:
3361:
3357:
3353:
3349:
3345:
3341:
3332:
3315:
3305:
3301:
3295:
3285:
3278:
3274:
3260:
3255:
3252:
3247:
3239:
3235:
3231:
3228:
3218:
3217:
3216:
3214:
3210:
3206:
3202:
3200:
3196:
3192:
3188:
3184:
3180:
3173:
3169:
3165:
3161:
3157:
3153:
3145:
3141:
3139:
3135:
3130:
3128:
3121:
3114:
3110:
3103:
3096:
3092:
3087:
3085:
3081:
3077:
3072:
3070:
3066:
3059:
3052:
3045:
3041:
3034:
3027:
3023:
3016:
3012:
3007:
3005:
3001:
2997:
2993:
2989:
2985:
2981:
2977:
2976:positive cone
2973:
2969:
2968:ordered group
2965:
2955:
2950:
2946:
2936:
2931:
2927:
2920:
2912:
2905:
2901:
2897:
2893:
2889:
2882:
2878:
2874:
2870:
2863:
2859:
2855:
2851:
2845:
2838:
2834:
2828:
2821:
2817:
2814:
2808:
2801:
2797:
2791:
2784:
2777:
2770:
2766:
2759:
2752:
2745:
2738:
2734:
2733:
2732:
2730:
2726:
2719:
2715:
2708:
2704:
2700:
2693:
2689:
2682:
2677:
2674:
2672:
2668:
2664:
2659:
2657:
2653:
2646:
2642:
2635:
2631:
2627:
2620:
2616:
2609:
2605:
2601:
2597:
2593:
2591:
2581:
2578:
2571:
2566:
2559:
2552:
2548:
2541:
2536:
2534:
2530:
2526:
2519:
2511:
2504:
2500:
2496:
2489:
2485:
2478:
2474:
2470:
2466:
2459:
2452:
2445:
2440:
2436:
2432:
2428:
2424:
2419:
2415:
2412:is AF. A map
2411:
2406:
2404:
2400:
2393:
2389:
2385:
2381:
2377:
2370:
2366:
2359:
2355:
2351:
2347:
2343:
2339:
2336:
2334:
2330:
2321:
2309:
2302:
2298:
2291:
2284:
2279:
2275:
2272:
2265:
2261:
2257:
2253:
2249:
2245:
2241:
2234:
2230:
2226:
2222:
2218:
2217:
2216:
2211:
2206:
2204:
2199:
2197:
2194:, {1, 2,...,
2193:
2189:
2185:
2178:
2174:
2170:
2166:
2159:
2152:
2133:
2122:
2118:
2111:
2108:
2105:
2102:
2089:
2081:
2077:
2073:
2070:
2064:
2058:
2045:
2044:
2043:
2041:
2034:
2029:
2024:
2005:
1999:
1996:
1993:
1990:
1985:
1981:
1977:
1972:
1968:
1954:
1945:
1939:
1926:
1925:
1924:
1922:
1918:
1914:
1907:
1903:
1899:
1894:
1892:
1885:
1878:
1871:
1864:
1857:
1850:
1846:
1839:
1835:
1828:
1824:
1817:
1810:
1796:
1792:
1785:
1779:
1775:
1771:
1766:
1759:
1755:
1748:
1741:
1736:
1729:
1726:
1723:
1716:
1712:
1708:
1704:
1697:
1693:
1686:
1679:
1675:
1671:
1667:
1663:
1656:
1653:
1652:
1651:
1649:
1642:
1638:
1619:
1611:
1606:
1596:
1591:
1578:
1570:
1562:
1554:
1550:
1546:
1540:
1532:
1528:
1517:
1516:
1515:
1498:
1491:
1487:
1482:
1476:
1471:
1468:
1465:
1461:
1457:
1454:
1447:
1446:
1445:
1442:
1440:
1436:
1429:
1428:ordered group
1425:
1421:
1415:. This makes
1414:
1410:
1406:
1402:
1395:
1393:
1389:
1382:
1378:
1374:
1367:
1364:that has the
1363:
1359:
1355:
1351:
1347:
1343:
1339:
1335:
1313:
1310:
1307:
1301:
1295:
1289:
1283:
1273:
1272:
1271:
1269:
1265:
1261:
1254:
1250:
1247:
1243:
1239:
1235:
1231:
1227:
1223:
1220:, denoted by
1219:
1218:
1213:
1209:
1205:
1201:
1197:
1178:
1175:
1166:
1158:
1155:
1152:
1148:
1138:
1130:
1126:
1119:
1116:
1111:
1108:
1103:
1097:
1085:
1077:
1076:
1075:
1073:
1067:
1063:
1059:
1054:
1050:
1046:
1042:
1038:
1034:
1029:
1025:
1021:
1014:
1010:
1006:
1002:
995:
977:
973:
971:
967:
963:
956:
950:
948:
944:
940:
935:
933:
929:
925:
921:
917:
913:
906:
901:
896:
892:
888:
881:
876:
868:
854:
848:
842:
836:
830:
823:
822:
821:
819:
815:
811:
804:
800:
799:
779:
768:
764:
758:
754:
747:
744:
737:
736:
735:
733:
729:
721:
717:
712:
708:
689:
686:
678:
675:
672:
668:
658:
654:
638:
634:
627:
624:
619:
616:
611:
608:
601:
600:
599:
597:
594:if it is the
593:
583:
581:
576:
573:Consider the
571:
569:
562:
555:
548:
541:
534:
530:
525:
521:
501:
496:
492:
488:
483:
479:
473:
470:
466:
460:
456:
448:
447:
446:
445:
440:
436:
432:
428:
424:
401:
397:
392:
386:
381:
377:
366:
362:
357:
351:
346:
342:
338:
328:
327:
326:
324:
321:. The number
320:
316:
312:
293:
288:
284:
280:
277:
274:
268:
255:
254:
253:
251:
244:
236:
234:
230:
226:
204:
197:
193:
188:
182:
178:
170:
169:
168:
166:
151:
149:
146:
142:
138:
131:
127:
123:
118:
116:
112:
108:
103:
101:
97:
93:
89:
82:
78:
75:
72:
68:
63:
61:
54:
50:
46:
42:
38:
34:
30:
26:
22:
4825:
4822:"AF-algebra"
4807:
4803:
4796:
4789:
4783:
4776:
4770:
4763:
4759:
4756:Effros, E.G.
4749:
4745:
4738:
4732:
4714:
4704:
4669:
4663:
4661:
4653:
4646:
4508:
4329:
4325:
4323:
4313:
4306:
4302:
4288:
4283:
4276:
4272:
4264:
4257:
4249:
4242:
4238:
4212:
4208:
4204:
4190:
4179:
4172:
4165:
4161:
4155:
4142:
4135:
4128:
4127:) such that
4124:
4117:
4110:
4106:
4103:directed set
4100:
4097:
4004:
3926:
3924:
3860:
3853:
3846:
3839:
3832:
3825:
3818:
3811:
3804:
3798:
3791:
3784:
3780:
3773:
3766:
3759:
3752:
3745:
3738:
3731:
3724:
3722:
3608:
3411:
3407:
3405:
3400:
3393:
3389:
3385:
3381:
3377:
3373:
3366:
3359:
3355:
3351:
3347:
3343:
3339:
3338:
3330:
3212:
3208:
3204:
3203:
3198:
3194:
3190:
3186:
3182:
3178:
3171:
3167:
3163:
3159:
3155:
3151:
3150:
3137:
3133:
3131:
3126:
3119:
3112:
3108:
3101:
3094:
3090:
3088:
3083:
3079:
3075:
3073:
3068:
3061:
3054:
3047:
3043:
3036:
3029:
3025:
3018:
3014:
3010:
3008:
3003:
2999:
2995:
2991:
2987:
2983:
2979:
2975:
2971:
2963:
2961:
2958:Riesz groups
2948:
2942:
2929:
2925:
2918:
2916:
2903:
2899:
2895:
2891:
2884:
2880:
2876:
2872:
2865:
2861:
2857:
2853:
2843:
2836:
2826:
2819:
2806:
2799:
2789:
2782:
2775:
2768:
2764:
2757:
2750:
2743:
2736:
2728:
2724:
2717:
2713:
2706:
2702:
2698:
2691:
2687:
2680:
2678:
2675:
2670:
2666:
2662:
2660:
2655:
2651:
2644:
2640:
2633:
2629:
2625:
2618:
2614:
2607:
2603:
2599:
2595:
2594:
2586:
2579:
2573:
2569:
2564:
2557:
2550:
2546:
2539:
2537:
2532:
2528:
2524:
2513:
2506:
2502:
2498:
2494:
2487:
2483:
2476:
2472:
2468:
2461:
2454:
2447:
2443:
2438:
2434:
2430:
2426:
2425:
2417:
2413:
2409:
2407:
2402:
2398:
2391:
2387:
2383:
2379:
2375:
2368:
2364:
2357:
2353:
2349:
2345:
2341:
2340:
2337:
2332:
2328:
2326:
2304:
2300:
2293:
2286:
2282:
2277:
2270:
2263:
2259:
2255:
2251:
2247:
2243:
2239:
2232:
2228:
2224:
2220:
2209:
2207:
2202:
2200:
2195:
2191:
2187:
2180:
2176:
2172:
2168:
2164:
2157:
2150:
2149:The triple (
2148:
2039:
2032:
2027:
2022:
2020:
1920:
1916:
1912:
1905:
1901:
1897:
1895:
1890:
1883:
1876:
1869:
1862:
1855:
1848:
1844:
1837:
1833:
1826:
1822:
1815:
1808:
1806:
1794:
1790:
1783:
1777:
1773:
1769:
1764:
1757:
1750:
1743:
1739:
1734:
1727:
1721:
1714:
1710:
1706:
1702:
1695:
1691:
1684:
1677:
1673:
1669:
1665:
1654:
1647:
1640:
1636:
1634:
1513:
1443:
1438:
1431:
1423:
1416:
1412:
1408:
1404:
1397:
1396:
1391:
1384:
1376:
1369:
1357:
1353:
1349:
1345:
1341:
1337:
1331:
1267:
1263:
1259:
1252:
1248:
1241:
1237:
1233:
1229:
1225:
1221:
1215:
1211:
1207:
1203:
1199:
1193:
1071:
1065:
1061:
1057:
1052:
1048:
1044:
1040:
1036:
1032:
1027:
1023:
1019:
1012:
1008:
997:
991:
974:
969:
958:
954:
951:
946:
942:
938:
936:
931:
927:
923:
919:
915:
908:
904:
899:
894:
890:
883:
879:
874:
872:
806:
802:
796:
794:
731:
724:
719:
715:
710:
706:
704:
596:direct limit
591:
589:
579:
572:
564:
557:
550:
543:
536:
523:
519:
517:
443:
438:
434:
430:
426:
422:
420:
322:
318:
314:
310:
308:
246:
239:
237:
232:
228:
221:
219:
164:
162:
140:
136:
129:
119:
104:
99:
95:
91:
80:
64:
52:
45:Ola Bratteli
31:that is the
24:
18:
4849:C*-algebras
4670:hyperfinite
4256:with scale
4236:CAR algebra
4005:with scale
3358:such that (
3335:The theorem
3076:Riesz group
2945:Riesz group
2203:contractive
1196:projections
1194:Denote the
994:K-theoretic
818:CAR algebra
705:where each
586:AF algebras
21:mathematics
4792:, 229-245.
4766:, 385-402.
4741:, 195-234.
4726:References
4649:conjecture
4311:Cantor set
3859:such that
3417:. Suppose
2970:. The set
2512:such that
2390:such that
2262:such that
429:matrix of
235:matrices.
148:metrizable
96:uniqueness
29:C*-algebra
16:C*-algebra
4832:EMS Press
4554:ρ
4482:ρ
4419:Γ
4309:) on the
4282:for some
4027:ϕ
4017:∪
3953:
3948:→
3897:
3892:→
3838:). Again
3693:Γ
3690:∈
3672:ϕ
3648:∈
3620:Γ
3463:
3458:→
3261:
3256:→
3205:Corollary
3067:for each
2112:≤
2106:≤
2074:∈
2053:Γ
1997:∈
1973:∗
1934:Γ
1462:⊕
1362:semigroup
1311:⊕
1244:for some
1176:⋯
1173:→
1145:→
1123:→
1120:⋯
1117:
1112:→
1090:∞
855:…
852:⇉
846:⇉
840:⇉
834:⇉
775:¯
755:∪
687:⋯
684:→
655:α
649:→
631:→
628:⋯
625:
620:→
457:∑
378:⊕
374:→
343:⊕
336:Φ
281:⊗
263:Φ
179:⊕
139:), where
92:existence
71:separable
47:. Later,
4843:Category
4779:, 29-44.
4688:factor.
4680:factor.
4295:spectrum
4218:) = {0,
4152:Examples
3197:) = ker(
3024:≥ 0 and
2998:implies
2962:A group
2742: :
2679:Let Φ: (
2632:)) and (
2254: :
2223: :
1756:), then
1683: :
1668: :
1514:one has
575:category
126:topology
115:Haagerup
77:amenable
37:sequence
4834:, 2001
4291:Gelfand
4232:
4220:
4201:
4187:
4105:. If Γ(
3399:, then
3340:Theorem
2871:}. Let
2453:). Let
1662:functor
1650:) are:
1270:)/~ by
150:space.
88:functor
4690:Powers
4682:Connes
4509:where
4328:, the
4269:simple
3925:whose
3723:where
3376:)) = (
3053:, and
2705:)) → (
2527:, and
2401:, and
2031:, the
1807:Since
1360:)/~ a
996:group
309:where
220:where
111:Connes
74:simple
4696:Notes
4668:, or
4109:) = {
3506:where
3177:) → (
3154:Let (
3152:Lemma
3013:: if
2935:= Φ.
2756:) →
2727:), Γ(
2701:), Γ(
2654:), Γ(
2628:), Γ(
2567:) = ∪
2467:into
2427:Lemma
2342:Lemma
2175:. If
2021:When
1902:scale
1426:) an
1228:, if
143:is a
35:of a
27:is a
23:, an
4709:1982
4608:and
4203:for
4141:...
4123:∈ Γ(
3866:= φ
3737:) =
3384:, Γ(
3372:, Γ(
3342:If (
3189:and
3046:= Σ
3028:≤ Σ
2902:and
2894:and
2669:and
2602:and
2486:) →
2441:= (∪
2437:AF,
2429:Let
2367:) →
2348:and
2331:and
2242:and
2198:}).
2163:, Γ(
1889:) =
1875:) =
1854:and
1709:and
1694:) →
1348:and
1340:and
1236:and
1210:and
992:The
877:= (∪
795:The
542:and
124:and
113:and
65:The
4808:126
4764:102
4739:171
4345:Ell
4297:is
4207:in
3945:lim
3932:is
3889:lim
3864:12*
3455:lim
3253:lim
3201:).
3166:: (
2978:of
2716:),
2690:),
2643:),
2617:),
2592:).
2280:= ∪
2276:If
2171:of
1904:of
1836:),
1798:)).
1737:= ∪
1441:).
1394:).
1251:in
1242:v*v
1234:vv*
1109:lim
1068:+ 1
968:of
893:in
801:of
617:lim
556:to
524:l k
439:l k
39:of
19:In
4845::
4830:,
4824:,
4790:45
4777:38
4672:,
4644:.
4286:.
4241:,
4171:,
4134:≥
3868:12
3857:12
3829:12
3824:=
3380:,
3365:,
3346:,
3211:,
3181:,
3170:,
3158:,
3129:.
3118:,
3100:,
3082:,
3074:A
3071:.
3060:≤
3017:,
3006:.
3002:∈
2994:∈
2879:→
2875::
2860:→
2856::
2803:1*
2788:→
2781::
2740:1*
2576:n*
2535:.
2523:=
2516:m*
2505:→
2501::
2475::
2397:=
2386:→
2382::
2356::
2269:=
2258:→
2227:→
2215::
2190:,
2179:=
2156:,
1923::
1893:.
1672:→
1639:↦
1240:=
1232:=
1224:~
1047:.
1039:×
1011:,
972:.
907:∩
592:AF
570:.
567:lk
425:×
317:=
245:→
231:×
117:.
4750:6
4686:∞
4678:1
4632:)
4629:A
4626:(
4621:0
4617:K
4596:)
4593:A
4590:(
4585:+
4581:T
4558:A
4533:)
4530:A
4527:(
4522:+
4518:T
4494:,
4491:)
4486:A
4478:,
4475:)
4472:A
4469:(
4464:+
4460:T
4456:,
4453:)
4450:A
4447:(
4442:1
4438:K
4434:,
4431:)
4428:)
4425:A
4422:(
4416:,
4411:+
4407:)
4403:A
4400:(
4395:0
4391:K
4387:,
4384:)
4381:A
4378:(
4373:0
4369:K
4365:(
4362:(
4359:=
4356:)
4353:A
4350:(
4326:A
4314:X
4307:X
4305:(
4303:C
4284:k
4279:k
4277:M
4273:Q
4265:A
4263:(
4261:0
4258:K
4250:A
4248:(
4246:0
4243:K
4239:A
4229:2
4226:/
4223:1
4216:2
4213:M
4209:Z
4205:a
4198:2
4195:/
4191:a
4183:2
4180:M
4178:(
4176:0
4173:K
4169:2
4166:M
4162:Q
4145:k
4143:v
4139:1
4136:v
4131:k
4129:u
4125:G
4120:k
4118:u
4113:k
4111:v
4107:G
4083:.
4080:]
4075:G
4071:u
4067:,
4064:0
4061:[
4058:=
4055:]
4050:k
4046:u
4042:,
4039:0
4036:[
4031:k
4021:k
3990:,
3987:)
3982:+
3977:k
3973:G
3969:,
3964:k
3960:G
3956:(
3930:0
3927:K
3910:,
3905:k
3901:A
3884:=
3881:A
3861:α
3854:α
3850:2
3847:A
3843:2
3840:u
3836:1
3833:u
3831:(
3826:φ
3822:2
3819:u
3815:1
3812:u
3808:1
3805:G
3802:1
3799:G
3795:1
3792:A
3788:1
3785:H
3781:Z
3777:1
3774:G
3770:1
3767:H
3763:1
3760:u
3756:1
3753:G
3749:1
3746:u
3741:G
3739:u
3735:1
3732:u
3730:(
3728:1
3725:φ
3708:,
3705:}
3702:)
3699:G
3696:(
3687:)
3684:v
3681:(
3676:1
3667:|
3661:+
3656:1
3652:H
3645:v
3642:{
3639:=
3636:)
3631:1
3627:H
3623:(
3594:.
3591:)
3584:k
3580:n
3574:+
3569:Z
3564:,
3557:k
3553:n
3547:Z
3542:(
3539:=
3536:)
3531:+
3526:k
3522:H
3518:,
3515:H
3512:(
3500:,
3497:)
3492:+
3487:k
3483:H
3479:,
3474:k
3470:H
3466:(
3450:=
3447:)
3442:+
3438:G
3434:,
3431:G
3428:(
3414:G
3412:u
3408:G
3401:A
3396:G
3394:u
3390:G
3386:G
3382:G
3378:G
3374:A
3370:0
3367:K
3363:0
3360:K
3356:A
3352:G
3348:G
3344:G
3316:,
3313:)
3306:k
3302:n
3296:+
3291:Z
3286:,
3279:k
3275:n
3269:Z
3264:(
3248:=
3245:)
3240:+
3236:G
3232:,
3229:G
3226:(
3213:G
3209:G
3199:ϕ
3195:σ
3191:ψ
3187:σ
3183:G
3179:G
3175:+
3172:Z
3168:Z
3164:ϕ
3160:G
3156:G
3134:Z
3127:A
3123:0
3120:K
3116:0
3113:K
3109:Z
3105:0
3102:K
3098:0
3095:K
3091:A
3084:G
3080:G
3078:(
3069:i
3064:i
3062:y
3057:i
3055:x
3050:i
3048:x
3044:x
3039:i
3037:x
3032:i
3030:y
3026:x
3021:i
3019:y
3015:x
3004:G
3000:g
2996:G
2992:g
2990:·
2988:k
2984:G
2980:G
2972:G
2964:G
2952:0
2949:K
2933:*
2930:φ
2926:k
2922:0
2919:K
2904:B
2900:A
2896:ψ
2892:φ
2887:n
2885:ψ
2881:A
2877:B
2873:ψ
2868:n
2866:φ
2862:B
2858:A
2854:φ
2847:1
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2840:0
2837:K
2832:.
2830:0
2827:K
2823:2
2820:A
2812:.
2810:2
2807:A
2800:β
2793:1
2790:B
2786:1
2783:A
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2776:φ
2772:1
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2765:B
2763:(
2761:0
2758:K
2754:1
2751:A
2749:(
2747:0
2744:K
2737:α
2729:B
2725:B
2723:(
2721:0
2718:K
2714:B
2712:(
2710:0
2707:K
2703:A
2699:A
2697:(
2695:0
2692:K
2688:A
2686:(
2684:0
2681:K
2671:B
2667:A
2656:B
2652:B
2650:(
2648:0
2645:K
2641:B
2639:(
2637:0
2634:K
2630:A
2626:A
2624:(
2622:0
2619:K
2615:A
2613:(
2611:0
2608:K
2604:B
2600:A
2589:n
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2583:0
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2574:β
2570:n
2565:B
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2561:0
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2507:B
2503:A
2499:φ
2495:B
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2491:0
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2462:B
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2361:0
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2354:ψ
2350:B
2346:A
2333:B
2329:A
2310:.
2307:n
2305:A
2301:A
2296:n
2294:A
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2289:n
2287:α
2283:n
2278:A
2273:.
2271:ψ
2267:*
2264:α
2260:B
2256:A
2252:α
2248:ψ
2244:B
2240:A
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2229:B
2225:A
2221:α
2213:0
2210:K
2196:s
2192:Z
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2173:A
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2161:0
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2154:0
2151:K
2134:.
2131:}
2128:]
2123:A
2119:1
2115:[
2109:x
2103:0
2098:|
2093:)
2090:A
2087:(
2082:0
2078:K
2071:x
2068:{
2065:=
2062:)
2059:A
2056:(
2040:A
2036:0
2033:K
2028:A
2023:A
2006:.
2003:}
2000:A
1994:p
1991:=
1986:2
1982:p
1978:=
1969:p
1963:|
1958:]
1955:p
1952:[
1949:{
1946:=
1943:)
1940:A
1937:(
1921:A
1917:A
1913:A
1911:(
1909:0
1906:K
1898:A
1891:Z
1887:4
1884:M
1882:(
1880:0
1877:K
1873:2
1870:M
1868:(
1866:0
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1859:4
1856:M
1852:2
1849:M
1841:0
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1834:A
1832:(
1830:∞
1827:M
1823:A
1821:(
1819:∞
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1812:∞
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1787:0
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1780:*
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1718:*
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1703:B
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1699:0
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1602:Z
1597:,
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1582:(
1579:=
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1563:A
1560:(
1555:0
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1469:=
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1458:=
1455:A
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1434:0
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1424:A
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1413:q
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1403:(
1400:0
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1354:P
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1305:[
1302:=
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1296:q
1293:[
1290:+
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1268:A
1266:(
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1256:∞
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1131:n
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1104:=
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1098:A
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1086:M
1072:A
1070:(
1066:n
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1053:n
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1028:n
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1016:0
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1009:A
1000:0
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961:n
959:A
955:n
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939:A
932:A
928:S
924:S
920:A
916:J
911:n
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807:α
803:A
780:.
769:n
765:A
759:n
748:=
745:A
732:A
727:i
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720:i
716:α
711:i
707:A
690:,
679:1
676:+
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639:i
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612:=
609:A
580:N
565:r
560:l
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489:=
484:k
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433:(
427:s
423:t
402:l
398:m
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229:i
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205:,
198:k
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165:A
141:X
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135:(
133:0
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56:0
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