6536:
1030:. For example, all finite-dimensional projections (or subspaces) are finite (since isometries between Hilbert spaces leave the dimension fixed), but the identity operator on an infinite-dimensional Hilbert space is not finite in the von Neumann algebra of all bounded operators on it, since it is isometrically isomorphic to a proper subset of itself. However it is possible for infinite dimensional subspaces to be finite.
462:
together with a
Hilbert space and a suitable faithful unital action on the Hilbert space. The concrete and abstract definitions of a von Neumann algebra are similar to the concrete and abstract definitions of a C*-algebra, which can be defined either as norm-closed *-algebras of operators on a Hilbert space, or as
1419:. (The Connes spectrum is a closed subgroup of the positive reals, so these are the only possibilities.) The only trace on type III factors takes value â on all non-zero positive elements, and any two non-zero projections are equivalent. At one time type III factors were considered to be intractable objects, but
2367:
The
Hilbert space tensor product of two Hilbert spaces is the completion of their algebraic tensor product. One can define a tensor product of von Neumann algebras (a completion of the algebraic tensor product of the algebras considered as rings), which is again a von Neumann algebra, and act on the
461:
of some other Banach space called the predual. The predual of a von
Neumann algebra is in fact unique up to isomorphism. Some authors use "von Neumann algebra" for the algebras together with a Hilbert space action, and "W*-algebra" for the abstract concept, so a von Neumann algebra is a W*-algebra
2368:
tensor product of the corresponding
Hilbert spaces. The tensor product of two finite algebras is finite, and the tensor product of an infinite algebra and a non-zero algebra is infinite. The type of the tensor product of two von Neumann algebras (I, II, or III) is the maximum of their types. The
1745:
Any factor has a trace such that the trace of a non-zero projection is non-zero and the trace of a projection is infinite if and only if the projection is infinite. Such a trace is unique up to rescaling. For factors that are separable or finite, two projections are equivalent if and only if they
1099:
showed that every factor has one of 3 types as described below. The type classification can be extended to von
Neumann algebras that are not factors, and a von Neumann algebra is of type X if it can be decomposed as a direct integral of type X factors; for example, every commutative von Neumann
2607:
1190:, isomorphism classes of factors of type I correspond exactly to the cardinal numbers. Since many authors consider von Neumann algebras only on separable Hilbert spaces, it is customary to call the bounded operators on a Hilbert space of finite dimension
2979:. This is analogous to the two approaches to measure and integration, where one has the choice to construct measures of sets first and define integrals later, or construct integrals first and define set measures as integrals of characteristic functions.
2728:. Popa's work on fundamental groups of non-amenable factors represents another significant advance. The theory of factors "beyond the hyperfinite" is rapidly expanding at present, with many new and surprising results; it has close links with
2481:(ITPFI stands for "infinite tensor product of finite type I factors"). The type of the infinite tensor product can vary dramatically as the states are changed; for example, the infinite tensor product of an infinite number of type I
1092:
of factors. This decomposition is essentially unique. Thus, the problem of classifying isomorphism classes of von
Neumann algebras on separable Hilbert spaces can be reduced to that of classifying isomorphism classes of factors.
2695:
Von
Neumann algebras of type I are always amenable, but for the other types there are an uncountable number of different non-amenable factors, which seem very hard to classify, or even distinguish from each other. Nevertheless,
2468:
showed that one should choose a state on each of the von
Neumann algebras, use this to define a state on the algebraic tensor product, which can be used to produce a Hilbert space and a (reasonably small) von Neumann algebra.
1331:
factor is defined to be the fundamental group of its tensor product with the infinite (separable) factor of type I. For many years it was an open problem to find a type II factor whose fundamental group was not the group of
571:
of two von
Neumann algebras acting on two Hilbert spaces is defined to be the von Neumann algebra generated by their algebraic tensor product, considered as operators on the Hilbert space tensor product of the Hilbert
2441:
1299:
has a semifinite trace, unique up to rescaling, and the set of traces of projections is . The set of real numbers λ such that there is an automorphism rescaling the trace by a factor of λ is called the
2507:
2683:
properties of Î can be formulated entirely in terms of bimodules and therefore make sense for the von
Neumann algebra itself. For example, Connes and Jones gave a definition of an analogue of
540:
5943:
1848:
Given an abstract separable factor, one can ask for a classification of its modules, meaning the separable Hilbert spaces that it acts on. The answer is given as follows: every such module
244:
588:: every finitely generated submodule of a projective module is itself projective. There have been several attempts to axiomatize the underlying rings of von Neumann algebras, including
5726:
148:
5816:
200:
2096:: this means the algebra contains an ascending sequence of finite dimensional subalgebras with dense union. (Warning: some authors use "hyperfinite" to mean "AFD and finite".)
5853:
1885:
if it has a cyclic separating vector. Each factor has a standard representation, which is unique up to isomorphism. The standard representation has an antilinear involution
272:
5892:
5660:
2866:
through right multiplication. One can show that this is the von Neumann algebra generated by the operators corresponding to multiplication from the left with an element
6019:
2247:
calling this a classification is a little misleading, as it is known that there is no easy way to classify the corresponding ergodic flows.) The ones of type I and II
1495:. (Here "normal" means that it preserves suprema when applied to increasing nets of self adjoint operators; or equivalently to increasing sequences of projections.)
2636:
with module actions of two commuting von Neumann algebras. Bimodules have a much richer structure than that of modules. Any bimodule over two factors always gives a
1375:
found an uncountable family of such groups with non-isomorphic von Neumann group algebras, thus showing the existence of uncountably many different separable type II
4196:
969:
292:
1053:-closure of the subspace generated by the indicator functions. Similarly, a von Neumann algebra is generated by its projections; this is a consequence of the
4770:
6425:
6050:
5104:
4298:
5749:
2900:
of a von Neumann algebra by a discrete (or more generally locally compact) group can be defined, and is a von Neumann algebra. Special cases are the
2464:
The tensor product of an infinite number of von Neumann algebras, if done naively, is usually a ridiculously large non-separable algebra. Instead
395:
of bounded operators (on a Hilbert space) containing the identity. In this definition the weak (operator) topology can be replaced by many other
3931:
2893:
The tensor product of two von Neumann algebras, or of a countable number with states, is a von Neumann algebra as described in the section above.
1746:
have the same trace. The type of a factor can be read off from the possible values of this trace over the projections of the factor, as follows:
6261:
5226:
4758:
2034:-dimension can be anything in . There is in general no canonical way to normalize it; the factor may have outer automorphisms multiplying the
1407:, where λ is a real number in the interval . More precisely, if the Connes spectrum (of its modular group) is 1 then the factor is of type III
6088:
6045:
1664:|). The Banach space of trace class operators is itself the dual of the C*-algebra of compact operators (which is not a von Neumann algebra).
801:
if there is a partial isometry mapping the first isomorphically onto the other that is an element of the von Neumann algebra (informally, if
2640:
since one of the factors is always contained in the commutant of the other. There is also a subtle relative tensor product operation due to
2378:
979:
are partially ordered by inclusion, and this induces a partial order †of projections. There is also a natural partial order on the set of
3953:
486:
Some of the terminology in von Neumann algebra theory can be confusing, and the terms often have different meanings outside the subject.
3022:
1371:
factor is the von Neumann group algebra of a countable infinite discrete group such that every non-trivial conjugacy class is infinite.
6251:
5159:
2976:
2890:
is a union of finite subgroups (for example, the group of all permutations of the integers fixing all but a finite number of elements).
3936:
3709:
1149:
otherwise. Factors of types I and II may be either finite or properly infinite, but factors of type III are always properly infinite.
3958:
3317:. This paper gives their basic properties and the division into types I, II, and III, and in particular finds factors not of type I.
6378:
6233:
5131:
4468:
2602:{\displaystyle x\mapsto {\rm {Tr}}{\begin{pmatrix}{1 \over \lambda +1}&0\\0&{\lambda \over \lambda +1}\\\end{pmatrix}}x.}
6209:
4765:
4597:
4361:
4283:
3946:
1483:
acts on, as this determines the ultraweak topology. However the predual can also be defined without using the Hilbert space that
5478:
4176:
1674:
1836:) is a normal state. This construction can be reversed to give an action on a Hilbert space from a normal state: this is the
445:
The first two definitions describe a von Neumann algebra concretely as a set of operators acting on some given Hilbert space.
5420:
4029:
3827:
1461:
is (as a Banach space) the dual of its predual. The predual is unique in the sense that any other Banach space whose dual is
5236:
4024:
670:
Due to this analogy, the theory of von Neumann algebras has been called noncommutative measure theory, while the theory of
435:
94:
5319:
1874:-dimension is additive, and a module is isomorphic to a subspace of another module if and only if it has smaller or equal
505:
4786:
1265:
6101:
1552:
1524:
is nonconstructive and uses the axiom of choice in an essential way; it is very hard to exhibit explicit elements of
6190:
6081:
4960:
4743:
4485:
4181:
3632:. This shows that some apparently topological properties in von Neumann algebras can be defined purely algebraically.
3485:
3449:
3428:
3282:
3179:
3160:
3098:
511:
5908:
3377:. This studies when factors are isomorphic, and in particular shows that all approximately finite factors of type II
564:
by a set of bounded operators on a Hilbert space is the smallest von Neumann algebra containing all those operators.
212:
6460:
5555:
4878:
4721:
3999:
1866:) (not its dimension as a complex vector space) such that modules are isomorphic if and only if they have the same
1411:, if the Connes spectrum is all integral powers of λ for 0 < λ < 1, then the type is III
6105:
5408:
5344:
4895:
3968:
2190:: any completely positive linear map from any self adjoint closed subspace containing 1 of any unital C*-algebra
987:
is a factor, †is a total order on equivalence classes of projections, described in the section on traces below.
5189:
1423:
has led to a good structure theory. In particular, any type III factor can be written in a canonical way as the
5682:
5600:
5403:
5082:
4861:
3882:
3766:
115:
2923:
can be defined. These examples generalise von Neumann group algebras and the group-measure space construction.
1104:. Every von Neumann algebra can be written uniquely as a sum of von Neumann algebras of types I, II, and III.
1054:
6565:
6256:
5779:
5441:
5219:
5174:
5164:
4433:
4191:
3702:
3469:
2770:
2107:
1284:; there are an uncountable number of other factors of these types that are the subject of intensive study.
6539:
6312:
6246:
6074:
5276:
5266:
5194:
5121:
4997:
4666:
4500:
4438:
3817:
2336:
2324:
1257:
5271:
2975:
provides an alternative axiomatization to probability theory. In this case the method goes by the name of
422:
The second definition is that a von Neumann algebra is a subalgebra of the bounded operators closed under
310:, in a series of papers written in the 1930s and 1940s (F.J. Murray & J. von Neumann
6276:
5614:
5604:
5204:
4590:
4354:
4328:
4248:
3802:
3464:
3459:
2049:-dimension can be 0 or â. Any two non-zero modules are isomorphic, and all non-zero modules are standard.
1711:
617:
167:
6521:
6475:
6399:
6281:
5973:
5775:
5437:
5251:
5144:
5139:
5034:
5007:
4972:
4824:
4717:
4541:
4454:
4303:
4201:
4081:
3002:
2758:
1420:
498:
203:
1399:. Since the identity operator is always infinite in those factors, they were sometimes called type III
6570:
6516:
6332:
5831:
5731:
5425:
5398:
5381:
5199:
5044:
4713:
4308:
4171:
4004:
3989:
3797:
3761:
3494:
von Neumann, J. (1930), "Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren",
2774:
2700:
has shown that the class of non-amenable factors coming from the group-measure space construction is
2684:
1391:
factors are factors that do not contain any nonzero finite projections at all. In their first paper
1341:
423:
4495:
3388:(1967), "Representations of Uniformly Hyperfinite Algebras and Their Associated von Neumann Rings",
253:
6560:
6368:
6266:
6169:
5948:
5241:
5231:
5149:
5087:
5014:
4968:
4883:
4708:
4561:
4490:
3900:
3890:
3771:
3695:
605:
577:
400:
2780:
The bounded operators on any Hilbert space form a von Neumann algebra, indeed a factor, of type I.
6465:
6241:
5870:
5633:
5214:
5154:
4263:
4238:
4056:
4045:
3756:
3348:. This is a continuation of the previous paper, that studies properties of the trace of a factor.
3198:
2953:
2725:
2352:
2351:. Type III factors occur in the remaining cases where there is no invariant measure, but only an
1200:, and the bounded operators on a separable infinite-dimensional Hilbert space, a factor of type I
948:
generally hold if one were to require unitary equivalence in the definition of ~, i.e. if we say
675:
158:
47:
5982:
494:
is a von Neumann algebra with trivial center, i.e. a center consisting only of scalar operators.
6496:
6440:
6404:
5256:
5169:
4950:
4866:
4702:
4696:
4583:
4459:
4428:
4347:
4114:
4104:
4099:
3807:
2784:
2733:
2664:
2660:
2210:
There is no generally accepted term for the class of algebras above; Connes has suggested that
389:
59:
5496:
1706:
is a weight with Ï(1) finite (or rather the extension of Ï to the whole algebra by linearity).
1479:
The definition of the predual given above seems to depend on the choice of Hilbert space that
1248:. If the identity operator in a type II factor is finite, the factor is said to be of type II
6203:
6040:
6035:
5510:
5458:
5415:
5339:
5292:
5029:
4691:
4658:
4631:
4510:
4464:
4407:
3859:
3277:, Proc. Sympos. Pure Math., vol. 50, Providence, RI.: Amer. Math. Soc., pp. 57â60,
2961:
2957:
2945:
2937:
2882:
is infinite (for example, a non-abelian free group), and is the hyperfinite factor of type II
2697:
2680:
2320:
2312:
2273:
1692:
1476:
showed that the existence of a predual characterizes von Neumann algebras among C* algebras.
303:
98:
6199:
5329:
3574:. This discusses infinite tensor products of Hilbert spaces and the algebras acting on them.
2769:) is not a von Neumann algebra; for example, the Ï-algebra of measurable sets might be the
2473:
studied the case where all the factors are finite matrix algebras; these factors are called
6479:
5978:
5184:
5179:
4890:
4774:
4680:
4393:
4273:
4252:
4166:
4051:
4014:
3503:
3226:
3083:
2941:
2916:
806:
449:
showed that von Neumann algebras can also be defined abstractly as C*-algebras that have a
404:
82:
6066:
1395:
were unable to decide whether or not they existed; the first examples were later found by
8:
6445:
6097:
5821:
5622:
5579:
5393:
5116:
4846:
4653:
4389:
3812:
1061:
774:
601:
585:
154:
151:
24:
5767:
3507:
3230:
2749:
The essentially bounded functions on a Ï-finite measure space form a commutative (type I
6470:
6337:
6055:
5966:
5589:
5559:
5376:
5334:
4941:
4851:
4796:
4643:
4206:
4135:
4066:
3910:
3872:
3653:
3624:
3595:
3548:
3519:
3405:
3369:
3340:
3309:
3262:
3216:
3149:
3136:
2724:
factors, a result first proved by Leeming Ge for free group factors using Voiculescu's
2217:
The amenable factors have been classified: there is a unique one of each of the types I
1353:
508:
of finite factors (meaning the von Neumann algebra has a faithful normal tracial state
408:
277:
5897:
5867:
5828:
274:
is a von Neumann algebra, non-commutative if the Hilbert space has dimension at least
6450:
5736:
5209:
4990:
4933:
4913:
4515:
4412:
4384:
4313:
4288:
3973:
3895:
3675:
3523:
3481:
3445:
3437:
3424:
3278:
3175:
3156:
3094:
3048:
2773:
on an uncountable set. A fundamental approximation theorem can be represented by the
1186:
bounded operators on some Hilbert space; since there is one Hilbert space for every
1107:
There are several other ways to divide factors into classes that are sometimes used:
396:
90:
63:
3661:. This discusses how to write a von Neumann algebra as a sum or integral of factors.
860:
The equivalence relation ~ thus defined is additive in the following sense: Suppose
738:. Informally these are the closed subspaces that can be described using elements of
6455:
6373:
6342:
6322:
6307:
6302:
6297:
5366:
5361:
5349:
5261:
5246:
5109:
5049:
5024:
4955:
4945:
4808:
4525:
4402:
4318:
4019:
3867:
3822:
3746:
3645:
3616:
3587:
3540:
3511:
3397:
3385:
3361:
3330:
3301:
3254:
3128:
3070:
3044:
2965:
2949:
2905:
2729:
2709:
2277:
1901:
1837:
1352:
showed that the fundamental group can be trivial for certain groups, including the
1077:
818:
463:
247:
102:
74:
6134:
4753:
3106:
2932:
Von Neumann algebras have found applications in diverse areas of mathematics like
1340:
then showed that the von Neumann group algebra of a countable discrete group with
1088:
showed, every von Neumann algebra on a separable Hilbert space is isomorphic to a
833:
and is an element of the von Neumann algebra. Another way of stating this is that
6317:
6271:
6219:
6214:
6185:
5386:
5371:
5297:
5099:
5092:
5059:
5019:
4985:
4977:
4905:
4873:
4738:
4670:
4505:
4293:
4278:
4186:
4149:
4145:
4109:
4071:
4009:
3994:
3963:
3905:
3864:
3851:
3776:
3718:
3687:
3416:
3080:
2997:
2897:
2616:
are isomorphic to ArakiâWoods factors, but there are uncountably many of type III
2285:
2268:
2007:-dimension can be anything in . It is normalized so that the standard module has
1912:-dimension 1, while for infinite factors the standard module is the module with
1424:
1187:
1089:
638:
635:
554:
78:
6144:
3561:
1453:, which is the Banach space of all ultraweakly continuous linear functionals on
411:
operator topologies. The *-algebras of bounded operators that are closed in the
6506:
6358:
6159:
5956:
5904:
5564:
5430:
5077:
5067:
4686:
4638:
4243:
4222:
4140:
4130:
3941:
3848:
3781:
3741:
3075:
2737:
2705:
1403:
in the past, but recently that notation has been superseded by the notation III
1333:
817:
if the corresponding subspaces are equivalent, or in other words if there is a
345:
Introductory accounts of von Neumann algebras are given in the online notes of
86:
1679:
Weights and their special cases states and traces are discussed in detail in (
1292:
has a unique finite tracial state, and the set of traces of projections is .
1033:
Orthogonal projections are noncommutative analogues of indicator functions in
6554:
6511:
6435:
6164:
6149:
6139:
5961:
5574:
5528:
5463:
5314:
5309:
5302:
4923:
4856:
4829:
4648:
4621:
4398:
4370:
2663:
of Î, then, regarding Î as the diagonal subgroup of Î Ă Î, the corresponding
2645:
2328:
2305:
1976:-dimension can be any of 0, 1, 2, 3, ..., â. The standard representation of
1611:
627:
412:
162:
157:
on the real line is a commutative von Neumann algebra, whose elements act as
51:
20:
3607:
von Neumann, J. (1943), "On Some Algebraical Properties of Operator Rings",
2704:
from the class coming from group von Neumann algebras of free groups. Later
2172:: this means that there is a projection of norm 1 from bounded operators on
1820:
If a von Neumann algebra acts on a Hilbert space containing a norm 1 vector
6501:
6154:
6124:
5473:
5468:
4928:
4918:
4791:
4781:
4626:
4606:
4061:
3915:
3856:
3189:
Les algÚbres d'opérateurs dans l'espace hilbertien: algÚbres de von Neumann
2713:
2641:
2485:
factors can have any type depending on the choice of states. In particular
1337:
581:
580:
about the topology on a von Neumann algebra, we can consider it a (unital)
454:
608:. (The von Neumann algebra itself is in general not von Neumann regular.)
546:
von Neumann algebras are the direct integral of properly infinite factors.
6430:
6420:
6327:
6129:
5762:
5678:
5584:
5569:
5549:
5523:
5488:
5039:
5002:
4675:
4520:
4258:
3843:
3679:
3238:
2988:
2933:
2061:
and others proved that the following conditions on a von Neumann algebra
1653:
1544:
1344:(the trivial representation is isolated in the dual space), such as SL(3,
623:
593:
589:
31:
3531:
von Neumann, J. (1936), "On a Certain Topology for Rings of Operators",
2814:. Equivalent subrepresentations correspond to equivalent projections in
2720:
factors, i.e. ones that cannot be factored as tensor products of type II
710:; they are exactly the operators which give an orthogonal projection of
6363:
6195:
5518:
5499: ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (H
5483:
5324:
5072:
4834:
3751:
3657:
3628:
3599:
3552:
3515:
3409:
3373:
3344:
3313:
3266:
3140:
2972:
2757:
functions. For certain non-Ï-finite measure spaces, usually considered
1535:. For example, exotic positive linear forms on the von Neumann algebra
1415:, and if the Connes spectrum is all positive reals then the type is III
1349:
671:
631:
549:
A von Neumann algebra that acts on a separable Hilbert space is called
458:
416:
67:
55:
4575:
2436:{\displaystyle (M\otimes N)^{\prime }=M^{\prime }\otimes N^{\prime },}
5594:
4839:
4803:
3736:
3722:
2637:
2454:
2158:
is a weak pointwise limit of completely positive maps of finite rank.
2038:-dimension by constants. The standard representation is the one with
597:
431:
427:
392:
43:
3649:
3620:
3591:
3544:
3401:
3365:
3335:
3305:
3258:
3132:
1182:. Any factor of type I is isomorphic to the von Neumann algebra of
5860:
5744:
5670:
5630:
5533:
5356:
4323:
4268:
3636:
von Neumann, J. (1949), "On Rings of Operators. Reduction Theory",
2920:
1487:
acts on, by defining it to be the space generated by all positive
805:"knows" that the subspaces are isomorphic). This induces a natural
642:
3352:
Murray, F.J.; von Neumann, J. (1943), "On rings of operators IV",
3321:
Murray, F.J.; von Neumann, J. (1937), "On rings of operators II",
3221:
2806:, whose projections correspond exactly to the closed subspaces of
2489:
found an uncountable family of non-isomorphic hyperfinite type III
2284:
transformation. In fact they are precisely the factors arising as
983:
of projections, induced by the partial order †of projections. If
972:
gives a sufficient condition for Murray-von Neumann equivalence.
5665:
4748:
4339:
2281:
749:
It can be shown that the closure of the image of any operator in
450:
3035:
Connes, A (May 1978). "On the cohomology of operator algebras".
1080:
consists only of multiples of the identity operator is called a
3292:
Murray, F.J.; von Neumann, J. (1936), "On rings of operators",
3061:
Araki, H.; Woods, E. J. (1968), "A classification of factors",
2651:
Bimodules are also important for the von Neumann group algebra
730:. This establishes a 1:1 correspondence between projections of
3288:
A historical account of the discovery of von Neumann algebras.
1509:(which consists of all norm-continuous linear functionals on
1224:
can be "halved" in the sense that there are two projections
419:, so in particular any von Neumann algebra is a C*-algebra.
385:
There are three common ways to define von Neumann algebras.
6096:
2648:, reconciles these two seemingly different points of view.
2114:
with values in a normal dual Banach bimodule are all inner.
1691:Ï on a von Neumann algebra is a linear map from the set of
1216:
if there are no minimal projections but there are non-zero
714:
onto some closed subspace. A subspace of the Hilbert space
108:
Two basic examples of von Neumann algebras are as follows:
3119:
Connes, A. (1976), "Classification of Injective Factors",
2362:
641:. Every commutative von Neumann algebra is isomorphic to
453:; in other words the von Neumann algebra, considered as a
1908:-dimension is normalized so that the standard module has
1900:. For finite factors the standard module is given by the
373:
gives an encyclopedic account of the theory. The book by
3671:. Reprints von Neumann's papers on von Neumann algebras.
19:"operator ring" redirects here. Not to be confused with
3273:
Murray, F. J. (2006), "The rings of operators papers",
1618:
on the closed subspace of bounded continuous functions
306:
developed the basic theory, under the original name of
2675:) is naturally a bimodule for two commuting copies of
2532:
1288:
proved the fundamental result that a factor of type II
1276:. These are the unique hyperfinite factors of types II
761:. Also, the closure of the image under an operator of
655:, ÎŒ) and conversely, for every Ï-finite measure space
442:) says that the first two definitions are equivalent.
5985:
5911:
5873:
5834:
5782:
5685:
5636:
3578:
von Neumann, J. (1940), "On rings of operators III",
2644:
on bimodules. The theory of subfactors, initiated by
2510:
2381:
2132:
the weak operator closed convex hull of the elements
1348:), has a countable fundamental group. Subsequently,
1233:
1217:
514:
280:
256:
215:
170:
118:
3275:
The legacy of John von Neumann (Hempstead, NY, 1988)
2993:
Pages displaying wikidata descriptions as a fallback
2612:
All hyperfinite von Neumann algebras not of type III
1904:applied to the unique normal tracial state and the
780:
611:
388:The first and most common way is to define them as
73:Von Neumann algebras were originally introduced by
6426:Spectral theory of ordinary differential equations
6013:
5937:
5886:
5847:
5810:
5720:
5654:
4299:Spectral theory of ordinary differential equations
3717:
3603:. This shows the existence of factors of type III.
3148:
3005: â Mathematical method in functional analysis
2601:
2435:
2266:All amenable factors can be constructed using the
2243:correspond to certain ergodic flows. (For type III
2198:can be extended to a completely positive map from
1256:. The best understood factors of type II are the
785:The basic theory of projections was worked out by
534:
286:
266:
238:
194:
142:
4197:SchröderâBernstein theorems for operator algebras
3351:
3320:
3291:
2252:
1547:; they correspond to exotic *-homomorphisms into
1392:
1285:
1273:
1096:
970:SchröderâBernstein theorems for operator algebras
786:
319:
315:
311:
16:*-algebra of bounded operators on a Hilbert space
6552:
2991: â algebraic generalization of a W*-algebra
2497:, by taking an infinite tensor product of type I
2053:
3667:Collected Works, Volume III: Rings of Operators
535:{\displaystyle \tau :M\rightarrow \mathbb {C} }
5938:{\displaystyle S\left(\mathbb {R} ^{n}\right)}
1668:
239:{\displaystyle {\mathcal {B}}({\mathcal {H}})}
6082:
6046:Mathematical formulation of quantum mechanics
4591:
4355:
3703:
3527:. The original paper on von Neumann algebras.
3329:(2), American Mathematical Society: 208â248,
3193:, the first book about von Neumann algebras.)
1923:-dimensions of modules are given as follows:
3665:von Neumann, John (1961), Taub, A.H. (ed.),
2255:, and the remaining ones were classified by
2239:, for 0 < λ †1, and the ones of type III
3664:
3635:
3606:
3577:
3559:
3530:
3493:
2846:is the algebra of all bounded operators on
2623:
2465:
1513:) but is generally smaller. The proof that
1396:
1085:
1055:spectral theorem for self-adjoint operators
773:. (These results are a consequence of the
584:, or just a ring. Von Neumann algebras are
439:
339:
335:
331:
327:
323:
299:
298:Von Neumann algebras were first studied by
6089:
6075:
4598:
4584:
4362:
4348:
3710:
3696:
3674:
3213:W*-algebras and noncommutative integration
3060:
2878:) if every non-trivial conjugacy class of
2795:then the bounded operators commuting with
2708:proved that group von Neumann algebras of
2470:
1648:) of bounded operators on a Hilbert space
1060:The projections of a finite factor form a
426:(the *-operation) and equal to its double
350:
5921:
5721:{\displaystyle B_{p,q}^{s}(\mathbb {R} )}
5711:
3334:
3220:
3088:
3074:
2915:The von Neumann algebras of a measurable
1629:) cannot be represented as a function in
1315:and an infinite type I factor has type II
1311:The tensor product of a factor of type II
726:if it is the image of some projection in
630:is analogous to that between commutative
528:
362:
185:
143:{\displaystyle L^{\infty }(\mathbb {R} )}
133:
6379:Group algebra of a locally compact group
3556:. This defines the ultrastrong topology.
3475:
3436:
3210:
2347:) measure, invariant under an action of
1843:
1680:
1252:; otherwise, it is said to be of type II
370:
358:
105:definition as an algebra of symmetries.
5811:{\displaystyle L^{\lambda ,p}(\Omega )}
4605:
3186:
3169:
2690:
2632:(or correspondence) is a Hilbert space
2501:factors, each with the state given by:
2370:commutation theorem for tensor products
2363:Tensor products of von Neumann algebras
1640:The predual of the von Neumann algebra
1586:) of integrable functions. The dual of
1566:The predual of the von Neumann algebra
1319:, and conversely any factor of type II
1170:such that there is no other projection
990:A projection (or subspace belonging to
434:of some subalgebra closed under *. The
354:
338:), reprinted in the collected works of
6553:
6051:Ordinary Differential Equations (ODEs)
5165:BanachâSteinhaus (Uniform boundedness)
3478:Theory of Operator Algebras I, II, III
3457:
3384:
3272:
3237:
3146:
3118:
3034:
2687:for von Neumann algebras in this way.
2486:
2263:case which was completed by Haagerup.
2256:
2058:
1675:Noncommutative measure and integration
1574:) of essentially bounded functions on
1372:
679:
553:. Note that such algebras are rarely
374:
6070:
4579:
4343:
4030:Spectral theory of normal C*-algebras
3828:Spectral theory of normal C*-algebras
3691:
3415:
3196:
1473:
1220:. This implies that every projection
604:of a finite von Neumann algebra is a
446:
366:
346:
101:definition is equivalent to a purely
4025:Spectral theory of compact operators
2753:) von Neumann algebra acting on the
2493:factors for 0 < λ < 1, called
1961:-dimension 1 (and complex dimension
1715:is a weight with Ï(1) = 1.
436:von Neumann double commutant theorem
2150:: this means the identity map from
2136:contains an element commuting with
1741:is a trace with Ï(1) = 1.
1382:
694:in a von Neumann algebra for which
195:{\displaystyle L^{2}(\mathbb {R} )}
161:by pointwise multiplication on the
13:
5879:
5840:
5802:
5646:
4369:
4177:CohenâHewitt factorization theorem
3063:Publ. Res. Inst. Math. Sci. Ser. A
2977:GelfandâNaimarkâSegal construction
2655:of a discrete group Î. Indeed, if
2522:
2519:
2425:
2412:
2399:
2170:HakedaâTomiyama extension property
1957:, ..., â. The standard module has
1323:can be constructed like this. The
1207:
1145:if the projection 1 is finite and
753:and the kernel of any operator in
259:
228:
218:
124:
14:
6582:
5543:Subsets / set operations
5320:Differentiation in Fréchet spaces
4486:Compact operator on Hilbert space
4182:Extensions of symmetric operators
3207:; incomplete notes from a course.
2304:). Type I factors occur when the
1505:is a closed subspace of the dual
1162:if there is a minimal projection
1153:
6535:
6534:
6461:Topological quantum field theory
4000:Positive operator-valued measure
2902:group-measure space construction
2315:and the action transitive. When
2296:on abelian von Neumann algebras
2269:group-measure space construction
2090:approximately finite dimensional
1656:operators with the trace norm ||
781:Comparison theory of projections
612:Commutative von Neumann algebras
377:discusses more advanced topics.
5848:{\displaystyle \ell ^{\infty }}
4284:RayleighâFaberâKrahn inequality
2927:
2858:) commuting with the action of
2253:Murray & von Neumann (1943)
1602:) For example, a functional on
1393:Murray & von Neumann (1936)
1286:Murray & von Neumann (1937)
1274:Murray & von Neumann (1936)
1097:Murray & von Neumann (1936)
787:Murray & von Neumann (1936)
6008:
5989:
5805:
5799:
5715:
5707:
5649:
5643:
5237:Lomonosov's invariant subspace
5160:BanachâSchauder (open mapping)
3037:Journal of Functional Analysis
3028:
3023:An Introduction To II1 Factors
3016:
2835:is also a von Neumann algebra.
2514:
2395:
2382:
2015:-dimension is also called the
1806:(usually normalized to be 1).
1795:(usually normalized to be 1).
1434:
829:isometrically to the image of
685:
524:
481:
380:
267:{\displaystyle {\mathcal {H}}}
233:
223:
189:
181:
137:
129:
1:
6257:Uniform boundedness principle
4192:Limiting absorption principle
3562:"On infinite direct products"
3009:
2771:countable-cocountable algebra
2331:. Type II factors occur when
2214:should be the standard term.
2065:on a separable Hilbert space
2054:Amenable von Neumann algebras
1520:is (usually) not the same as
1465:is canonically isomorphic to
1431:factor and the real numbers.
1234:Murrayâvon Neumann equivalent
789:. Two subspaces belonging to
765:of any subspace belonging to
734:and subspaces that belong to
5122:Singular value decomposition
3818:Singular value decomposition
3241:(1969), "Uncountably many II
3049:10.1016/0022-1236(78)90088-5
2874:. It is a factor (of type II
1772:(usually normalized to be 1/
1134:if it has type I or II, and
667:) is a von Neumann algebra.
77:, motivated by his study of
7:
5887:{\displaystyle L^{\infty }}
5655:{\displaystyle ba(\Sigma )}
5524:Radially convex/Star-shaped
4249:Hearing the shape of a drum
3932:Decomposition of a spectrum
3465:Encyclopedia of Mathematics
3421:C*-algebras and W*-algebras
2982:
2799:form a von Neumann algebra
2743:
2355:: these factors are called
2288:by free ergodic actions of
2124:: for any bounded operator
1941:-dimension can be any of 0/
1791:, ....,â for some positive
1669:Weights, states, and traces
1652:is the Banach space of all
1553:StoneâÄech compactification
1127:) if it has type II or III.
975:The subspaces belonging to
809:on projections by defining
618:Abelian von Neumann algebra
369:. The three volume work by
204:square-integrable functions
10:
6587:
6400:Invariant subspace problem
6014:{\displaystyle W(X,L^{p})}
4455:Hilbert projection theorem
3837:Special Elements/Operators
3681:Operators on Hilbert space
1704:positive linear functional
1672:
1594:) is strictly larger than
1212:A factor is said to be of
1158:A factor is said to be of
1067:
1002:if there is no projection
849:for some partial isometry
651:) for some measure space (
615:
322:; J. von Neumann
66:. It is a special type of
18:
6530:
6489:
6413:
6392:
6351:
6290:
6232:
6178:
6120:
6113:
6028:
5613:
5560:Algebraic interior (core)
5542:
5451:
5285:
5175:CauchyâSchwarz inequality
5130:
5058:
4904:
4818:Function space Topologies
4817:
4731:
4614:
4534:
4478:
4447:
4434:CauchyâSchwarz inequality
4421:
4377:
4309:Superstrong approximation
4231:
4215:
4172:Banach algebra cohomology
4159:
4123:
4092:
4038:
4005:Projection-valued measure
3990:Borel functional calculus
3982:
3924:
3881:
3836:
3790:
3762:Projection-valued measure
3729:
2840:von Neumann group algebra
2775:Kaplansky density theorem
2259:, except for the type III
1119:) if it has type I, and
626:von Neumann algebras and
622:The relationship between
6369:Spectrum of a C*-algebra
3901:Spectrum of a C*-algebra
3772:Spectrum of a C*-algebra
3560:von Neumann, J. (1938),
3151:Non-commutative geometry
3076:10.2977/prims/1195195263
2681:representation theoretic
2624:Bimodules and subfactors
2471:Araki & Woods (1968)
1457:. As the name suggests,
1439:Any von Neumann algebra
1026:) that is equivalent to
722:the von Neumann algebra
606:von Neumann regular ring
560:The von Neumann algebra
159:multiplication operators
95:double commutant theorem
6466:Noncommutative geometry
4329:WienerâKhinchin theorem
4264:Kuznetsov trace formula
4239:Almost Mathieu operator
4057:Banach function algebra
4046:Amenable Banach algebra
3803:GelfandâNaimark theorem
3757:Noncommutative topology
3323:Trans. Amer. Math. Soc.
3211:Kostecki, R.P. (2013),
2954:noncommutative geometry
2821:. The double commutant
2353:invariant measure class
1916:-dimension equal to â.
1367:An example of a type II
825:that maps the image of
676:noncommutative topology
6522:TomitaâTakesaki theory
6497:Approximation property
6441:Calculus of variations
6015:
5939:
5888:
5849:
5812:
5722:
5656:
4825:BanachâMazur compactum
4615:Types of Banach spaces
4304:SturmâLiouville theory
4202:ShermanâTakeda theorem
4082:TomitaâTakesaki theory
3857:Hermitian/Self-adjoint
3808:Gelfand representation
3458:Shtern, A.I. (2001) ,
3197:Jones, V.F.R. (2003),
3089:Blackadar, B. (2005),
3003:TomitaâTakesaki theory
2785:unitary representation
2734:geometric group theory
2685:Kazhdan's property (T)
2665:induced representation
2661:unitary representation
2603:
2437:
2106:: this means that the
1824:, then the functional
1491:linear functionals on
1421:TomitaâTakesaki theory
1342:Kazhdan's property (T)
1072:A von Neumann algebra
536:
430:, or equivalently the
288:
268:
240:
196:
144:
60:weak operator topology
6517:BanachâMazur distance
6480:Generalized functions
6041:Finite element method
6036:Differential operator
6016:
5940:
5889:
5850:
5813:
5723:
5657:
5497:Convex series related
5293:Abstract Wiener space
5220:hyperplane separation
4775:Minkowski functionals
4659:Polarization identity
4465:Polarization identity
4408:Orthogonal complement
3798:GelfandâMazur theorem
3638:Annals of Mathematics
3609:Annals of Mathematics
3580:Annals of Mathematics
3533:Annals of Mathematics
3476:Takesaki, M. (1979),
3460:"von Neumann algebra"
3390:Annals of Mathematics
3354:Annals of Mathematics
3294:Annals of Mathematics
3247:Annals of Mathematics
3121:Annals of Mathematics
2962:differential geometry
2958:representation theory
2946:local quantum physics
2938:statistical mechanics
2604:
2438:
1844:Modules over a factor
1802:: for some positive
1673:Further information:
557:in the norm topology.
537:
289:
269:
241:
197:
145:
83:group representations
6566:Von Neumann algebras
6262:Kakutani fixed-point
6247:Riesz representation
5983:
5909:
5871:
5832:
5780:
5683:
5634:
5623:Absolute continuity
5277:Schauder fixed-point
5267:Riesz representation
5227:Kakutani fixed-point
5195:Freudenthal spectral
4681:L-semi-inner product
4439:Riesz representation
4394:L-semi-inner product
4274:Proto-value function
4253:Dirichlet eigenvalue
4167:Abstract index group
4052:Approximate identity
4015:Rigged Hilbert space
3891:KreinâRutman theorem
3737:Involution/*-algebra
3669:, NY: Pergamon Press
3200:von Neumann algebras
3187:Dixmier, J. (1957),
3172:Von Neumann algebras
3170:Dixmier, J. (1981),
3108:corrected manuscript
2942:quantum field theory
2917:equivalence relation
2842:of a discrete group
2691:Non-amenable factors
2508:
2379:
2094:approximately finite
1578:is the Banach space
1166:, i.e. a projection
1138:if it has type III.
813:to be equivalent to
807:equivalence relation
674:is sometimes called
602:affiliated operators
512:
504:is one which is the
278:
254:
213:
168:
155:measurable functions
116:
6446:Functional calculus
6405:Mahler's conjecture
6384:Von Neumann algebra
6098:Functional analysis
5706:
5444:measurable function
5394:Functional calculus
5257:Parseval's identity
5170:Bessel's inequality
5117:Polar decomposition
4896:Uniform convergence
4654:Inner product space
4460:Parseval's identity
4429:Bessel's inequality
4077:Von Neumann algebra
3813:Polar decomposition
3508:1930MatAn.102..685E
3231:2013arXiv1307.4818P
3147:Connes, A. (1994),
2791:on a Hilbert space
2251:were classified by
1881:A module is called
1840:for normal states.
1722:is a weight with Ï(
1695:(those of the form
1610:) that extends the
1295:A factor of type II
1266:hyperfinite type II
1258:hyperfinite type II
1141:A factor is called
1130:A factor is called
1111:A factor is called
1062:continuous geometry
981:equivalence classes
944:. Additivity would
775:polar decomposition
502:von Neumann algebra
250:on a Hilbert space
152:essentially bounded
36:von Neumann algebra
25:operator assistance
6471:Riemann hypothesis
6170:Topological vector
6056:Validated numerics
6011:
5967:Sobolev inequality
5935:
5884:
5845:
5808:
5737:Bounded variation
5718:
5686:
5671:Banach coordinate
5652:
5590:Minkowski addition
5252:M. Riesz extension
4732:Banach spaces are:
4207:Unbounded operator
4136:Essential spectrum
4115:SchurâHorn theorem
4105:BauerâFike theorem
4100:AlonâBoppana bound
4093:Finite-Dimensional
4067:Nuclear C*-algebra
3911:Spectral asymmetry
3516:10.1007/BF01782352
3191:, Gauthier-Villars
3185:(A translation of
3155:, Academic Press,
2730:rigidity phenomena
2599:
2587:
2466:von Neumann (1938)
2433:
2011:-dimension 1. The
1768:for some positive
1397:von Neumann (1940)
1354:semidirect product
1218:finite projections
1194:a factor of type I
1100:algebra has type I
1086:von Neumann (1949)
795:Murrayâvon Neumann
532:
340:von Neumann (1961)
308:rings of operators
300:von Neumann (1930)
284:
264:
236:
192:
140:
6548:
6547:
6451:Integral operator
6228:
6227:
6064:
6063:
5776:MorreyâCampanato
5758:compact Hausdorff
5605:Relative interior
5459:Absolutely convex
5426:Projection-valued
5035:Strictly singular
4961:on Hilbert spaces
4722:of Hilbert spaces
4573:
4572:
4516:Sesquilinear form
4469:Parallelogram law
4413:Orthonormal basis
4337:
4336:
4314:Transfer operator
4289:Spectral geometry
3974:Spectral abscissa
3954:Approximate point
3896:Normal eigenvalue
3676:Wassermann, A. J.
3640:, Second Series,
3611:, Second Series,
3582:, Second Series,
3535:, Second Series,
3392:, Second Series,
3386:Powers, Robert T.
3356:, Second Series,
3296:, Second Series,
3249:, Second Series,
3123:, Second Series,
3091:Operator algebras
2966:dynamical systems
2919:and a measurable
2710:hyperbolic groups
2583:
2551:
2343:) or infinite (II
2017:coupling constant
1693:positive elements
1551:and describe the
1545:free ultrafilters
1325:fundamental group
1302:fundamental group
1147:properly infinite
1123:(or occasionally
1115:(or occasionally
1000:finite projection
964:for some unitary
952:is equivalent to
837:is equivalent to
544:properly infinite
464:Banach *-algebras
397:common topologies
353:and the books by
351:Wassermann (1991)
287:{\displaystyle 2}
248:bounded operators
91:quantum mechanics
64:identity operator
62:and contains the
48:bounded operators
6578:
6571:John von Neumann
6538:
6537:
6456:Jones polynomial
6374:Operator algebra
6118:
6117:
6091:
6084:
6077:
6068:
6067:
6020:
6018:
6017:
6012:
6007:
6006:
5974:TriebelâLizorkin
5944:
5942:
5941:
5936:
5934:
5930:
5929:
5924:
5893:
5891:
5890:
5885:
5883:
5882:
5854:
5852:
5851:
5846:
5844:
5843:
5817:
5815:
5814:
5809:
5798:
5797:
5727:
5725:
5724:
5719:
5714:
5705:
5700:
5661:
5659:
5658:
5653:
5514:
5492:
5474:Balanced/Circled
5272:Robinson-Ursescu
5190:EberleinâĆ mulian
5110:Spectral theorem
4906:Linear operators
4703:Uniformly smooth
4600:
4593:
4586:
4577:
4576:
4403:Prehilbert space
4364:
4357:
4350:
4341:
4340:
4319:Transform theory
4039:Special algebras
4020:Spectral theorem
3983:Spectral Theorem
3823:Spectral theorem
3712:
3705:
3698:
3689:
3688:
3684:
3670:
3660:
3631:
3602:
3573:
3555:
3526:
3490:
3472:
3454:
3433:
3412:
3376:
3347:
3338:
3316:
3287:
3269:
3233:
3224:
3206:
3205:
3192:
3184:
3165:
3154:
3143:
3115:
3113:
3103:
3079:
3078:
3053:
3052:
3032:
3026:
3020:
2994:
2950:free probability
2830:
2826:
2819:
2810:invariant under
2804:
2608:
2606:
2605:
2600:
2592:
2591:
2584:
2582:
2568:
2552:
2550:
2536:
2526:
2525:
2451:
2442:
2440:
2439:
2434:
2429:
2428:
2416:
2415:
2403:
2402:
2286:crossed products
1996:-dimension is â.
1902:GNS construction
1898:
1870:-dimension. The
1852:can be given an
1838:GNS construction
1816:Type III: {0,â}.
1726:) = Ï(
1528:that are not in
1383:Type III factors
998:is said to be a
819:partial isometry
769:also belongs to
659:, the *-algebra
639:Hausdorff spaces
541:
539:
538:
533:
531:
440:von Neumann 1930
363:Blackadar (2005)
302:in 1929; he and
293:
291:
290:
285:
273:
271:
270:
265:
263:
262:
245:
243:
242:
237:
232:
231:
222:
221:
201:
199:
198:
193:
188:
180:
179:
149:
147:
146:
141:
136:
128:
127:
79:single operators
75:John von Neumann
6586:
6585:
6581:
6580:
6579:
6577:
6576:
6575:
6561:Operator theory
6551:
6550:
6549:
6544:
6526:
6490:Advanced topics
6485:
6409:
6388:
6347:
6313:HilbertâSchmidt
6286:
6277:GelfandâNaimark
6224:
6174:
6109:
6095:
6065:
6060:
6024:
6002:
5998:
5984:
5981:
5980:
5979:Wiener amalgam
5949:SegalâBargmann
5925:
5920:
5919:
5915:
5910:
5907:
5906:
5878:
5874:
5872:
5869:
5868:
5839:
5835:
5833:
5830:
5829:
5787:
5783:
5781:
5778:
5777:
5732:BirnbaumâOrlicz
5710:
5701:
5690:
5684:
5681:
5680:
5635:
5632:
5631:
5609:
5565:Bounding points
5538:
5512:
5490:
5447:
5298:Banach manifold
5281:
5205:GelfandâNaimark
5126:
5100:Spectral theory
5068:Banach algebras
5060:Operator theory
5054:
5015:Pseudo-monotone
4998:HilbertâSchmidt
4978:Densely defined
4900:
4813:
4727:
4610:
4604:
4574:
4569:
4562:SegalâBargmann
4530:
4501:HilbertâSchmidt
4491:Densely defined
4474:
4443:
4417:
4373:
4368:
4338:
4333:
4294:Spectral method
4279:Ramanujan graph
4227:
4211:
4187:Fredholm theory
4155:
4150:Shilov boundary
4146:Structure space
4124:Generalizations
4119:
4110:Numerical range
4088:
4072:Uniform algebra
4034:
4010:Riesz projector
3995:Min-max theorem
3978:
3964:Direct integral
3920:
3906:Spectral radius
3877:
3832:
3786:
3777:Spectral radius
3725:
3719:Spectral theory
3716:
3650:10.2307/1969463
3621:10.2307/1969106
3592:10.2307/1968823
3545:10.2307/1968692
3488:
3452:
3438:Schwartz, Jacob
3431:
3402:10.2307/1970364
3381:are isomorphic.
3380:
3366:10.2307/1969107
3336:10.2307/1989620
3306:10.2307/1968693
3285:
3259:10.2307/1970730
3244:
3203:
3182:
3163:
3133:10.2307/1971057
3111:
3105:
3101:
3057:
3056:
3033:
3029:
3021:
3017:
3012:
2998:Central carrier
2992:
2985:
2930:
2910:Krieger factors
2898:crossed product
2886:if in addition
2885:
2877:
2828:
2824:
2817:
2802:
2783:If we have any
2752:
2746:
2723:
2719:
2693:
2626:
2619:
2615:
2586:
2585:
2572:
2567:
2565:
2559:
2558:
2553:
2540:
2535:
2528:
2527:
2518:
2517:
2509:
2506:
2505:
2500:
2492:
2484:
2449:
2424:
2420:
2411:
2407:
2398:
2394:
2380:
2377:
2376:
2365:
2357:Krieger factors
2346:
2342:
2262:
2250:
2246:
2242:
2238:
2234:
2230:
2226:
2222:
2120:has Schwartz's
2056:
2029:
2002:
1971:
1932:
1896:
1861:
1846:
1812:
1801:
1782:
1755:
1677:
1671:
1624:
1617:
1543:) are given by
1534:
1519:
1504:
1471:
1452:
1437:
1430:
1425:crossed product
1418:
1414:
1410:
1406:
1402:
1385:
1378:
1370:
1330:
1322:
1318:
1314:
1307:
1298:
1291:
1283:
1279:
1269:
1261:
1255:
1251:
1210:
1208:Type II factors
1203:
1199:
1188:cardinal number
1156:
1136:purely infinite
1103:
1090:direct integral
1070:
1052:
943:
936:
929:
922:
915:
908:
901:
894:
887:
880:
873:
866:
783:
746:"knows" about.
688:
636:locally compact
620:
614:
542:). Similarly,
527:
513:
510:
509:
506:direct integral
484:
383:
371:Takesaki (1979)
359:Schwartz (1967)
279:
276:
275:
258:
257:
255:
252:
251:
227:
226:
217:
216:
214:
211:
210:
184:
175:
171:
169:
166:
165:
132:
123:
119:
117:
114:
113:
97:shows that the
28:
17:
12:
11:
5:
6584:
6574:
6573:
6568:
6563:
6546:
6545:
6543:
6542:
6531:
6528:
6527:
6525:
6524:
6519:
6514:
6509:
6507:Choquet theory
6504:
6499:
6493:
6491:
6487:
6486:
6484:
6483:
6473:
6468:
6463:
6458:
6453:
6448:
6443:
6438:
6433:
6428:
6423:
6417:
6415:
6411:
6410:
6408:
6407:
6402:
6396:
6394:
6390:
6389:
6387:
6386:
6381:
6376:
6371:
6366:
6361:
6359:Banach algebra
6355:
6353:
6349:
6348:
6346:
6345:
6340:
6335:
6330:
6325:
6320:
6315:
6310:
6305:
6300:
6294:
6292:
6288:
6287:
6285:
6284:
6282:BanachâAlaoglu
6279:
6274:
6269:
6264:
6259:
6254:
6249:
6244:
6238:
6236:
6230:
6229:
6226:
6225:
6223:
6222:
6217:
6212:
6210:Locally convex
6207:
6193:
6188:
6182:
6180:
6176:
6175:
6173:
6172:
6167:
6162:
6157:
6152:
6147:
6142:
6137:
6132:
6127:
6121:
6115:
6111:
6110:
6094:
6093:
6086:
6079:
6071:
6062:
6061:
6059:
6058:
6053:
6048:
6043:
6038:
6032:
6030:
6026:
6025:
6023:
6022:
6010:
6005:
6001:
5997:
5994:
5991:
5988:
5976:
5971:
5970:
5969:
5959:
5957:Sequence space
5954:
5946:
5933:
5928:
5923:
5918:
5914:
5902:
5901:
5900:
5895:
5881:
5877:
5858:
5857:
5856:
5842:
5838:
5819:
5807:
5804:
5801:
5796:
5793:
5790:
5786:
5773:
5765:
5760:
5747:
5742:
5734:
5729:
5717:
5713:
5709:
5704:
5699:
5696:
5693:
5689:
5676:
5668:
5663:
5651:
5648:
5645:
5642:
5639:
5628:
5619:
5617:
5611:
5610:
5608:
5607:
5597:
5592:
5587:
5582:
5577:
5572:
5567:
5562:
5552:
5546:
5544:
5540:
5539:
5537:
5536:
5531:
5526:
5521:
5516:
5508:
5494:
5486:
5481:
5476:
5471:
5466:
5461:
5455:
5453:
5449:
5448:
5446:
5445:
5435:
5434:
5433:
5428:
5423:
5413:
5412:
5411:
5406:
5401:
5391:
5390:
5389:
5384:
5379:
5374:
5372:GelfandâPettis
5369:
5364:
5354:
5353:
5352:
5347:
5342:
5337:
5332:
5322:
5317:
5312:
5307:
5306:
5305:
5295:
5289:
5287:
5283:
5282:
5280:
5279:
5274:
5269:
5264:
5259:
5254:
5249:
5244:
5239:
5234:
5229:
5224:
5223:
5222:
5212:
5207:
5202:
5197:
5192:
5187:
5182:
5177:
5172:
5167:
5162:
5157:
5152:
5147:
5145:BanachâAlaoglu
5142:
5140:AndersonâKadec
5136:
5134:
5128:
5127:
5125:
5124:
5119:
5114:
5113:
5112:
5107:
5097:
5096:
5095:
5090:
5080:
5078:Operator space
5075:
5070:
5064:
5062:
5056:
5055:
5053:
5052:
5047:
5042:
5037:
5032:
5027:
5022:
5017:
5012:
5011:
5010:
5000:
4995:
4994:
4993:
4988:
4980:
4975:
4965:
4964:
4963:
4953:
4948:
4938:
4937:
4936:
4931:
4926:
4916:
4910:
4908:
4902:
4901:
4899:
4898:
4893:
4888:
4887:
4886:
4881:
4871:
4870:
4869:
4864:
4854:
4849:
4844:
4843:
4842:
4832:
4827:
4821:
4819:
4815:
4814:
4812:
4811:
4806:
4801:
4800:
4799:
4789:
4784:
4779:
4778:
4777:
4766:Locally convex
4763:
4762:
4761:
4751:
4746:
4741:
4735:
4733:
4729:
4728:
4726:
4725:
4718:Tensor product
4711:
4705:
4700:
4694:
4689:
4683:
4678:
4673:
4663:
4662:
4661:
4656:
4646:
4641:
4639:Banach lattice
4636:
4635:
4634:
4624:
4618:
4616:
4612:
4611:
4603:
4602:
4595:
4588:
4580:
4571:
4570:
4568:
4567:
4559:
4553:compact &
4538:
4536:
4532:
4531:
4529:
4528:
4523:
4518:
4513:
4508:
4503:
4498:
4496:Hermitian form
4493:
4488:
4482:
4480:
4476:
4475:
4473:
4472:
4462:
4457:
4451:
4449:
4445:
4444:
4442:
4441:
4436:
4431:
4425:
4423:
4419:
4418:
4416:
4415:
4410:
4405:
4396:
4387:
4381:
4379:
4378:Basic concepts
4375:
4374:
4371:Hilbert spaces
4367:
4366:
4359:
4352:
4344:
4335:
4334:
4332:
4331:
4326:
4321:
4316:
4311:
4306:
4301:
4296:
4291:
4286:
4281:
4276:
4271:
4266:
4261:
4256:
4246:
4244:Corona theorem
4241:
4235:
4233:
4229:
4228:
4226:
4225:
4223:Wiener algebra
4219:
4217:
4213:
4212:
4210:
4209:
4204:
4199:
4194:
4189:
4184:
4179:
4174:
4169:
4163:
4161:
4157:
4156:
4154:
4153:
4143:
4141:Pseudospectrum
4138:
4133:
4131:Dirac spectrum
4127:
4125:
4121:
4120:
4118:
4117:
4112:
4107:
4102:
4096:
4094:
4090:
4089:
4087:
4086:
4085:
4084:
4074:
4069:
4064:
4059:
4054:
4048:
4042:
4040:
4036:
4035:
4033:
4032:
4027:
4022:
4017:
4012:
4007:
4002:
3997:
3992:
3986:
3984:
3980:
3979:
3977:
3976:
3971:
3966:
3961:
3956:
3951:
3950:
3949:
3944:
3939:
3928:
3926:
3922:
3921:
3919:
3918:
3913:
3908:
3903:
3898:
3893:
3887:
3885:
3879:
3878:
3876:
3875:
3870:
3862:
3854:
3846:
3840:
3838:
3834:
3833:
3831:
3830:
3825:
3820:
3815:
3810:
3805:
3800:
3794:
3792:
3788:
3787:
3785:
3784:
3782:Operator space
3779:
3774:
3769:
3764:
3759:
3754:
3749:
3744:
3742:Banach algebra
3739:
3733:
3731:
3730:Basic concepts
3727:
3726:
3715:
3714:
3707:
3700:
3692:
3686:
3685:
3672:
3662:
3644:(2): 401â485,
3633:
3615:(4): 709â715,
3604:
3575:
3557:
3539:(1): 111â115,
3528:
3502:(1): 370â427,
3491:
3486:
3473:
3455:
3450:
3434:
3429:
3413:
3396:(1): 138â171,
3382:
3378:
3360:(4): 716â808,
3349:
3318:
3300:(1): 116â229,
3289:
3283:
3270:
3253:(2): 372â377,
3242:
3235:
3208:
3194:
3180:
3167:
3161:
3144:
3116:
3099:
3086:
3055:
3054:
3043:(2): 248â253.
3027:
3014:
3013:
3011:
3008:
3007:
3006:
3000:
2995:
2984:
2981:
2971:For instance,
2929:
2926:
2925:
2924:
2913:
2904:of Murray and
2894:
2891:
2883:
2875:
2836:
2781:
2778:
2750:
2745:
2742:
2738:ergodic theory
2721:
2717:
2706:Narutaka Ozawa
2692:
2689:
2625:
2622:
2620:that are not.
2617:
2613:
2610:
2609:
2598:
2595:
2590:
2581:
2578:
2575:
2571:
2566:
2564:
2561:
2560:
2557:
2554:
2549:
2546:
2543:
2539:
2534:
2533:
2531:
2524:
2521:
2516:
2513:
2498:
2495:Powers factors
2490:
2482:
2444:
2443:
2432:
2427:
2423:
2419:
2414:
2410:
2406:
2401:
2397:
2393:
2390:
2387:
2384:
2364:
2361:
2344:
2340:
2319:is diffuse or
2260:
2248:
2244:
2240:
2236:
2232:
2228:
2224:
2218:
2208:
2207:
2181:
2159:
2141:
2115:
2097:
2055:
2052:
2051:
2050:
2045:Type III: The
2043:
2027:
2024:
2019:of the module
2000:
1997:
1969:
1966:
1928:
1857:
1856:-dimension dim
1845:
1842:
1818:
1817:
1814:
1810:
1807:
1799:
1796:
1780:
1777:
1751:
1743:
1742:
1735:
1716:
1707:
1700:
1670:
1667:
1666:
1665:
1638:
1622:
1615:
1532:
1517:
1502:
1469:
1450:
1436:
1433:
1428:
1416:
1412:
1408:
1404:
1400:
1384:
1381:
1376:
1368:
1334:positive reals
1328:
1320:
1316:
1312:
1305:
1304:of the type II
1296:
1289:
1281:
1277:
1267:
1259:
1253:
1249:
1209:
1206:
1201:
1195:
1155:
1154:Type I factors
1152:
1151:
1150:
1139:
1128:
1101:
1069:
1066:
1050:
1049:) is the ||·||
941:
934:
927:
920:
913:
906:
899:
892:
885:
878:
871:
864:
782:
779:
687:
684:
628:measure spaces
616:Main article:
613:
610:
586:semihereditary
574:
573:
569:tensor product
565:
558:
547:
530:
526:
523:
520:
517:
495:
483:
480:
399:including the
382:
379:
355:Dixmier (1981)
304:Francis Murray
296:
295:
283:
261:
235:
230:
225:
220:
207:
191:
187:
183:
178:
174:
139:
135:
131:
126:
122:
87:ergodic theory
15:
9:
6:
4:
3:
2:
6583:
6572:
6569:
6567:
6564:
6562:
6559:
6558:
6556:
6541:
6533:
6532:
6529:
6523:
6520:
6518:
6515:
6513:
6512:Weak topology
6510:
6508:
6505:
6503:
6500:
6498:
6495:
6494:
6492:
6488:
6481:
6477:
6474:
6472:
6469:
6467:
6464:
6462:
6459:
6457:
6454:
6452:
6449:
6447:
6444:
6442:
6439:
6437:
6436:Index theorem
6434:
6432:
6429:
6427:
6424:
6422:
6419:
6418:
6416:
6412:
6406:
6403:
6401:
6398:
6397:
6395:
6393:Open problems
6391:
6385:
6382:
6380:
6377:
6375:
6372:
6370:
6367:
6365:
6362:
6360:
6357:
6356:
6354:
6350:
6344:
6341:
6339:
6336:
6334:
6331:
6329:
6326:
6324:
6321:
6319:
6316:
6314:
6311:
6309:
6306:
6304:
6301:
6299:
6296:
6295:
6293:
6289:
6283:
6280:
6278:
6275:
6273:
6270:
6268:
6265:
6263:
6260:
6258:
6255:
6253:
6250:
6248:
6245:
6243:
6240:
6239:
6237:
6235:
6231:
6221:
6218:
6216:
6213:
6211:
6208:
6205:
6201:
6197:
6194:
6192:
6189:
6187:
6184:
6183:
6181:
6177:
6171:
6168:
6166:
6163:
6161:
6158:
6156:
6153:
6151:
6148:
6146:
6143:
6141:
6138:
6136:
6133:
6131:
6128:
6126:
6123:
6122:
6119:
6116:
6112:
6107:
6103:
6099:
6092:
6087:
6085:
6080:
6078:
6073:
6072:
6069:
6057:
6054:
6052:
6049:
6047:
6044:
6042:
6039:
6037:
6034:
6033:
6031:
6027:
6021:
6003:
5999:
5995:
5992:
5986:
5977:
5975:
5972:
5968:
5965:
5964:
5963:
5960:
5958:
5955:
5953:
5952:
5947:
5945:
5931:
5926:
5916:
5912:
5903:
5899:
5896:
5894:
5875:
5866:
5865:
5864:
5863:
5859:
5855:
5836:
5827:
5826:
5825:
5824:
5820:
5818:
5794:
5791:
5788:
5784:
5774:
5772:
5771:
5766:
5764:
5761:
5759:
5757:
5753:
5748:
5746:
5743:
5741:
5740:
5735:
5733:
5730:
5728:
5702:
5697:
5694:
5691:
5687:
5677:
5675:
5674:
5669:
5667:
5664:
5662:
5640:
5637:
5629:
5627:
5626:
5621:
5620:
5618:
5616:
5612:
5606:
5602:
5598:
5596:
5593:
5591:
5588:
5586:
5583:
5581:
5578:
5576:
5575:Extreme point
5573:
5571:
5568:
5566:
5563:
5561:
5557:
5553:
5551:
5548:
5547:
5545:
5541:
5535:
5532:
5530:
5527:
5525:
5522:
5520:
5517:
5515:
5509:
5506:
5502:
5498:
5495:
5493:
5487:
5485:
5482:
5480:
5477:
5475:
5472:
5470:
5467:
5465:
5462:
5460:
5457:
5456:
5454:
5452:Types of sets
5450:
5443:
5439:
5436:
5432:
5429:
5427:
5424:
5422:
5419:
5418:
5417:
5414:
5410:
5407:
5405:
5402:
5400:
5397:
5396:
5395:
5392:
5388:
5385:
5383:
5380:
5378:
5375:
5373:
5370:
5368:
5365:
5363:
5360:
5359:
5358:
5355:
5351:
5348:
5346:
5343:
5341:
5338:
5336:
5333:
5331:
5328:
5327:
5326:
5323:
5321:
5318:
5316:
5315:Convex series
5313:
5311:
5310:Bochner space
5308:
5304:
5301:
5300:
5299:
5296:
5294:
5291:
5290:
5288:
5284:
5278:
5275:
5273:
5270:
5268:
5265:
5263:
5262:Riesz's lemma
5260:
5258:
5255:
5253:
5250:
5248:
5247:Mazur's lemma
5245:
5243:
5240:
5238:
5235:
5233:
5230:
5228:
5225:
5221:
5218:
5217:
5216:
5213:
5211:
5208:
5206:
5203:
5201:
5200:GelfandâMazur
5198:
5196:
5193:
5191:
5188:
5186:
5183:
5181:
5178:
5176:
5173:
5171:
5168:
5166:
5163:
5161:
5158:
5156:
5153:
5151:
5148:
5146:
5143:
5141:
5138:
5137:
5135:
5133:
5129:
5123:
5120:
5118:
5115:
5111:
5108:
5106:
5103:
5102:
5101:
5098:
5094:
5091:
5089:
5086:
5085:
5084:
5081:
5079:
5076:
5074:
5071:
5069:
5066:
5065:
5063:
5061:
5057:
5051:
5048:
5046:
5043:
5041:
5038:
5036:
5033:
5031:
5028:
5026:
5023:
5021:
5018:
5016:
5013:
5009:
5006:
5005:
5004:
5001:
4999:
4996:
4992:
4989:
4987:
4984:
4983:
4981:
4979:
4976:
4974:
4970:
4966:
4962:
4959:
4958:
4957:
4954:
4952:
4949:
4947:
4943:
4939:
4935:
4932:
4930:
4927:
4925:
4922:
4921:
4920:
4917:
4915:
4912:
4911:
4909:
4907:
4903:
4897:
4894:
4892:
4889:
4885:
4882:
4880:
4877:
4876:
4875:
4872:
4868:
4865:
4863:
4860:
4859:
4858:
4855:
4853:
4850:
4848:
4845:
4841:
4838:
4837:
4836:
4833:
4831:
4828:
4826:
4823:
4822:
4820:
4816:
4810:
4807:
4805:
4802:
4798:
4795:
4794:
4793:
4790:
4788:
4785:
4783:
4780:
4776:
4772:
4769:
4768:
4767:
4764:
4760:
4757:
4756:
4755:
4752:
4750:
4747:
4745:
4742:
4740:
4737:
4736:
4734:
4730:
4723:
4719:
4715:
4712:
4710:
4706:
4704:
4701:
4699:) convex
4698:
4695:
4693:
4690:
4688:
4684:
4682:
4679:
4677:
4674:
4672:
4668:
4664:
4660:
4657:
4655:
4652:
4651:
4650:
4647:
4645:
4644:Grothendieck
4642:
4640:
4637:
4633:
4630:
4629:
4628:
4625:
4623:
4620:
4619:
4617:
4613:
4608:
4601:
4596:
4594:
4589:
4587:
4582:
4581:
4578:
4566:
4565:
4560:
4558:
4556:
4552:
4548:
4544:
4540:
4539:
4537:
4533:
4527:
4524:
4522:
4519:
4517:
4514:
4512:
4509:
4507:
4504:
4502:
4499:
4497:
4494:
4492:
4489:
4487:
4484:
4483:
4481:
4477:
4470:
4466:
4463:
4461:
4458:
4456:
4453:
4452:
4450:
4448:Other results
4446:
4440:
4437:
4435:
4432:
4430:
4427:
4426:
4424:
4420:
4414:
4411:
4409:
4406:
4404:
4400:
4399:Hilbert space
4397:
4395:
4391:
4390:Inner product
4388:
4386:
4383:
4382:
4380:
4376:
4372:
4365:
4360:
4358:
4353:
4351:
4346:
4345:
4342:
4330:
4327:
4325:
4322:
4320:
4317:
4315:
4312:
4310:
4307:
4305:
4302:
4300:
4297:
4295:
4292:
4290:
4287:
4285:
4282:
4280:
4277:
4275:
4272:
4270:
4267:
4265:
4262:
4260:
4257:
4254:
4250:
4247:
4245:
4242:
4240:
4237:
4236:
4234:
4230:
4224:
4221:
4220:
4218:
4214:
4208:
4205:
4203:
4200:
4198:
4195:
4193:
4190:
4188:
4185:
4183:
4180:
4178:
4175:
4173:
4170:
4168:
4165:
4164:
4162:
4160:Miscellaneous
4158:
4151:
4147:
4144:
4142:
4139:
4137:
4134:
4132:
4129:
4128:
4126:
4122:
4116:
4113:
4111:
4108:
4106:
4103:
4101:
4098:
4097:
4095:
4091:
4083:
4080:
4079:
4078:
4075:
4073:
4070:
4068:
4065:
4063:
4060:
4058:
4055:
4053:
4049:
4047:
4044:
4043:
4041:
4037:
4031:
4028:
4026:
4023:
4021:
4018:
4016:
4013:
4011:
4008:
4006:
4003:
4001:
3998:
3996:
3993:
3991:
3988:
3987:
3985:
3981:
3975:
3972:
3970:
3967:
3965:
3962:
3960:
3957:
3955:
3952:
3948:
3945:
3943:
3940:
3938:
3935:
3934:
3933:
3930:
3929:
3927:
3925:Decomposition
3923:
3917:
3914:
3912:
3909:
3907:
3904:
3902:
3899:
3897:
3894:
3892:
3889:
3888:
3886:
3884:
3880:
3874:
3871:
3869:
3866:
3863:
3861:
3858:
3855:
3853:
3850:
3847:
3845:
3842:
3841:
3839:
3835:
3829:
3826:
3824:
3821:
3819:
3816:
3814:
3811:
3809:
3806:
3804:
3801:
3799:
3796:
3795:
3793:
3789:
3783:
3780:
3778:
3775:
3773:
3770:
3768:
3765:
3763:
3760:
3758:
3755:
3753:
3750:
3748:
3745:
3743:
3740:
3738:
3735:
3734:
3732:
3728:
3724:
3720:
3713:
3708:
3706:
3701:
3699:
3694:
3693:
3690:
3683:
3682:
3677:
3673:
3668:
3663:
3659:
3655:
3651:
3647:
3643:
3639:
3634:
3630:
3626:
3622:
3618:
3614:
3610:
3605:
3601:
3597:
3593:
3589:
3586:(1): 94â161,
3585:
3581:
3576:
3571:
3567:
3566:Compos. Math.
3563:
3558:
3554:
3550:
3546:
3542:
3538:
3534:
3529:
3525:
3521:
3517:
3513:
3509:
3505:
3501:
3497:
3492:
3489:
3487:3-540-42248-X
3483:
3479:
3474:
3471:
3467:
3466:
3461:
3456:
3453:
3451:0-677-00670-5
3447:
3443:
3439:
3435:
3432:
3430:3-540-63633-1
3426:
3422:
3418:
3414:
3411:
3407:
3403:
3399:
3395:
3391:
3387:
3383:
3375:
3371:
3367:
3363:
3359:
3355:
3350:
3346:
3342:
3337:
3332:
3328:
3324:
3319:
3315:
3311:
3307:
3303:
3299:
3295:
3290:
3286:
3284:0-8218-4219-6
3280:
3276:
3271:
3268:
3264:
3260:
3256:
3252:
3248:
3240:
3236:
3232:
3228:
3223:
3218:
3214:
3209:
3202:
3201:
3195:
3190:
3183:
3181:0-444-86308-7
3177:
3173:
3168:
3164:
3162:0-12-185860-X
3158:
3153:
3152:
3145:
3142:
3138:
3134:
3130:
3127:(1): 73â115,
3126:
3122:
3117:
3110:
3109:
3102:
3100:3-540-28486-9
3096:
3092:
3087:
3085:
3082:
3077:
3072:
3069:(1): 51â130,
3068:
3064:
3059:
3058:
3050:
3046:
3042:
3038:
3031:
3024:
3019:
3015:
3004:
3001:
2999:
2996:
2990:
2987:
2986:
2980:
2978:
2974:
2969:
2967:
2963:
2959:
2955:
2951:
2947:
2943:
2939:
2935:
2922:
2918:
2914:
2911:
2907:
2903:
2899:
2895:
2892:
2889:
2881:
2873:
2869:
2865:
2861:
2857:
2853:
2849:
2845:
2841:
2837:
2834:
2827:
2820:
2813:
2809:
2805:
2798:
2794:
2790:
2786:
2782:
2779:
2776:
2772:
2768:
2764:
2760:
2756:
2748:
2747:
2741:
2739:
2735:
2731:
2727:
2715:
2711:
2707:
2703:
2699:
2688:
2686:
2682:
2678:
2674:
2670:
2666:
2662:
2658:
2654:
2649:
2647:
2646:Vaughan Jones
2643:
2639:
2635:
2631:
2621:
2596:
2593:
2588:
2579:
2576:
2573:
2569:
2562:
2555:
2547:
2544:
2541:
2537:
2529:
2511:
2504:
2503:
2502:
2496:
2488:
2487:Powers (1967)
2480:
2479:ITPFI factors
2476:
2472:
2467:
2462:
2460:
2456:
2452:
2430:
2421:
2417:
2408:
2404:
2391:
2388:
2385:
2375:
2374:
2373:
2371:
2360:
2358:
2354:
2350:
2338:
2334:
2330:
2329:measure space
2326:
2322:
2318:
2314:
2310:
2307:
2306:measure space
2303:
2299:
2295:
2291:
2287:
2283:
2280:for a single
2279:
2275:
2271:
2270:
2264:
2258:
2257:Connes (1976)
2254:
2221:
2215:
2213:
2205:
2201:
2197:
2193:
2189:
2185:
2182:
2179:
2175:
2171:
2167:
2163:
2160:
2157:
2153:
2149:
2145:
2142:
2139:
2135:
2131:
2127:
2123:
2119:
2116:
2113:
2109:
2105:
2101:
2098:
2095:
2091:
2087:
2083:
2079:
2076:
2075:
2074:
2072:
2068:
2064:
2060:
2059:Connes (1976)
2048:
2044:
2042:-dimension â.
2041:
2037:
2033:
2025:
2022:
2018:
2014:
2010:
2006:
1998:
1995:
1991:
1987:
1983:
1979:
1975:
1967:
1964:
1960:
1956:
1952:
1948:
1944:
1940:
1937:finite): The
1936:
1931:
1926:
1925:
1924:
1922:
1919:The possible
1917:
1915:
1911:
1907:
1903:
1899:
1892:
1888:
1884:
1879:
1877:
1873:
1869:
1865:
1860:
1855:
1851:
1841:
1839:
1835:
1831:
1827:
1823:
1815:
1808:
1805:
1797:
1794:
1790:
1786:
1778:
1775:
1771:
1767:
1763:
1759:
1754:
1749:
1748:
1747:
1740:
1739:tracial state
1736:
1733:
1729:
1725:
1721:
1717:
1714:
1713:
1708:
1705:
1701:
1698:
1694:
1690:
1686:
1685:
1684:
1682:
1681:Takesaki 1979
1676:
1663:
1659:
1655:
1651:
1647:
1643:
1639:
1636:
1632:
1628:
1621:
1613:
1612:Dirac measure
1609:
1605:
1601:
1597:
1593:
1589:
1585:
1581:
1577:
1573:
1569:
1565:
1564:
1563:
1560:
1558:
1554:
1550:
1546:
1542:
1538:
1531:
1527:
1523:
1516:
1512:
1508:
1501:
1496:
1494:
1490:
1486:
1482:
1477:
1475:
1468:
1464:
1460:
1456:
1449:
1446:
1442:
1432:
1426:
1422:
1398:
1394:
1390:
1380:
1374:
1373:McDuff (1969)
1365:
1363:
1359:
1355:
1351:
1347:
1343:
1339:
1335:
1326:
1309:
1303:
1293:
1287:
1275:
1271:
1263:
1247:
1243:
1239:
1235:
1231:
1227:
1223:
1219:
1215:
1205:
1198:
1193:
1189:
1185:
1181:
1177:
1173:
1169:
1165:
1161:
1148:
1144:
1140:
1137:
1133:
1129:
1126:
1122:
1118:
1114:
1110:
1109:
1108:
1105:
1098:
1094:
1091:
1087:
1083:
1079:
1075:
1065:
1063:
1058:
1056:
1048:
1044:
1040:
1036:
1031:
1029:
1025:
1021:
1017:
1013:
1009:
1005:
1001:
997:
993:
988:
986:
982:
978:
973:
971:
967:
963:
959:
955:
951:
947:
940:
933:
926:
919:
912:
905:
898:
891:
884:
877:
870:
863:
858:
856:
852:
848:
844:
840:
836:
832:
828:
824:
820:
816:
812:
808:
804:
800:
796:
792:
788:
778:
776:
772:
768:
764:
760:
756:
752:
747:
745:
741:
737:
733:
729:
725:
721:
717:
713:
709:
705:
701:
697:
693:
683:
681:
677:
673:
668:
666:
662:
658:
654:
650:
646:
645:
640:
637:
633:
629:
625:
619:
609:
607:
603:
599:
595:
591:
587:
583:
579:
570:
566:
563:
559:
556:
552:
548:
545:
521:
518:
515:
507:
503:
501:
496:
493:
489:
488:
487:
479:
477:
473:
469:
465:
460:
456:
452:
448:
443:
441:
437:
433:
429:
425:
420:
418:
414:
413:norm topology
410:
406:
402:
398:
394:
391:
390:weakly closed
386:
378:
376:
375:Connes (1994)
372:
368:
364:
360:
356:
352:
348:
343:
341:
337:
333:
329:
325:
321:
317:
313:
309:
305:
301:
281:
249:
208:
205:
176:
172:
164:
163:Hilbert space
160:
156:
153:
120:
111:
110:
109:
106:
104:
100:
96:
92:
88:
84:
80:
76:
71:
69:
65:
61:
57:
53:
52:Hilbert space
49:
45:
41:
37:
33:
26:
22:
21:ring operator
6502:Balanced set
6476:Distribution
6414:Applications
6383:
6267:KreinâMilman
6252:Closed graph
6029:Applications
5950:
5861:
5822:
5769:
5755:
5751:
5738:
5672:
5624:
5511:Linear cone
5504:
5500:
5489:Convex cone
5382:PaleyâWiener
5242:MackeyâArens
5232:KreinâMilman
5185:Closed range
5180:Closed graph
5150:BanachâMazur
5030:Self-adjoint
4934:sesquilinear
4667:Polynomially
4607:Banach space
4563:
4554:
4550:
4546:
4542:
4511:Self-adjoint
4422:Main results
4232:Applications
4076:
4062:Disk algebra
3916:Spectral gap
3791:Main results
3680:
3666:
3641:
3637:
3612:
3608:
3583:
3579:
3569:
3565:
3536:
3532:
3499:
3495:
3477:
3463:
3442:W-* Algebras
3441:
3423:, Springer,
3420:
3393:
3389:
3357:
3353:
3326:
3322:
3297:
3293:
3274:
3250:
3246:
3239:McDuff, Dusa
3212:
3199:
3188:
3171:
3150:
3124:
3120:
3107:
3093:, Springer,
3090:
3066:
3062:
3040:
3036:
3030:
3018:
2970:
2931:
2928:Applications
2909:
2901:
2887:
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2871:
2867:
2863:
2859:
2855:
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2839:
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2807:
2800:
2796:
2792:
2788:
2766:
2762:
2759:pathological
2754:
2726:free entropy
2701:
2694:
2679:. Important
2676:
2672:
2668:
2656:
2652:
2650:
2633:
2629:
2627:
2611:
2494:
2478:
2474:
2463:
2458:
2453:denotes the
2447:
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2372:states that
2369:
2366:
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2169:
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2161:
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2151:
2148:semidiscrete
2147:
2143:
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2133:
2129:
2125:
2121:
2117:
2111:
2103:
2099:
2093:
2089:
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2081:
2077:
2070:
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2057:
2046:
2039:
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2020:
2016:
2012:
2008:
2004:
1993:
1989:
1985:
1981:
1977:
1973:
1962:
1958:
1954:
1950:
1946:
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1938:
1934:
1929:
1920:
1918:
1913:
1909:
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1882:
1880:
1878:-dimension.
1875:
1871:
1867:
1863:
1858:
1853:
1849:
1847:
1833:
1829:
1825:
1821:
1819:
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1529:
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1506:
1499:
1498:The predual
1497:
1492:
1488:
1484:
1480:
1478:
1474:Sakai (1971)
1466:
1462:
1458:
1454:
1447:
1444:
1440:
1438:
1427:of a type II
1388:
1386:
1366:
1361:
1357:
1345:
1327:of a type II
1324:
1310:
1301:
1294:
1245:
1241:
1237:
1236:and satisfy
1229:
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1175:
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1171:
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826:
822:
814:
810:
802:
798:
794:
793:are called (
790:
784:
770:
766:
762:
758:
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750:
748:
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739:
735:
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727:
723:
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711:
707:
703:
699:
695:
691:
689:
669:
664:
660:
656:
652:
648:
643:
621:
594:AW*-algebras
590:Baer *-rings
575:
568:
561:
550:
543:
499:
491:
485:
475:
471:
467:
466:such that ||
455:Banach space
447:Sakai (1971)
444:
421:
387:
384:
367:Sakai (1971)
347:Jones (2003)
344:
307:
297:
209:The algebra
107:
72:
39:
35:
29:
6431:Heat kernel
6421:Hardy space
6328:Trace class
6242:HahnâBanach
6204:Topological
5750:Continuous
5585:Linear span
5570:Convex hull
5550:Affine hull
5409:holomorphic
5345:holomorphic
5325:Derivatives
5215:HahnâBanach
5155:BanachâSaks
5073:C*-algebras
5040:Trace class
5003:Functionals
4891:Ultrastrong
4804:Quasinormed
4521:Trace class
4259:Heat kernel
3959:Compression
3844:Isospectral
3025:ens-lyon.fr
2989:AW*-algebra
2934:knot theory
2906:von Neumann
2787:of a group
2477:factors or
2475:ArakiâWoods
2278:von Neumann
2108:derivations
2082:hyperfinite
1654:trace class
1435:The predual
1272:, found by
757:belongs to
718:is said to
708:projections
706:are called
686:Projections
680:Connes 1994
672:C*-algebras
632:C*-algebras
624:commutative
482:Terminology
417:C*-algebras
405:ultrastrong
381:Definitions
32:mathematics
6555:Categories
6364:C*-algebra
6179:Properties
5503:), and (Hw
5404:continuous
5340:functional
5088:C*-algebra
4973:Continuous
4835:Dual space
4809:Stereotype
4787:Metrizable
4714:Projective
3937:Continuous
3752:C*-algebra
3747:B*-algebra
3496:Math. Ann.
3245:factors",
3010:References
2973:C*-algebra
2698:Voiculescu
2339:finite (II
2337:equivalent
2335:admits an
2325:equivalent
2321:non-atomic
2166:property E
2122:property P
2071:equivalent
1889:such that
1730:) for all
1562:Examples:
1350:Sorin Popa
1132:semifinite
1121:continuous
799:equivalent
742:, or that
690:Operators
578:forgetting
424:involution
393:*-algebras
68:C*-algebra
40:W*-algebra
6338:Unbounded
6333:Transpose
6291:Operators
6220:Separable
6215:Reflexive
6200:Algebraic
6186:Barrelled
5962:Sobolev W
5905:Schwartz
5880:∞
5841:∞
5837:ℓ
5803:Ω
5789:λ
5647:Σ
5529:Symmetric
5464:Absorbing
5377:regulated
5357:Integrals
5210:Goldstine
5045:Transpose
4982:Fredholm
4852:Ultraweak
4840:Dual norm
4771:Seminorms
4739:Barrelled
4709:Injective
4697:Uniformly
4671:Reflexive
3723:-algebras
3524:121141866
3470:EMS Press
3417:Sakai, S.
3222:1307.4818
2638:subfactor
2574:λ
2570:λ
2542:λ
2515:↦
2455:commutant
2426:′
2418:⊗
2413:′
2400:′
2389:⊗
2327:to as a
2188:injective
1379:factors.
1232:that are
1010:(meaning
720:belong to
598:*-algebra
582:*-algebra
562:generated
555:separable
551:separable
525:→
516:τ
457:, is the
432:commutant
428:commutant
409:ultraweak
125:∞
112:The ring
103:algebraic
44:*-algebra
6540:Category
6352:Algebras
6234:Theorems
6191:Complete
6160:Schwartz
6106:glossary
5898:weighted
5768:Hilbert
5745:Bs space
5615:Examples
5580:Interior
5556:Relative
5534:Zonotope
5513:(subset)
5491:(subset)
5442:Strongly
5421:Lebesgue
5416:Measures
5286:Analysis
5132:Theorems
5083:Spectrum
5008:positive
4991:operator
4929:operator
4919:Bilinear
4884:operator
4867:operator
4847:Operator
4744:Complete
4692:Strictly
4535:Examples
4324:Weyl law
4269:Lax pair
4216:Examples
4050:With an
3969:Discrete
3947:Residual
3883:Spectrum
3868:operator
3860:operator
3852:operator
3767:Spectrum
3678:(1991),
3440:(1967),
3419:(1971),
2983:See also
2921:groupoid
2744:Examples
2702:disjoint
2630:bimodule
2323:, it is
2212:amenable
2104:amenable
2069:are all
1883:standard
1660:||= Tr(|
1389:type III
1387:Lastly,
1360:by SL(2,
1308:factor.
1264:and the
1113:discrete
1022:≠
1014:≤
916:, then
99:analytic
54:that is
6343:Unitary
6323:Nuclear
6308:Compact
6303:Bounded
6298:Adjoint
6272:Minâmax
6165:Sobolev
6150:Nuclear
6140:Hilbert
6135:Fréchet
6100: (
5763:Hardy H
5666:c space
5603:)
5558:)
5479:Bounded
5367:Dunford
5362:Bochner
5335:Gateaux
5330:Fréchet
5105:of ODEs
5050:Unitary
5025:Nuclear
4956:Compact
4946:Bounded
4914:Adjoint
4754:Fréchet
4749:F-space
4720: (
4716:)
4669:)
4649:Hilbert
4622:Asplund
4549:) with
4526:Unitary
4385:Adjoint
3865:Unitary
3658:1969463
3629:1969106
3600:1968823
3553:1968692
3504:Bibcode
3410:1970364
3374:1969107
3345:1989620
3314:1968693
3267:1970730
3227:Bibcode
3141:1971057
3084:0244773
2716:type II
2659:is any
2282:ergodic
2168:or the
2026:Type II
1999:Type II
1809:Type II
1798:Type II
1764:, ....,
1445:predual
1280:and II
1214:type II
1068:Factors
572:spaces.
451:predual
246:of all
58:in the
6318:Normal
6155:Orlicz
6145:Hölder
6125:Banach
6114:Spaces
6102:topics
5679:Besov
5519:Radial
5484:Convex
5469:Affine
5438:Weakly
5431:Vector
5303:bundle
5093:radius
5020:Normal
4986:kernel
4951:Closed
4874:Strong
4792:Normed
4782:Mackey
4627:Banach
4609:topics
4506:Normal
3849:Normal
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3114:, 2013
3097:
2964:, and
2712:yield
2642:Connes
2446:where
2313:atomic
2274:Murray
2030:: The
2003:: The
1992:; its
1968:Type I
1927:Type I
1779:Type I
1776:or 1).
1750:Type I
1699:) to .
1689:weight
1489:normal
1443:has a
1338:Connes
1336:, but
1270:factor
1262:factor
1160:type I
1143:finite
1084:. As
1082:factor
1078:center
1076:whose
968:. The
596:. The
500:finite
492:factor
401:strong
93:. His
56:closed
6130:Besov
5754:with
5601:Quasi
5595:Polar
5399:Borel
5350:quasi
4879:polar
4862:polar
4676:Riesz
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3942:Point
3654:JSTOR
3625:JSTOR
3596:JSTOR
3549:JSTOR
3520:S2CID
3406:JSTOR
3370:JSTOR
3341:JSTOR
3310:JSTOR
3263:JSTOR
3217:arXiv
3204:(PDF)
3137:JSTOR
3112:(PDF)
2714:prime
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1783:: 0,
1756:: 0,
1720:trace
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843:E=uu*
474:|| ||
470:||=||
50:on a
42:is a
6478:(or
6196:Dual
5752:C(K)
5387:weak
4924:form
4857:Weak
4830:Dual
4797:norm
4759:tame
4632:list
4479:Maps
4401:and
4392:and
3873:Unit
3721:and
3482:ISBN
3446:ISBN
3425:ISBN
3279:ISBN
3176:ISBN
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3095:ISBN
2908:and
2896:The
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2134:uTu*
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1949:, 2/
1945:, 1/
1228:and
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1018:and
958:u*Eu
902:and
874:and
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634:and
592:and
567:The
478:||.
459:dual
415:are
365:and
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332:1943
328:1940
324:1938
320:1943
316:1937
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3646:doi
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