4694:
2260:
The theory of
Hilbert algebras can be used to deduce the commutation theorems of Murray and von Neumann; equally well the main results on Hilbert algebras can also be deduced directly from the commutation theorems for traces. The theory of Hilbert algebras was generalised by Takesaki as a tool for proving commutation theorems for semifinite weights in
2259:
naturally lead to examples of
Hilbert algebras. Every von Neumann algebra endowed with a semifinite trace has a canonical "completed" or "full" Hilbert algebra associated with it; and conversely a completed Hilbert algebra of exactly this form can be canonically associated with every Hilbert algebra.
991:
2939:
1929:
883:
838:
1680:
3001:
3292:
2134:
is again valid in this case. This result can be proved directly by a variety of methods, but follows immediately from the result for finite traces, by repeated use of the following elementary fact:
1766:
1354:
2856:
1999:
1064:
1514:
406:
1475:
639:
3520:
315:
1276:
1400:
878:
702:
3402:
3378:
3342:
3318:
3147:
3115:
3054:
2780:
2748:
2443:
2389:
2358:
2298:
2125:
459:
3456:
2247:
The theory of
Hilbert algebras was introduced by Godement (under the name "unitary algebras"), Segal and Dixmier to formalize the classical method of defining the trace for
529:
1433:
1153:
2049:
2657:
2591:
2504:
2875:
3534:
There is a one-one correspondence between von
Neumann algebras on H with faithful semifinite trace and full Hilbert algebras with Hilbert space completion H.
715:
1850:
4583:
4419:
4246:
986:{\displaystyle \lambda (\Gamma )^{\prime \prime }=\rho (\Gamma )^{\prime },\,\,\rho (\Gamma )^{\prime \prime }=\lambda (\Gamma )^{\prime }.}
1608:
1552:
is required when there is no invariant measure in the equivalence class, even though the equivalence class of the measure is preserved by
534:
an orthogonal direct sum for the real part of the inner product. This is just the real orthogonal decomposition for the ±1 eigenspaces of
1301:
4738:
4409:
101:
3250:
1000:
4733:
4723:
4536:
4391:
2954:
4367:
3827:
3485:
1211:
1710:
1373:
4259:
2788:
4348:
4239:
4216:
4198:
4180:
4105:
3932:
3907:
3881:
3848:
3806:
1940:
40:
3414:
4618:
2455:
The
HilbertâSchmidt operators on an infinite-dimensional Hilbert space form a Hilbert algebra with inner product (
1480:
4263:
1192:, Ό) be a probability space and let Πbe a countable discrete group of measure-preserving transformations of (
369:
124:
1438:
605:
4718:
4414:
128:
278:
4697:
4470:
4404:
4232:
2252:
1528:
One of the most important cases of the groupâmeasure space construction is when Î is the group of integers
1124:
88:
and other closely related representations. In particular this framework led to an abstract version of the
4434:
1091:
661:
One of the simplest cases of the commutation theorem, where it can easily be seen directly, is that of a
234:
847:
671:
4679:
4633:
4557:
4439:
3560:
2261:
2242:
1549:
136:
4674:
4490:
3822:, Lecture Notes in Mathematics, vol. (AlgĂšbres d'OpĂ©rateurs), Springer-Verlag, pp. 19â143,
3383:
3359:
3323:
3299:
3128:
3096:
3020:
2761:
2729:
2424:
2370:
2339:
2279:
2091:
425:
3945:(1951), "Mémoire sur la théorie des caractÚres dans les groupes localement compacts unimodulaires",
4526:
4424:
4327:
2214:
496:
4728:
4623:
4399:
1545:
1106:
1540:
must preserve the probability measure Ό. Semifinite traces are required to handle the case when
1412:
4654:
4598:
4562:
2256:
709:
705:
97:
85:
74:
4114:
Rieffel, M.A.; van Daele, A. (1977), "A bounded operator approach to TomitaâTakesaki theory",
2018:
4361:
4175:, London Mathematical Society Lecture Note Series, vol. 35, Cambridge University Press,
4061:
4024:
3989:
2562:
81:
47:
4357:
4637:
2934:{\displaystyle \lambda ({\mathfrak {A}})^{\prime \prime }=\rho ({\mathfrak {A}})^{\prime }}
2642:
2576:
2489:
2248:
1188:, it was one of von Neumann's original motivations for studying von Neumann algebras. Let (
4224:
8:
4603:
4541:
4255:
3555:
3550:
666:
160:
67:
32:
4628:
4495:
4156:
4085:
4050:
4013:
3978:
230:
89:
1121:, Ό) and the constant function 1 is a cyclic-separating trace vector. It follows that
4608:
4212:
4194:
4176:
4101:
3928:
3903:
3877:
3844:
3823:
3802:
3074:
The proof relies on the notion of "bounded elements" in the
Hilbert space completion
1576:
1083:
3836:
4613:
4531:
4500:
4480:
4465:
4460:
4455:
4148:
4123:
4077:
4065:
4040:
4028:
4005:
3993:
3970:
3247:
The commutation theorem follows immediately from the last assertion. In particular
1924:{\displaystyle M_{0}=\left\{a\in M\mid \tau \left(a^{*}a\right)<\infty \right\}}
1784:
1174:
59:
51:
4292:
4475:
4429:
4377:
4372:
4343:
3916:
3891:
3869:
3857:
3633:
2528:
is a von
Neumann algebra with faithful semifinite trace Ï, then the *-subalgebra
1367:
113:
4302:
3921:
3896:
1821:
If in addition Ï is non-zero on every non-zero projection, then Ï is called a
4664:
4516:
4317:
3958:
3942:
1516:
is a cyclic-separating trace vector. Moreover the modular conjugation operator
1357:
1185:
1070:
109:
63:
55:
4712:
4669:
4593:
4322:
4307:
4297:
152:
36:
4128:
4659:
4312:
4282:
4136:
3862:
Les algÚbres d'opérateurs dans l'espace hilbertien: algÚbres de von
Neumann
3815:
2754:
denote the extension of the involution to a conjugate-linear involution of
2620:
2559:
662:
105:
93:
78:
139:
was developed, heralding a new era in the theory of von
Neumann algebras.
4588:
4578:
4485:
4287:
4168:
132:
20:
4100:, London Mathematical Society Monographs, vol. 14, Academic Press,
2865:. In this case the commutation theorem for Hilbert algebras states that
229:. It is called a trace vector because the last condition means that the
4521:
4353:
4160:
4089:
4054:
4017:
3982:
2713:
Any dense *-subalgebra of a
Hilbert algebra is also a Hilbert algebra.
3799:
Operator Algebras and Quantum Statistical Mechanics 1, Second Edition
1067:
833:{\displaystyle (\lambda (g)f)(x)=f(g^{-1}x),\,\,(\rho (g)f)(x)=f(xg)}
192:
28:
4152:
4081:
4045:
4009:
3974:
1675:{\displaystyle \tau (\lambda a+\mu b)=\lambda \tau (a)+\mu \tau (b)}
3628:
can be identified with the space of square integrable functions on
54:
in the 1930s and applies to the von Neumann algebra generated by a
1798:Ï is completely additive on orthogonal families of projections in
1532:, i.e. the case of a single invertible measurable transformation
1844:, Ï) be the Hilbert space completion of the inner product space
1184:
The third class of examples combines the above two. Coming from
4139:(1953), "A non-commutative extension of abstract integration",
1066:
Exactly the same results remain true if Î is allowed to be any
123:
It was not until the late 1960s, prompted partly by results in
1813:
is as orthogonal direct sum of projections with finite trace (
1196:, Ό). The group therefore acts unitarily on the Hilbert space
2996:{\displaystyle M=\lambda ({\mathfrak {A}})^{\prime \prime },}
2758:. Define a representation λ and an anti-representation Ï of
1559:
4254:
3528:
3014:
2869:
2085:
419:
3645:
It should not be confused with the von Neumann algebra on
16:
Identifies the commutant of a specific von Neumann algebra
3961:(1954), "Théorie des caractÚres. I. AlgÚbres unitaires",
3478:) and â otherwise, yields a faithful semifinite trace on
2674:) is a Hilbert algebra with the usual inner product from
2600:) is a Hilbert algebra with the usual inner product from
2513:) is a Hilbert algebra with the usual inner product from
1073:. The von Neumann algebra λ(Î)' ' is usually called the
142:
2535:
defined above is a Hilbert algebra with inner product (
2264:; they can be dispensed with when dealing with states.
2081:= I. The commutation theorem of Murray and von Neumann
3161:). In this case it is straightforward to prove that:
3006:
the von Neumann algebra generated by the operators λ(
2985:
2900:
953:
904:
3488:
3417:
3386:
3362:
3326:
3302:
3253:
3131:
3099:
3023:
2957:
2878:
2791:
2764:
2732:
2645:
2579:
2492:
2427:
2373:
2342:
2282:
2094:
2021:
1943:
1853:
1713:
1611:
1483:
1441:
1415:
1376:
1304:
1214:
1127:
1003:
886:
850:
718:
674:
608:
499:
428:
372:
281:
468:
One of the easiest ways to see this is to introduce
112:. Their work was put in final form in the 1950s by
4584:Spectral theory of ordinary differential equations
3920:
3895:
3514:
3450:
3396:
3372:
3336:
3312:
3286:
3141:
3109:
3048:
2995:
2933:
2850:
2774:
2742:
2651:
2585:
2498:
2437:
2383:
2352:
2292:
2119:
2043:
1993:
1923:
1761:{\displaystyle \tau \left(uau^{*}\right)=\tau (a)}
1760:
1674:
1544:(or more generally Î) only preserves an infinite
1508:
1469:
1427:
1394:
1349:{\displaystyle H_{1}=H\otimes \ell ^{2}(\Gamma ),}
1348:
1270:
1147:
1058:
985:
872:
832:
696:
633:
523:
453:
400:
309:
73:Another important application is in the theory of
4060:
4023:
3988:
2851:{\displaystyle \lambda (a)x=ax,\,\,\rho (a)x=xa.}
4710:
4113:
3796:
3747:
3593:
3578:
2861:These actions extend continuously to actions on
1286:and normalises the Abelian von Neumann algebra
3820:Sur la thĂ©orie non commutative de lâintĂ©gration
3681:
3679:
2478:, Ό) is an infinite measure space, the algebra
1994:{\displaystyle (a,b)=\tau \left(b^{*}a\right).}
3287:{\displaystyle M=\lambda ({\mathfrak {B}})''.}
2073:and extends to a conjugate-linear isometry of
1059:{\displaystyle Jf(g)={\overline {f(g^{-1})}}.}
4240:
4039:(2), American Mathematical Society: 208â248,
92:for unimodular locally compact groups due to
3676:
2782:on itself by left and right multiplication:
1509:{\displaystyle \Omega =1\otimes \delta _{1}}
1402:is defined to be the von Neumann algebra on
249:-module. It is called separating because if
3607:
3605:
3603:
3601:
3344:as a dense *-subalgebra. It is said to be
3063:These results were proved independently by
1082:Another important example is provided by a
84:, where the theory has been applied to the
4247:
4233:
2750:with respect to the inner product and let
2012:and can be identified with its image. Let
102:Plancherel theorem for spherical functions
4127:
4098:C* algebras and their automorphism groups
4044:
3835:
3770:
3589:
3587:
2820:
2819:
1560:Commutation theorem for semifinite traces
935:
934:
880:and the commutation theorem implies that
781:
780:
401:{\displaystyle JMJ\subseteq M^{\prime }.}
4537:Group algebra of a locally compact group
4206:
4188:
4095:
3957:
3941:
3898:Les C*-algÚbres et leurs représentations
3781:
3736:
3696:
3685:
3598:
3064:
1470:{\displaystyle U_{g}\otimes \lambda (g)}
634:{\displaystyle JM^{\prime }J\subseteq M}
241:. It is called cyclic since Ω generates
3915:
3890:
3868:
3856:
3758:
3611:
328:defines a conjugate-linear isometry of
4711:
3814:
3671:
3584:
3515:{\displaystyle M_{0}={\mathfrak {B}}.}
2155:are two von Neumann algebras such that
583:In particular Ω is a trace vector for
415:of Murray and von Neumann states that
310:{\displaystyle Ja\Omega =a^{*}\Omega }
4228:
4167:
4147:(3), Annals of Mathematics: 401â457,
4135:
4076:(4), Annals of Mathematics: 716â808,
4004:(1), Annals of Mathematics: 116â229,
3797:Bratteli, O.; Robinson, D.W. (1987),
3708:
3068:
2308:* and an inner product (,) such that
1271:{\displaystyle U_{g}f(x)=f(g^{-1}x),}
486:denotes the self-adjoint elements in
233:corresponding to Ω defines a tracial
143:Commutation theorem for finite traces
4068:(1943), "On rings of operators IV",
4031:(1937), "On rings of operators II",
135:, that the more general non-tracial
46:The first such result was proved by
4173:Trace ideals and their applications
3969:(1), Annals of Mathematics: 47â62,
3504:
3389:
3365:
3329:
3320:forms a Hilbert algebra containing
3305:
3268:
3190:is given by the bounded operator Ï(
3168:is also a bounded element, denoted
3134:
3102:
2972:
2916:
2887:
2767:
2735:
2726:be the Hilbert space completion of
2430:
2376:
2345:
2285:
2236:
1548:measure; and the full force of the
665:Î acting on the finite-dimensional
472:, the closure of the real subspace
13:
3296:The space of all bounded elements
3153:extends to a bounded operator on
3041:
2982:
2926:
2897:
2394:* is the adjoint, in other words (
2112:
1934:with respect to the inner product
1913:
1484:
1395:{\displaystyle M=A\rtimes \Gamma }
1389:
1337:
1173:), the von Neumann algebra of all
975:
967:
950:
942:
926:
918:
901:
893:
873:{\displaystyle \ell ^{2}(\Gamma )}
864:
697:{\displaystyle \ell ^{2}(\Gamma )}
688:
644:follows by reversing the roles of
617:
446:
390:
304:
288:
14:
4750:
4739:Theorems in representation theory
3996:(1936), "On rings of operators",
2698:) denotes the closed subspace of
4693:
4692:
4619:Topological quantum field theory
2417:the linear span of all products
1363:groupâmeasure space construction
351:It is immediately verified that
4734:Theorems in functional analysis
4724:Representation theory of groups
3775:
3764:
3752:
3741:
3730:
3713:
3397:{\displaystyle {\mathfrak {B}}}
3373:{\displaystyle {\mathfrak {B}}}
3337:{\displaystyle {\mathfrak {A}}}
3313:{\displaystyle {\mathfrak {B}}}
3142:{\displaystyle {\mathfrak {A}}}
3110:{\displaystyle {\mathfrak {A}}}
3049:{\displaystyle JMJ=M^{\prime }}
2775:{\displaystyle {\mathfrak {A}}}
2743:{\displaystyle {\mathfrak {A}}}
2438:{\displaystyle {\mathfrak {A}}}
2384:{\displaystyle {\mathfrak {A}}}
2363:left multiplication by a fixed
2353:{\displaystyle {\mathfrak {A}}}
2293:{\displaystyle {\mathfrak {A}}}
2257:representation theory of groups
2120:{\displaystyle JMJ=M^{\prime }}
2008:acts by left multiplication on
1524:' can be explicitly identified.
454:{\displaystyle JMJ=M^{\prime }}
167:with a unit vector Ω such that
4209:Theory of Operator Algebras II
3702:
3690:
3665:
3639:
3616:
3572:
3451:{\displaystyle \tau (x)=(a,a)}
3445:
3433:
3427:
3421:
3274:
3263:
2978:
2967:
2922:
2911:
2893:
2882:
2830:
2824:
2801:
2795:
1956:
1944:
1755:
1749:
1669:
1663:
1651:
1645:
1633:
1615:
1464:
1458:
1435:and the normalising operators
1340:
1334:
1262:
1243:
1234:
1228:
1208:, Ό) according to the formula
1044:
1028:
1016:
1010:
971:
964:
946:
939:
922:
915:
897:
890:
867:
861:
827:
818:
809:
803:
800:
794:
788:
782:
774:
755:
746:
740:
737:
731:
725:
719:
691:
685:
227:cyclic-separating trace vector
125:algebraic quantum field theory
25:commutation theorem for traces
1:
4415:Uniform boundedness principle
4191:Theory of Operator Algebras I
3841:Treatise on Analysis, Vol. II
3790:
3380:must actually already lie in
2717:
2449:
2267:
1568:be a von Neumann algebra and
599:'. So the opposite inclusion
524:{\displaystyle H=K\oplus iK,}
129:quantum statistical mechanics
3748:Bratteli & Robinson 1987
3719:Dixmier uses the adjectives
3594:Rieffel & van Daele 1977
3579:Bratteli & Robinson 1987
2071:modular conjugation operator
1828:If Ï is a faithful trace on
1048:
346:modular conjugation operator
7:
3544:
2187:for a family of projections
1092:Abelian von Neumann algebra
655:
10:
4755:
4558:Invariant subspace problem
2702:-biinvariant functions in
2623:, the convolution algebra
2565:, the convolution algebra
2240:
1428:{\displaystyle A\otimes I}
1162:maximal Abelian subalgebra
712:are given by the formulas
332:with square the identity,
27:explicitly identifies the
4688:
4647:
4571:
4550:
4509:
4448:
4390:
4336:
4278:
4271:
3632:x Î with respect to the
2253:HilbertâSchmidt operators
1409:generated by the algebra
1076:group von Neumann algebra
225:The vector Ω is called a
116:as part of the theory of
64:measurable transformation
4527:Spectrum of a C*-algebra
3566:
3216:' is generated by the Ï(
2215:strong operator topology
2044:{\displaystyle Ja=a^{*}}
2004:The von Neumann algebra
1107:multiplication operators
997:is given by the formula
538:. On the other hand for
359:commute on the subspace
272:It follows that the map
4624:Noncommutative geometry
4129:10.2140/pjm.1977.69.187
4096:Pedersen, G.K. (1979),
4033:Trans. Amer. Math. Soc.
3352:because any element in
1595:is a functional Ï from
1360:of Hilbert spaces. The
710:unitary representations
706:regular representations
568:is self-adjoint. Hence
564:Ω, Ω) is real, because
75:unitary representations
4680:TomitaâTakesaki theory
4655:Approximation property
4599:Calculus of variations
3561:TomitaâTakesaki theory
3516:
3452:
3404:. The functional Ï on
3398:
3374:
3338:
3314:
3288:
3143:
3111:
3050:
2997:
2935:
2852:
2776:
2744:
2653:
2587:
2500:
2439:
2391:is a bounded operator;
2385:
2354:
2294:
2262:TomitaâTakesaki theory
2255:. Applications in the
2243:TomitaâTakesaki theory
2121:
2045:
1995:
1925:
1762:
1676:
1550:TomitaâTakesaki theory
1510:
1471:
1429:
1396:
1350:
1272:
1149:
1060:
987:
874:
834:
704:by the left and right
698:
635:
525:
455:
402:
344:is usually called the
311:
137:TomitaâTakesaki theory
86:regular representation
82:locally compact groups
4675:BanachâMazur distance
4638:Generalized functions
4207:Takesaki, M. (2002),
4189:Takesaki, M. (1979),
3938:(English translation)
3887:(English translation)
3517:
3453:
3399:
3375:
3339:
3315:
3289:
3144:
3112:
3051:
2998:
2936:
2853:
2777:
2745:
2654:
2652:{\displaystyle \cap }
2588:
2586:{\displaystyle \cap }
2563:locally compact group
2501:
2499:{\displaystyle \cap }
2440:
2386:
2355:
2295:
2249:trace class operators
2122:
2046:
1996:
1926:
1763:
1677:
1511:
1472:
1430:
1397:
1351:
1273:
1150:
1148:{\displaystyle A'=A,}
1061:
988:
875:
835:
699:
636:
560:, the inner product (
526:
456:
403:
312:
131:due to the school of
48:Francis Joseph Murray
39:in the presence of a
4719:Von Neumann algebras
4420:Kakutani fixed-point
4405:Riesz representation
3947:J. Math. Pures Appl.
3902:, Gauthier-Villars,
3874:Von Neumann algebras
3486:
3415:
3384:
3360:
3356:bounded relative to
3324:
3300:
3251:
3129:
3097:
3021:
2955:
2876:
2789:
2762:
2730:
2643:
2577:
2490:
2425:
2371:
2340:
2280:
2092:
2069:is again called the
2019:
1941:
1851:
1711:
1609:
1481:
1439:
1413:
1374:
1370:von Neumann algebra
1302:
1212:
1125:
1001:
884:
848:
716:
672:
606:
497:
426:
370:
279:
4604:Functional calculus
4563:Mahler's conjecture
4542:Von Neumann algebra
4256:Functional analysis
4211:, Springer-Verlag,
4193:, Springer-Verlag,
3801:, Springer-Verlag,
3761:, Appendix A54âA61.
3556:Affiliated operator
3551:von Neumann algebra
2198:in the commutant of
1809:each projection in
1587:(or sometimes just
1583:. By definition, a
667:inner product space
413:commutation theorem
161:von Neumann algebra
98:Forrest Stinespring
68:probability measure
33:von Neumann algebra
4629:Riemann hypothesis
4328:Topological vector
3864:, Gauthier-Villars
3843:, Academic Press,
3699:, pp. 324â325
3653:and the operators
3512:
3448:
3394:
3370:
3334:
3310:
3284:
3139:
3107:
3046:
2993:
2931:
2848:
2772:
2740:
2649:
2583:
2496:
2435:
2381:
2350:
2290:
2117:
2041:
1991:
1921:
1793:unitary invariance
1758:
1672:
1577:positive operators
1506:
1467:
1425:
1392:
1346:
1268:
1145:
1056:
983:
870:
830:
694:
631:
521:
490:. It follows that
451:
398:
307:
265:Ω= (0), and hence
231:matrix coefficient
104:associated with a
90:Plancherel theorem
62:associated with a
4706:
4705:
4609:Integral operator
4386:
4385:
3927:, North Holland,
3876:, North Holland,
3829:978-3-540-09512-5
3540:
3539:
3059:
3058:
2944:
2943:
2130:
2129:
1175:bounded operators
1084:probability space
1051:
464:
463:
245:as a topological
183:' Ω is dense in
4746:
4696:
4695:
4614:Jones polynomial
4532:Operator algebra
4276:
4275:
4249:
4242:
4235:
4226:
4225:
4221:
4203:
4185:
4163:
4132:
4131:
4116:Pacific J. Math.
4110:
4092:
4057:
4048:
4020:
3985:
3954:
3937:
3926:
3912:
3901:
3886:
3865:
3853:
3832:
3811:
3785:
3784:, pp. 52â53
3779:
3773:
3768:
3762:
3756:
3750:
3745:
3739:
3734:
3728:
3717:
3711:
3706:
3700:
3694:
3688:
3683:
3674:
3669:
3663:
3643:
3637:
3620:
3614:
3609:
3596:
3591:
3582:
3581:, pp. 81â82
3576:
3529:
3521:
3519:
3518:
3513:
3508:
3507:
3498:
3497:
3457:
3455:
3454:
3449:
3403:
3401:
3400:
3395:
3393:
3392:
3379:
3377:
3376:
3371:
3369:
3368:
3343:
3341:
3340:
3335:
3333:
3332:
3319:
3317:
3316:
3311:
3309:
3308:
3293:
3291:
3290:
3285:
3280:
3272:
3271:
3148:
3146:
3145:
3140:
3138:
3137:
3116:
3114:
3113:
3108:
3106:
3105:
3055:
3053:
3052:
3047:
3045:
3044:
3015:
3002:
3000:
2999:
2994:
2989:
2988:
2976:
2975:
2940:
2938:
2937:
2932:
2930:
2929:
2920:
2919:
2904:
2903:
2891:
2890:
2870:
2857:
2855:
2854:
2849:
2781:
2779:
2778:
2773:
2771:
2770:
2749:
2747:
2746:
2741:
2739:
2738:
2658:
2656:
2655:
2650:
2592:
2590:
2589:
2584:
2505:
2503:
2502:
2497:
2444:
2442:
2441:
2436:
2434:
2433:
2390:
2388:
2387:
2382:
2380:
2379:
2359:
2357:
2356:
2351:
2349:
2348:
2300:with involution
2299:
2297:
2296:
2291:
2289:
2288:
2237:Hilbert algebras
2126:
2124:
2123:
2118:
2116:
2115:
2086:
2050:
2048:
2047:
2042:
2040:
2039:
2000:
1998:
1997:
1992:
1987:
1983:
1979:
1978:
1930:
1928:
1927:
1922:
1920:
1916:
1909:
1905:
1901:
1900:
1863:
1862:
1785:unitary operator
1767:
1765:
1764:
1759:
1742:
1738:
1737:
1736:
1703:
1702:
1681:
1679:
1678:
1673:
1602:into such that
1585:semifinite trace
1515:
1513:
1512:
1507:
1505:
1504:
1476:
1474:
1473:
1468:
1451:
1450:
1434:
1432:
1431:
1426:
1401:
1399:
1398:
1393:
1355:
1353:
1352:
1347:
1333:
1332:
1314:
1313:
1277:
1275:
1274:
1269:
1258:
1257:
1224:
1223:
1154:
1152:
1151:
1146:
1135:
1065:
1063:
1062:
1057:
1052:
1047:
1043:
1042:
1023:
992:
990:
989:
984:
979:
978:
957:
956:
930:
929:
908:
907:
879:
877:
876:
871:
860:
859:
839:
837:
836:
831:
770:
769:
703:
701:
700:
695:
684:
683:
640:
638:
637:
632:
621:
620:
591:is unaltered if
572:is unaltered if
530:
528:
527:
522:
460:
458:
457:
452:
450:
449:
420:
407:
405:
404:
399:
394:
393:
316:
314:
313:
308:
303:
302:
118:Hilbert algebras
100:and an abstract
60:dynamical system
52:John von Neumann
4754:
4753:
4749:
4748:
4747:
4745:
4744:
4743:
4709:
4708:
4707:
4702:
4684:
4648:Advanced topics
4643:
4567:
4546:
4505:
4471:HilbertâSchmidt
4444:
4435:GelfandâNaimark
4382:
4332:
4267:
4253:
4219:
4201:
4183:
4153:10.2307/1969729
4108:
4082:10.2307/1969107
4066:von Neumann, J.
4046:10.2307/1989620
4029:von Neumann, J.
4010:10.2307/1968693
3994:von Neumann, J.
3975:10.2307/1969832
3935:
3910:
3884:
3851:
3830:
3809:
3793:
3788:
3780:
3776:
3769:
3765:
3757:
3753:
3746:
3742:
3735:
3731:
3718:
3714:
3707:
3703:
3695:
3691:
3684:
3677:
3670:
3666:
3661:
3644:
3640:
3634:product measure
3627:
3621:
3617:
3610:
3599:
3592:
3585:
3577:
3573:
3569:
3547:
3503:
3502:
3493:
3489:
3487:
3484:
3483:
3416:
3413:
3412:
3410:
3388:
3387:
3385:
3382:
3381:
3364:
3363:
3361:
3358:
3357:
3328:
3327:
3325:
3322:
3321:
3304:
3303:
3301:
3298:
3297:
3273:
3267:
3266:
3252:
3249:
3248:
3157:, denoted by λ(
3133:
3132:
3130:
3127:
3126:
3101:
3100:
3098:
3095:
3094:
3065:Godement (1954)
3040:
3036:
3022:
3019:
3018:
2981:
2977:
2971:
2970:
2956:
2953:
2952:
2925:
2921:
2915:
2914:
2896:
2892:
2886:
2885:
2877:
2874:
2873:
2790:
2787:
2786:
2766:
2765:
2763:
2760:
2759:
2734:
2733:
2731:
2728:
2727:
2720:
2644:
2641:
2640:
2578:
2575:
2574:
2534:
2491:
2488:
2487:
2452:
2429:
2428:
2426:
2423:
2422:
2375:
2374:
2372:
2369:
2368:
2344:
2343:
2341:
2338:
2337:
2284:
2283:
2281:
2278:
2277:
2274:Hilbert algebra
2270:
2245:
2239:
2231:
2224:
2205:
2196:
2185:
2179:
2170:
2164:
2153:
2146:
2111:
2107:
2093:
2090:
2089:
2065:. The operator
2064:
2035:
2031:
2020:
2017:
2016:
1974:
1970:
1969:
1965:
1942:
1939:
1938:
1896:
1892:
1891:
1887:
1871:
1867:
1858:
1854:
1852:
1849:
1848:
1778:
1732:
1728:
1721:
1717:
1712:
1709:
1708:
1700:
1699:
1696:
1610:
1607:
1606:
1601:
1574:
1562:
1500:
1496:
1482:
1479:
1478:
1446:
1442:
1440:
1437:
1436:
1414:
1411:
1410:
1408:
1375:
1372:
1371:
1368:crossed product
1328:
1324:
1309:
1305:
1303:
1300:
1299:
1250:
1246:
1219:
1215:
1213:
1210:
1209:
1128:
1126:
1123:
1122:
1035:
1031:
1024:
1022:
1002:
999:
998:
974:
970:
949:
945:
925:
921:
900:
896:
885:
882:
881:
855:
851:
849:
846:
845:
762:
758:
717:
714:
713:
708:λ and Ï. These
679:
675:
673:
670:
669:
658:
616:
612:
607:
604:
603:
595:is replaced by
576:is replaced by
559:
548:
498:
495:
494:
485:
478:
445:
441:
427:
424:
423:
389:
385:
371:
368:
367:
340:. The operator
298:
294:
280:
277:
276:
174:Ω is dense in
145:
114:Jacques Dixmier
17:
12:
11:
5:
4752:
4742:
4741:
4736:
4731:
4729:Ergodic theory
4726:
4721:
4704:
4703:
4701:
4700:
4689:
4686:
4685:
4683:
4682:
4677:
4672:
4667:
4665:Choquet theory
4662:
4657:
4651:
4649:
4645:
4644:
4642:
4641:
4631:
4626:
4621:
4616:
4611:
4606:
4601:
4596:
4591:
4586:
4581:
4575:
4573:
4569:
4568:
4566:
4565:
4560:
4554:
4552:
4548:
4547:
4545:
4544:
4539:
4534:
4529:
4524:
4519:
4517:Banach algebra
4513:
4511:
4507:
4506:
4504:
4503:
4498:
4493:
4488:
4483:
4478:
4473:
4468:
4463:
4458:
4452:
4450:
4446:
4445:
4443:
4442:
4440:BanachâAlaoglu
4437:
4432:
4427:
4422:
4417:
4412:
4407:
4402:
4396:
4394:
4388:
4387:
4384:
4383:
4381:
4380:
4375:
4370:
4368:Locally convex
4365:
4351:
4346:
4340:
4338:
4334:
4333:
4331:
4330:
4325:
4320:
4315:
4310:
4305:
4300:
4295:
4290:
4285:
4279:
4273:
4269:
4268:
4252:
4251:
4244:
4237:
4229:
4223:
4222:
4217:
4204:
4199:
4186:
4181:
4165:
4133:
4111:
4106:
4093:
4058:
4021:
3986:
3955:
3939:
3933:
3913:
3908:
3888:
3882:
3866:
3854:
3849:
3833:
3828:
3812:
3807:
3792:
3789:
3787:
3786:
3774:
3771:Dieudonné 1976
3763:
3751:
3740:
3729:
3712:
3701:
3689:
3675:
3664:
3657:
3638:
3625:
3615:
3597:
3583:
3570:
3568:
3565:
3564:
3563:
3558:
3553:
3546:
3543:
3542:
3541:
3538:
3537:
3511:
3506:
3501:
3496:
3492:
3447:
3444:
3441:
3438:
3435:
3432:
3429:
3426:
3423:
3420:
3408:
3391:
3367:
3331:
3307:
3283:
3279:
3276:
3270:
3265:
3262:
3259:
3256:
3245:
3244:
3235:) commute for
3225:
3211:
3181:
3136:
3104:
3089:is said to be
3081:An element of
3061:
3060:
3057:
3056:
3043:
3039:
3035:
3032:
3029:
3026:
3004:
3003:
2992:
2987:
2984:
2980:
2974:
2969:
2966:
2963:
2960:
2946:
2945:
2942:
2941:
2928:
2924:
2918:
2913:
2910:
2907:
2902:
2899:
2895:
2889:
2884:
2881:
2859:
2858:
2847:
2844:
2841:
2838:
2835:
2832:
2829:
2826:
2823:
2818:
2815:
2812:
2809:
2806:
2803:
2800:
2797:
2794:
2769:
2737:
2719:
2716:
2715:
2714:
2711:
2648:
2609:
2582:
2552:
2532:
2522:
2495:
2472:
2451:
2448:
2447:
2446:
2432:
2415:
2392:
2378:
2361:
2347:
2287:
2276:is an algebra
2269:
2266:
2251:starting from
2238:
2235:
2234:
2233:
2229:
2222:
2203:
2192:
2183:
2175:
2168:
2160:
2151:
2144:
2132:
2131:
2128:
2127:
2114:
2110:
2106:
2103:
2100:
2097:
2062:
2052:
2051:
2038:
2034:
2030:
2027:
2024:
2002:
2001:
1990:
1986:
1982:
1977:
1973:
1968:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1932:
1931:
1919:
1915:
1912:
1908:
1904:
1899:
1895:
1890:
1886:
1883:
1880:
1877:
1874:
1870:
1866:
1861:
1857:
1823:faithful trace
1819:
1818:
1815:semifiniteness
1807:
1796:
1776:
1757:
1754:
1751:
1748:
1745:
1741:
1735:
1731:
1727:
1724:
1720:
1716:
1706:
1697:and λ, Ό ℠0 (
1694:
1671:
1668:
1665:
1662:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1620:
1617:
1614:
1599:
1572:
1561:
1558:
1526:
1525:
1520:and commutant
1503:
1499:
1495:
1492:
1489:
1486:
1466:
1463:
1460:
1457:
1454:
1449:
1445:
1424:
1421:
1418:
1406:
1391:
1388:
1385:
1382:
1379:
1358:tensor product
1345:
1342:
1339:
1336:
1331:
1327:
1323:
1320:
1317:
1312:
1308:
1267:
1264:
1261:
1256:
1253:
1249:
1245:
1242:
1239:
1236:
1233:
1230:
1227:
1222:
1218:
1186:ergodic theory
1182:
1144:
1141:
1138:
1134:
1131:
1080:
1071:discrete group
1055:
1050:
1046:
1041:
1038:
1034:
1030:
1027:
1021:
1018:
1015:
1012:
1009:
1006:
982:
977:
973:
969:
966:
963:
960:
955:
952:
948:
944:
941:
938:
933:
928:
924:
920:
917:
914:
911:
906:
903:
899:
895:
892:
889:
869:
866:
863:
858:
854:
829:
826:
823:
820:
817:
814:
811:
808:
805:
802:
799:
796:
793:
790:
787:
784:
779:
776:
773:
768:
765:
761:
757:
754:
751:
748:
745:
742:
739:
736:
733:
730:
727:
724:
721:
693:
690:
687:
682:
678:
657:
654:
642:
641:
630:
627:
624:
619:
615:
611:
557:
546:
532:
531:
520:
517:
514:
511:
508:
505:
502:
483:
476:
466:
465:
462:
461:
448:
444:
440:
437:
434:
431:
409:
408:
397:
392:
388:
384:
381:
378:
375:
318:
317:
306:
301:
297:
293:
290:
287:
284:
223:
222:
209:Ω, Ω) for all
199:
191:' denotes the
178:
144:
141:
110:Roger Godement
56:discrete group
31:of a specific
15:
9:
6:
4:
3:
2:
4751:
4740:
4737:
4735:
4732:
4730:
4727:
4725:
4722:
4720:
4717:
4716:
4714:
4699:
4691:
4690:
4687:
4681:
4678:
4676:
4673:
4671:
4670:Weak topology
4668:
4666:
4663:
4661:
4658:
4656:
4653:
4652:
4650:
4646:
4639:
4635:
4632:
4630:
4627:
4625:
4622:
4620:
4617:
4615:
4612:
4610:
4607:
4605:
4602:
4600:
4597:
4595:
4594:Index theorem
4592:
4590:
4587:
4585:
4582:
4580:
4577:
4576:
4574:
4570:
4564:
4561:
4559:
4556:
4555:
4553:
4551:Open problems
4549:
4543:
4540:
4538:
4535:
4533:
4530:
4528:
4525:
4523:
4520:
4518:
4515:
4514:
4512:
4508:
4502:
4499:
4497:
4494:
4492:
4489:
4487:
4484:
4482:
4479:
4477:
4474:
4472:
4469:
4467:
4464:
4462:
4459:
4457:
4454:
4453:
4451:
4447:
4441:
4438:
4436:
4433:
4431:
4428:
4426:
4423:
4421:
4418:
4416:
4413:
4411:
4408:
4406:
4403:
4401:
4398:
4397:
4395:
4393:
4389:
4379:
4376:
4374:
4371:
4369:
4366:
4363:
4359:
4355:
4352:
4350:
4347:
4345:
4342:
4341:
4339:
4335:
4329:
4326:
4324:
4321:
4319:
4316:
4314:
4311:
4309:
4306:
4304:
4301:
4299:
4296:
4294:
4291:
4289:
4286:
4284:
4281:
4280:
4277:
4274:
4270:
4265:
4261:
4257:
4250:
4245:
4243:
4238:
4236:
4231:
4230:
4227:
4220:
4218:3-540-42248-X
4214:
4210:
4205:
4202:
4200:3-540-42914-X
4196:
4192:
4187:
4184:
4182:0-521-22286-9
4178:
4174:
4170:
4166:
4162:
4158:
4154:
4150:
4146:
4142:
4141:Ann. of Math.
4138:
4134:
4130:
4125:
4121:
4117:
4112:
4109:
4107:0-12-549450-5
4103:
4099:
4094:
4091:
4087:
4083:
4079:
4075:
4071:
4070:Ann. of Math.
4067:
4063:
4059:
4056:
4052:
4047:
4042:
4038:
4034:
4030:
4026:
4022:
4019:
4015:
4011:
4007:
4003:
3999:
3998:Ann. of Math.
3995:
3991:
3987:
3984:
3980:
3976:
3972:
3968:
3964:
3963:Ann. of Math.
3960:
3956:
3952:
3948:
3944:
3940:
3936:
3934:0-7204-0762-1
3930:
3925:
3924:
3918:
3914:
3911:
3909:0-7204-0762-1
3905:
3900:
3899:
3893:
3889:
3885:
3883:0-444-86308-7
3879:
3875:
3871:
3867:
3863:
3859:
3855:
3852:
3850:0-12-215502-5
3846:
3842:
3838:
3837:Dieudonné, J.
3834:
3831:
3825:
3821:
3817:
3813:
3810:
3808:3-540-17093-6
3804:
3800:
3795:
3794:
3783:
3782:Godement 1954
3778:
3772:
3767:
3760:
3755:
3749:
3744:
3738:
3737:Pedersen 1979
3733:
3726:
3722:
3716:
3710:
3705:
3698:
3697:Takesaki 1979
3693:
3687:
3686:Takesaki 2002
3682:
3680:
3673:
3668:
3660:
3656:
3652:
3649:generated by
3648:
3642:
3635:
3631:
3624:
3619:
3613:
3608:
3606:
3604:
3602:
3595:
3590:
3588:
3580:
3575:
3571:
3562:
3559:
3557:
3554:
3552:
3549:
3548:
3536:
3535:
3531:
3530:
3527:
3526:
3525:
3522:
3509:
3499:
3494:
3490:
3481:
3477:
3473:
3469:
3465:
3461:
3442:
3439:
3436:
3430:
3424:
3418:
3407:
3355:
3351:
3347:
3294:
3281:
3277:
3260:
3257:
3254:
3242:
3238:
3234:
3230:
3226:
3223:
3219:
3215:
3212:
3209:
3205:
3201:
3197:
3193:
3189:
3185:
3182:
3179:
3175:
3171:
3167:
3164:
3163:
3162:
3160:
3156:
3152:
3124:
3120:
3117:) if the map
3093:(relative to
3092:
3088:
3084:
3079:
3077:
3072:
3070:
3066:
3037:
3033:
3030:
3027:
3024:
3017:
3016:
3013:
3012:
3011:
3009:
2990:
2964:
2961:
2958:
2951:
2950:
2949:
2908:
2905:
2879:
2872:
2871:
2868:
2867:
2866:
2864:
2845:
2842:
2839:
2836:
2833:
2827:
2821:
2816:
2813:
2810:
2807:
2804:
2798:
2792:
2785:
2784:
2783:
2757:
2753:
2725:
2712:
2709:
2705:
2701:
2697:
2693:
2689:
2685:
2681:
2677:
2673:
2669:
2665:
2661:
2646:
2638:
2634:
2630:
2626:
2622:
2618:
2614:
2610:
2607:
2603:
2599:
2595:
2580:
2572:
2568:
2564:
2561:
2557:
2553:
2550:
2546:
2542:
2538:
2531:
2527:
2523:
2520:
2516:
2512:
2508:
2493:
2485:
2481:
2477:
2473:
2470:
2466:
2462:
2458:
2454:
2453:
2420:
2416:
2413:
2409:
2405:
2401:
2397:
2393:
2366:
2362:
2335:
2331:
2327:
2323:
2319:
2315:
2311:
2310:
2309:
2307:
2303:
2275:
2265:
2263:
2258:
2254:
2250:
2244:
2228:
2221:
2218:
2216:
2211:
2208:
2207:increasing to
2202:
2199:
2195:
2191:
2188:
2182:
2178:
2174:
2167:
2163:
2159:
2156:
2150:
2143:
2140:
2137:
2136:
2135:
2108:
2104:
2101:
2098:
2095:
2088:
2087:
2084:
2083:
2082:
2080:
2076:
2072:
2068:
2061:
2057:
2036:
2032:
2028:
2025:
2022:
2015:
2014:
2013:
2011:
2007:
1988:
1984:
1980:
1975:
1971:
1966:
1962:
1959:
1953:
1950:
1947:
1937:
1936:
1935:
1917:
1910:
1906:
1902:
1897:
1893:
1888:
1884:
1881:
1878:
1875:
1872:
1868:
1864:
1859:
1855:
1847:
1846:
1845:
1843:
1839:
1835:
1831:
1826:
1824:
1816:
1812:
1808:
1805:
1801:
1797:
1794:
1790:
1786:
1782:
1775:
1771:
1752:
1746:
1743:
1739:
1733:
1729:
1725:
1722:
1718:
1714:
1707:
1704:
1701:semilinearity
1693:
1689:
1685:
1666:
1660:
1657:
1654:
1648:
1642:
1639:
1636:
1630:
1627:
1624:
1621:
1618:
1612:
1605:
1604:
1603:
1598:
1594:
1590:
1586:
1582:
1578:
1571:
1567:
1557:
1555:
1551:
1547:
1543:
1539:
1535:
1531:
1523:
1519:
1501:
1497:
1493:
1490:
1487:
1477:. The vector
1461:
1455:
1452:
1447:
1443:
1422:
1419:
1416:
1405:
1386:
1383:
1380:
1377:
1369:
1365:
1364:
1359:
1343:
1329:
1325:
1321:
1318:
1315:
1310:
1306:
1297:
1293:
1289:
1285:
1281:
1265:
1259:
1254:
1251:
1247:
1240:
1237:
1231:
1225:
1220:
1216:
1207:
1203:
1199:
1195:
1191:
1187:
1183:
1180:
1176:
1172:
1168:
1164:
1163:
1158:
1142:
1139:
1136:
1132:
1129:
1120:
1116:
1112:
1108:
1105:, Ό) acts by
1104:
1100:
1096:
1093:
1089:
1085:
1081:
1078:
1077:
1072:
1069:
1053:
1039:
1036:
1032:
1025:
1019:
1013:
1007:
1004:
996:
993:The operator
980:
961:
958:
936:
931:
912:
909:
887:
856:
852:
843:
824:
821:
815:
812:
806:
797:
791:
785:
777:
771:
766:
763:
759:
752:
749:
743:
734:
728:
722:
711:
707:
680:
676:
668:
664:
660:
659:
653:
651:
647:
628:
625:
622:
613:
609:
602:
601:
600:
598:
594:
590:
586:
581:
579:
575:
571:
567:
563:
556:
552:
545:
541:
537:
518:
515:
512:
509:
506:
503:
500:
493:
492:
491:
489:
482:
475:
471:
442:
438:
435:
432:
429:
422:
421:
418:
417:
416:
414:
395:
386:
382:
379:
376:
373:
366:
365:
364:
362:
358:
354:
349:
347:
343:
339:
335:
331:
327:
323:
299:
295:
291:
285:
282:
275:
274:
273:
270:
268:
264:
260:
256:
252:
248:
244:
240:
236:
232:
228:
220:
216:
212:
208:
204:
200:
198:
194:
190:
186:
182:
179:
177:
173:
170:
169:
168:
166:
162:
158:
154:
153:Hilbert space
150:
140:
138:
134:
130:
126:
121:
119:
115:
111:
107:
103:
99:
95:
91:
87:
83:
80:
76:
71:
69:
66:preserving a
65:
61:
57:
53:
49:
44:
42:
38:
37:Hilbert space
34:
30:
26:
22:
4660:Balanced set
4634:Distribution
4572:Applications
4425:KreinâMilman
4410:Closed graph
4208:
4190:
4172:
4144:
4140:
4119:
4115:
4097:
4073:
4069:
4062:Murray, F.J.
4036:
4032:
4025:Murray, F.J.
4001:
3997:
3990:Murray, F.J.
3966:
3962:
3959:Godement, R.
3950:
3946:
3943:Godement, R.
3922:
3897:
3873:
3861:
3840:
3819:
3798:
3777:
3766:
3759:Dixmier 1977
3754:
3743:
3732:
3724:
3720:
3715:
3704:
3692:
3667:
3658:
3654:
3650:
3646:
3641:
3629:
3622:
3618:
3612:Dixmier 1957
3574:
3533:
3532:
3523:
3479:
3475:
3471:
3467:
3463:
3459:
3405:
3353:
3349:
3345:
3295:
3246:
3240:
3236:
3232:
3228:
3221:
3217:
3213:
3207:
3203:
3199:
3195:
3191:
3187:
3183:
3177:
3173:
3169:
3165:
3158:
3154:
3150:
3122:
3118:
3090:
3086:
3082:
3080:
3075:
3073:
3069:Segal (1953)
3062:
3007:
3005:
2948:Moreover if
2947:
2862:
2860:
2755:
2751:
2723:
2721:
2707:
2703:
2699:
2695:
2691:
2687:
2683:
2679:
2675:
2671:
2667:
2663:
2659:
2636:
2632:
2628:
2624:
2621:Gelfand pair
2616:
2612:
2605:
2601:
2597:
2593:
2570:
2566:
2555:
2548:
2544:
2540:
2536:
2529:
2525:
2518:
2514:
2510:
2506:
2483:
2479:
2475:
2468:
2464:
2460:
2456:
2421:is dense in
2418:
2411:
2407:
2403:
2399:
2395:
2364:
2333:
2329:
2325:
2321:
2317:
2313:
2305:
2301:
2273:
2271:
2246:
2226:
2219:
2212:
2209:
2206:
2200:
2197:
2193:
2189:
2186:
2180:
2176:
2172:
2165:
2161:
2157:
2154:
2148:
2141:
2138:
2133:
2078:
2074:
2070:
2066:
2059:
2055:
2053:
2009:
2005:
2003:
1933:
1841:
1837:
1833:
1829:
1827:
1822:
1820:
1814:
1810:
1803:
1799:
1792:
1788:
1780:
1773:
1769:
1698:
1691:
1687:
1683:
1596:
1592:
1588:
1584:
1580:
1569:
1565:
1563:
1553:
1541:
1537:
1533:
1529:
1527:
1521:
1517:
1403:
1362:
1361:
1295:
1291:
1287:
1283:
1279:
1205:
1201:
1197:
1193:
1189:
1178:
1170:
1166:
1161:
1160:
1156:
1118:
1114:
1110:
1102:
1098:
1094:
1087:
1075:
1074:
994:
841:
663:finite group
649:
645:
643:
596:
592:
588:
584:
582:
577:
573:
569:
565:
561:
554:
550:
543:
539:
535:
533:
487:
480:
473:
469:
467:
412:
410:
360:
356:
352:
350:
345:
341:
337:
333:
329:
325:
321:
319:
271:
266:
262:
258:
254:
250:
246:
242:
238:
226:
224:
218:
214:
210:
206:
202:
196:
188:
184:
180:
175:
171:
164:
156:
148:
146:
122:
117:
106:Gelfand pair
94:Irving Segal
72:
45:
35:acting on a
24:
18:
4589:Heat kernel
4579:Hardy space
4486:Trace class
4400:HahnâBanach
4362:Topological
4164:(Section 5)
4137:Segal, I.E.
4122:: 187â221,
3923:C* algebras
3917:Dixmier, J.
3892:Dixmier, J.
3870:Dixmier, J.
3858:Dixmier, J.
3672:Connes 1979
3411:defined by
2077:satisfying
1575:the set of
363:Ω, so that
133:Rudolf Haag
21:mathematics
4713:Categories
4522:C*-algebra
4337:Properties
3816:Connes, A.
3791:References
3709:Simon 1979
2718:Properties
2560:unimodular
2268:Definition
2241:See also:
1546:equivalent
1298:, Ό). Let
1090:, Ό). The
253:Ω = 0 for
79:unimodular
58:or by the
4496:Unbounded
4491:Transpose
4449:Operators
4378:Separable
4373:Reflexive
4358:Algebraic
4344:Barrelled
4169:Simon, B.
3419:τ
3346:completed
3261:λ
3220:)'s with
3172:*, and λ(
3042:′
2986:′
2983:′
2965:λ
2927:′
2909:ρ
2901:′
2898:′
2880:λ
2822:ρ
2793:λ
2647:∩
2581:∩
2494:∩
2113:′
2037:∗
1976:∗
1963:τ
1914:∞
1898:∗
1885:τ
1882:∣
1876:∈
1804:normality
1747:τ
1734:∗
1715:τ
1661:τ
1658:μ
1643:τ
1640:λ
1628:μ
1619:λ
1613:τ
1498:δ
1494:⊗
1485:Ω
1456:λ
1453:⊗
1420:⊗
1390:Γ
1387:⋊
1338:Γ
1326:ℓ
1322:⊗
1252:−
1068:countable
1049:¯
1037:−
976:′
968:Γ
962:λ
954:′
951:′
943:Γ
937:ρ
927:′
919:Γ
913:ρ
905:′
902:′
894:Γ
888:λ
865:Γ
853:ℓ
786:ρ
764:−
723:λ
689:Γ
677:ℓ
626:⊆
618:′
510:⊕
479:Ω, where
447:′
391:′
383:⊆
305:Ω
300:∗
289:Ω
205:Ω, Ω) = (
193:commutant
29:commutant
4698:Category
4510:Algebras
4392:Theorems
4349:Complete
4318:Schwartz
4264:glossary
4171:(1979),
3919:(1977),
3894:(1969),
3872:(1981),
3860:(1957),
3839:(1976),
3818:(1979),
3725:maximale
3545:See also
3278:″
3243:bounded.
3231:) and Ï(
3224:bounded;
3010:), then
2682:); here
2463:) = Tr (
2450:Examples
1556:(or Î).
1155:so that
1133:′
656:Examples
187:, where
4501:Unitary
4481:Nuclear
4466:Compact
4461:Bounded
4456:Adjoint
4430:Minâmax
4323:Sobolev
4308:Nuclear
4298:Hilbert
4293:Fréchet
4258: (
4161:1969729
4090:1969107
4055:1989620
4018:1968693
3983:1969832
3953:: 1â110
3721:achevée
3176:*) = λ(
3091:bounded
2619:) is a
2543:) = Ï(
2328:*) for
2213:in the
1536:. Here
261:, then
108:due to
4476:Normal
4313:Orlicz
4303:Hölder
4283:Banach
4272:Spaces
4260:topics
4215:
4197:
4179:
4159:
4104:
4088:
4053:
4016:
3981:
3931:
3906:
3880:
3847:
3826:
3805:
3524:Thus:
2217:, then
1832:, let
4288:Besov
4157:JSTOR
4086:JSTOR
4072:, 2,
4051:JSTOR
4014:JSTOR
4000:, 2,
3979:JSTOR
3567:Notes
3482:with
3149:into
2558:is a
2402:) = (
2320:) = (
1591:) on
1589:trace
1159:is a
1079:of Î.
269:= 0.
235:state
151:be a
41:trace
23:, a
4636:(or
4354:Dual
4213:ISBN
4195:ISBN
4177:ISBN
4102:ISBN
3929:ISBN
3904:ISBN
3878:ISBN
3845:ISBN
3824:ISBN
3803:ISBN
3350:full
3194:) =
3067:and
2722:Let
2611:If (
2474:If (
2367:in
2054:for
1911:<
1779:and
1768:for
1682:for
1564:Let
1278:for
840:for
648:and
587:and
549:and
542:in
411:The
355:and
320:for
155:and
147:Let
127:and
96:and
50:and
4149:doi
4124:doi
4078:doi
4041:doi
4006:doi
3971:doi
3723:or
3458:if
3348:or
3206:on
3180:)*;
3125:of
3085:in
2554:If
2524:If
2336:in
2324:*,
2058:in
1787:in
1772:in
1690:in
1579:in
1366:or
1282:in
1177:on
1165:of
1109:on
844:in
580:'.
553:in
353:JMJ
324:in
263:aM'
257:in
237:on
217:in
195:of
163:on
120:.
77:of
70:.
43:.
19:In
4715::
4262:â
4155:,
4145:57
4143:,
4120:69
4118:,
4084:,
4074:44
4064:;
4049:,
4037:41
4035:,
4027:;
4012:,
4002:37
3992:;
3977:,
3967:59
3965:,
3951:30
3949:,
3678:^
3600:^
3586:^
3470:)*
3462:=
3239:,
3227:λ(
3202:*)
3198:λ(
3188:ax
3186:â
3166:Jx
3123:xa
3121:â
3078:.
3071:.
2710:).
2615:,
2608:).
2551:).
2539:,
2521:).
2486:)
2471:).
2459:,
2419:xy
2414:);
2406:,
2398:,
2396:xy
2332:,
2316:,
2272:A
2225:=
2171:=
2147:â
2139:If
1836:=
1825:.
1817:).
1806:);
1795:);
1783:a
1705:);
1686:,
1356:a
1290:=
1200:=
1113:=
1097:=
652:.
650:M'
585:M'
566:ab
562:ab
558:sa
555:M'
547:sa
484:sa
477:sa
348:.
336:=
213:,
207:ba
203:ab
159:a
4640:)
4364:)
4360:/
4356:(
4266:)
4248:e
4241:t
4234:v
4151::
4126::
4080::
4043::
4008::
3973::
3727:.
3662:.
3659:g
3655:U
3651:A
3647:H
3636:.
3630:X
3626:1
3623:H
3510:.
3505:B
3500:=
3495:0
3491:M
3480:M
3476:a
3474:(
3472:λ
3468:a
3466:(
3464:λ
3460:x
3446:)
3443:a
3440:,
3437:a
3434:(
3431:=
3428:)
3425:x
3422:(
3409:+
3406:M
3390:B
3366:B
3354:H
3330:A
3306:B
3282:.
3275:)
3269:B
3264:(
3258:=
3255:M
3241:y
3237:x
3233:y
3229:x
3222:x
3218:x
3214:M
3210:;
3208:H
3204:J
3200:x
3196:J
3192:x
3184:a
3178:x
3174:x
3170:x
3159:x
3155:H
3151:H
3135:A
3119:a
3103:A
3087:H
3083:x
3076:H
3038:M
3034:=
3031:J
3028:M
3025:J
3008:a
2991:,
2979:)
2973:A
2968:(
2962:=
2959:M
2923:)
2917:A
2912:(
2906:=
2894:)
2888:A
2883:(
2863:H
2846:.
2843:a
2840:x
2837:=
2834:x
2831:)
2828:a
2825:(
2817:,
2814:x
2811:a
2808:=
2805:x
2802:)
2799:a
2796:(
2768:A
2756:H
2752:J
2736:A
2724:H
2708:G
2706:(
2704:L
2700:K
2696:K
2694:/
2692:G
2690:\
2688:K
2686:(
2684:L
2680:G
2678:(
2676:L
2672:K
2670:/
2668:G
2666:\
2664:K
2662:(
2660:L
2639:)
2637:K
2635:/
2633:G
2631:\
2629:K
2627:(
2625:L
2617:K
2613:G
2606:G
2604:(
2602:L
2598:G
2596:(
2594:L
2573:)
2571:G
2569:(
2567:L
2556:G
2549:a
2547:*
2545:b
2541:b
2537:a
2533:0
2530:M
2526:M
2519:X
2517:(
2515:L
2511:X
2509:(
2507:L
2484:X
2482:(
2480:L
2476:X
2469:a
2467:*
2465:b
2461:b
2457:a
2445:.
2431:A
2412:z
2410:*
2408:x
2404:y
2400:z
2377:A
2365:a
2360:;
2346:A
2334:b
2330:a
2326:a
2322:b
2318:b
2314:a
2312:(
2306:x
2304:â
2302:x
2286:A
2232:.
2230:2
2227:M
2223:1
2220:M
2210:I
2204:1
2201:M
2194:n
2190:p
2184:2
2181:M
2177:n
2173:p
2169:1
2166:M
2162:n
2158:p
2152:2
2149:M
2145:1
2142:M
2109:M
2105:=
2102:J
2099:M
2096:J
2079:J
2075:H
2067:J
2063:0
2060:M
2056:a
2033:a
2029:=
2026:a
2023:J
2010:H
2006:M
1989:.
1985:)
1981:a
1972:b
1967:(
1960:=
1957:)
1954:b
1951:,
1948:a
1945:(
1918:}
1907:)
1903:a
1894:a
1889:(
1879:M
1873:a
1869:{
1865:=
1860:0
1856:M
1842:M
1840:(
1838:L
1834:H
1830:M
1811:M
1802:(
1800:M
1791:(
1789:M
1781:u
1777:+
1774:M
1770:a
1756:)
1753:a
1750:(
1744:=
1740:)
1730:u
1726:a
1723:u
1719:(
1695:+
1692:M
1688:b
1684:a
1670:)
1667:b
1664:(
1655:+
1652:)
1649:a
1646:(
1637:=
1634:)
1631:b
1625:+
1622:a
1616:(
1600:+
1597:M
1593:M
1581:M
1573:+
1570:M
1566:M
1554:T
1542:T
1538:T
1534:T
1530:Z
1522:M
1518:J
1502:1
1491:1
1488:=
1465:)
1462:g
1459:(
1448:g
1444:U
1423:I
1417:A
1407:1
1404:H
1384:A
1381:=
1378:M
1344:,
1341:)
1335:(
1330:2
1319:H
1316:=
1311:1
1307:H
1296:X
1294:(
1292:L
1288:A
1284:H
1280:f
1266:,
1263:)
1260:x
1255:1
1248:g
1244:(
1241:f
1238:=
1235:)
1232:x
1229:(
1226:f
1221:g
1217:U
1206:X
1204:(
1202:L
1198:H
1194:X
1190:X
1181:.
1179:H
1171:H
1169:(
1167:B
1157:A
1143:,
1140:A
1137:=
1130:A
1119:X
1117:(
1115:L
1111:H
1103:X
1101:(
1099:L
1095:A
1088:X
1086:(
1054:.
1045:)
1040:1
1033:g
1029:(
1026:f
1020:=
1017:)
1014:g
1011:(
1008:f
1005:J
995:J
981:.
972:)
965:(
959:=
947:)
940:(
932:,
923:)
916:(
910:=
898:)
891:(
868:)
862:(
857:2
842:f
828:)
825:g
822:x
819:(
816:f
813:=
810:)
807:x
804:(
801:)
798:f
795:)
792:g
789:(
783:(
778:,
775:)
772:x
767:1
760:g
756:(
753:f
750:=
747:)
744:x
741:(
738:)
735:f
732:)
729:g
726:(
720:(
692:)
686:(
681:2
646:M
629:M
623:J
614:M
610:J
597:M
593:M
589:J
578:M
574:M
570:K
551:b
544:M
540:a
536:J
519:,
516:K
513:i
507:K
504:=
501:H
488:M
481:M
474:M
470:K
443:M
439:=
436:J
433:M
430:J
396:.
387:M
380:J
377:M
374:J
361:M
357:M
342:J
338:I
334:J
330:H
326:M
322:a
296:a
292:=
286:a
283:J
267:a
259:M
255:a
251:a
247:M
243:H
239:M
221:.
219:M
215:b
211:a
201:(
197:M
189:M
185:H
181:M
176:H
172:M
165:H
157:M
149:H
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