1562:
1764:
1322:
1581:
1557:{\displaystyle {\begin{bmatrix}T_{11}&T_{12}&\cdots &T_{1n}&\cdots \\T_{21}&T_{22}&\cdots &T_{2n}&\cdots \\\vdots &\vdots &\ddots &\vdots &\cdots \\T_{n1}&T_{n2}&\cdots &T_{nn}&\cdots \\\vdots &\vdots &\cdots &\vdots &\ddots \end{bmatrix}}.}
52:. One of von Neumann's goals in this paper was to reduce the classification of (what are now called) von Neumann algebras on separable Hilbert spaces to the classification of so-called factors. Factors are analogous to full matrix algebras over a field, and von Neumann wanted to prove a continuous analogue of the
1759:{\displaystyle {\begin{bmatrix}\lambda _{1}&0&\cdots &0&\cdots \\0&\lambda _{2}&\cdots &0&\cdots \\\vdots &\vdots &\ddots &\vdots &\cdots \\0&0&\cdots &\lambda _{n}&\cdots \\\vdots &\vdots &\cdots &\vdots &\ddots \end{bmatrix}}.}
537:
3006:
The previous two theorems provide a complete classification of
Abelian von Neumann algebras on separable Hilbert spaces. This classification takes into account the realization of the von Neumann algebra as an algebra of operators. If one considers the underlying von Neumann algebra independently of
907:
are allowed to vary from point to point without having a local triviality requirement (local in a measure-theoretic sense). One of the main theorems of the von
Neumann theory is to show that in fact the more general definition is equivalent to the simpler one given here.
2399:
2057:
2259:
1110:
1245:
884:
995:
2874:
433:
4318:
2471:
3561:
1925:
2998:
2703:
3230:
4055:
759:
3409:
One of the main theorems of von
Neumann and Murray in their original series of papers is a proof of the decomposition theorem: Any von Neumann algebra is a direct integral of factors. Precisely stated,
895:. This definition is apparently more restrictive than the one given by von Neumann and discussed in Dixmier's classic treatise on von Neumann algebras. In the more general definition, the Hilbert space
3110:
2613:
3863:
409:
1311:
2291:
1936:
3953:
3761:
3383:
3320:
340:
2068:
1006:
3648:
5028:
2927:
4208:
163:
1159:
5130:
605:
567:
4763:
789:
4508:
Theory of
Operator Algebras I, II, III", encyclopedia of mathematical sciences, Springer-Verlag, 2001–2003 (the first volume was published 1979 in 1. Edition)
927:
532:{\displaystyle \mathbf {H} _{n}=\left\{{\begin{matrix}\mathbb {C} ^{n}&{\mbox{ if }}n<\omega \\\ell ^{2}&{\mbox{ if }}n=\omega \end{matrix}}\right.}
2778:
4223:
889:
Given the local nature of our definition, many definitions applicable to single
Hilbert spaces apply to measurable families of Hilbert spaces as well.
4768:
4541:
5115:
3883:
are decomposable operators. This can be used to prove the basic result of von
Neumann: any von Neumann algebra admits a decomposition into factors.
1843:
63:
of matrices; in this case the results are easy to prove directly. The infinite-dimensional case is complicated by measure-theoretic technicalities.
5008:
2410:
4861:
4659:
4856:
1852:
911:
Note that the direct integral of a measurable family of
Hilbert spaces depends only on the measure class of the measure μ; more precisely:
2946:
1575:, having zero for all non-diagonal entries. Decomposable operators can be characterized as those which commute with diagonal matrices:
2639:
5013:
4513:
4479:
4466:
4440:
3156:
3995:
704:
3441:
4831:
3435:
is a measurable family of von
Neumann algebras and μ is standard, then the family of operator commutants is also measurable and
764:
consists of equivalence classes (with respect to almost everywhere equality) of measurable square integrable cross-sections of {
4800:
3063:
4790:
4785:
4778:
4714:
4598:
2557:
2394:{\displaystyle \phi :L_{\mu }^{\infty }(X)\rightarrow \operatorname {L} {\bigg (}\int _{X}^{\oplus }H_{x}\ d\mu (x){\bigg )}}
3814:
2052:{\displaystyle \int _{X}^{\oplus }\ T_{x}d\mu (x)\in \operatorname {L} {\bigg (}\int _{X}^{\oplus }H_{x}\ d\mu (x){\bigg )}}
355:
1265:
5191:
5023:
4534:
3007:
its realization (as a von
Neumann algebra), then its structure is determined by very simple measure-theoretic invariants.
53:
5181:
4649:
4161:, μ) (which, as stated, is unique in a measure theoretic sense), there is a measurable family of factor representations
3895:
5160:
4634:
2254:{\displaystyle {\bigg }{\bigg (}\int _{X}^{\oplus }\ s_{x}d\mu (x){\bigg )}=\int _{X}^{\oplus }\ T_{x}(s_{x})d\mu (x).}
1105:{\displaystyle \int _{X}^{\oplus }H_{x}\,\mathrm {d} \mu (x)\rightarrow \int _{X}^{\oplus }H_{x}\,\mathrm {d} \nu (x).}
5080:
3715:
3331:
3268:
2522:
5135:
5033:
4913:
293:
207:
171:: The terminology adopted by the literature on the subject is followed here, according to which a measurable space
4629:
5140:
5003:
4836:
4821:
4593:
2888:
4722:
4732:
4603:
4527:
4105:
has a unit, non-degeneracy is equivalent to unit-preserving. By the general correspondence that exists between
3690:
is a von
Neumann algebra whose center contains a sequence of minimal pairwise orthogonal non-zero projections {
3605:
5186:
5095:
5070:
4888:
4877:
4588:
574:
102:
71:
4946:
4936:
4931:
4167:
4639:
4109:
75:
3872:
as a direct sum of factors. This is a special case of the central decomposition theorem of von Neumann.
287:, which is locally equivalent to a trivial family in the following sense: There is a countable partition
4691:
4364:
4113:
3003:
The isomorphism φ is a measure class isomorphism, in that φ and its inverse preserve sets of measure 0.
118:
79:
2736:
This version of the spectral theorem does not explicitly state how the underlying standard Borel space
1240:{\displaystyle \int _{X}^{\oplus }H_{x}\,\mathrm {d} \mu (x)\cong \bigoplus _{k\in \mathbb {N} }H_{k}}
5105:
5084:
4998:
4883:
4846:
4404:
2496:
Decomposable operators are precisely those that are in the operator commutant of the abelian algebra
243:
4908:
4644:
4106:
3035:
1808:
206:; all Polish spaces of a given cardinality are isomorphic to each other (as Borel spaces). Given a
41:
21:
5038:
4967:
4898:
4742:
4704:
4504:
4424:
4123:), the theory for C*-algebras immediately provides a decomposition theory for representations of
4097:, the above results can be applied to measurable families of non-degenerate *-representations of
59:
Results on direct integrals can be viewed as generalizations of results about finite-dimensional
580:
3879:) as an algebra of scalar diagonal operators. In any such representation, all the operators in
545:
5145:
5120:
4805:
4727:
4509:
4475:
4462:
4436:
2880:
695:
192:
5150:
4851:
4699:
4654:
4578:
3584:
3262:
is a measurable family of von Neumann algebras, the direct integral of von Neumann algebras
1129:
95:
45:
879:{\displaystyle \langle s|t\rangle =\int _{X}\langle s(x)|t(x)\rangle \,\mathrm {d} \mu (x)}
5125:
5110:
5018:
4981:
4977:
4941:
4903:
4841:
4826:
4737:
4696:
4683:
4608:
4550:
4432:
4157:. Then corresponding to any central decomposition of W*(π) over a standard measure space (
4124:
990:{\displaystyle s\mapsto \left({\frac {\mathrm {d} \mu }{\mathrm {d} \nu }}\right)^{1/2}s}
677:. A cross-section is measurable if and only if its restriction to each partition element
4403:. This decomposition is essentially unique. This result is fundamental in the theory of
5075:
5054:
4972:
4962:
4773:
4680:
4613:
4573:
4494:
3116:
2869:{\displaystyle \int _{X}^{\oplus }H_{x}d\mu (x),\quad \int _{Y}^{\oplus }K_{y}d\nu (y)}
2515:
The spectral theorem has many variants. A particularly powerful version is as follows:
1572:
570:
215:
199:
1255:
For the example of a discrete measure on a countable set, any bounded linear operator
5175:
1125:
67:
37:
4313:{\displaystyle \pi (a)=\int _{X}^{\oplus }\pi _{x}(a)d\mu (x),\quad \forall a\in A.}
4893:
4747:
4688:
4484:
1769:
The above example motivates the general definition: A family of bounded operators {
203:
4501:
The Annals of Mathematics 2nd Ser., Vol. 50, No. 2 (Apr., 1949), pp. 401–485.
1153:} of separable Hilbert spaces can be considered as a measurable family. Moreover,
180:
4519:
5090:
4675:
247:
17:
3875:
In general, the structure theorem of Abelian von Neumann algebras represents Z(
4583:
4454:
4137:
be a separable C*-algebra and π a non-degenerate involutive representation of
4094:
188:
60:
33:
4568:
4554:
2271:
Examples of decomposable operators are those defined by scalar-valued (i.e.
5155:
5100:
2466:{\displaystyle \lambda \mapsto \int _{X}^{\oplus }\ \lambda _{x}d\mu (x)}
219:
4498:
4381:, consisting of quasi-equivalence classes of factor representations of
2740:
is obtained. There is a uniqueness result for the above decomposition.
4370:
One can show that the direct integral can be indexed on the so-called
2729:). Note that this asserts more than just the algebraic equivalence of
1920:{\displaystyle \operatorname {ess-sup} _{x\in X}\|T_{x}\|<\infty }
191:, regardless of whether or not the underlying σ-algebra comes from a
4145:. Let W*(π) be the von Neumann algebra generated by the operators π(
94:
spaces associated to a (σ-finite) countably additive measure μ on a
698:. Given a measurable family of Hilbert spaces, the direct integral
2993:{\displaystyle K_{\phi (x)}=H_{x}\quad {\mbox{almost everywhere}}}
4119:
and non-degenerate *-representations of the groups C*-algebra C*(
260:
be a Borel space equipped with a countably additive measure μ. A
3675:
is a von Neumann algebra, if the center is 1-dimensional we say
3142:
as a von Neumann algebra in the following sense: For almost all
4523:
2698:{\displaystyle U:H\rightarrow \int _{X}^{\oplus }H_{x}d\mu (x)}
2540:
such that it is unitarily equivalent as an operator algebra to
3225:{\displaystyle \operatorname {W^{*}} (\{S_{x}:S\in D\})=A_{x}}
4389:
and a measurable family of factor representations indexed on
4050:{\displaystyle \mathbf {A} =\int _{X}^{\oplus }A_{x}d\mu (x)}
3974:) is represented by the algebra of scalar diagonal operators
754:{\displaystyle \int _{X}^{\oplus }H_{x}\,\mathrm {d} \mu (x)}
3556:{\displaystyle {\bigg }'=\int _{X}^{\oplus }A'_{x}d\mu (x).}
1567:
For this example, of a discrete measure on a countable set,
917:. Suppose μ, ν are σ-finite countably additive measures on
526:
686:
is measurable. We will identify measurable cross-sections
3123:
of measurable operator families that pointwise generate {
3105:{\displaystyle A_{x}\subseteq \operatorname {L} (H_{x})}
3239:) denotes the von Neumann algebra generated by the set
2633:) as an operator algebra means that there is a unitary
2608:{\displaystyle \int _{X}^{\oplus }H_{x}d\mu (x).\quad }
2476:
is an involutive algebraic isomorphism onto its image.
921:
that have the same sets of measure 0. Then the mapping
36:. The theory is most developed for direct integrals of
3858:{\displaystyle A=\bigoplus _{i\in \mathbb {N} }AE_{i}}
3034:
be a measurable family of Hilbert spaces. A family of
2984:
1590:
1331:
507:
475:
457:
404:{\displaystyle H_{x}=\mathbf {H} _{n}\quad x\in X_{n}}
4226:
4170:
3998:
3898:
3817:
3718:
3608:
3444:
3334:
3271:
3159:
3066:
2949:
2891:
2781:
2642:
2560:
2413:
2294:
2071:
1939:
1855:
1584:
1325:
1306:{\displaystyle H=\bigoplus _{k\in \mathbb {N} }H_{k}}
1268:
1162:
1009:
930:
792:
707:
583:
548:
436:
358:
296:
121:
202:
it is isomorphic to the underlying Borel space of a
5063:
5047:
4991:
4955:
4924:
4870:
4814:
4756:
4713:
4668:
4622:
4561:
5131:Spectral theory of ordinary differential equations
4312:
4202:
4049:
3948:{\displaystyle H=\int _{X}^{\oplus }H_{x}d\mu (x)}
3947:
3857:
3755:
3642:
3555:
3377:
3314:
3224:
3104:
2992:
2921:
2868:
2697:
2607:
2465:
2393:
2253:
2051:
1919:
1758:
1556:
1305:
1239:
1104:
989:
878:
783:. This is a Hilbert space under the inner product
753:
599:
561:
531:
403:
334:
157:
90:The simplest example of a direct integral are the
5029:Schröder–Bernstein theorems for operator algebras
3495:
3447:
2386:
2336:
2181:
2131:
2124:
2074:
2044:
1994:
1821:is strongly measurable. This makes sense because
3756:{\displaystyle 1=\sum _{i\in \mathbb {N} }E_{i}}
3378:{\displaystyle \int _{X}^{\oplus }T_{x}d\mu (x)}
3315:{\displaystyle \int _{X}^{\oplus }A_{x}d\mu (x)}
2879:and μ, ν are standard measures, then there is a
2551:) acting on a direct integral of Hilbert spaces
335:{\displaystyle \{X_{n}\}_{1\leq n\leq \omega }}
4535:
2511:Decomposition of Abelian von Neumann algebras
195:(in most examples it does). A Borel space is
101:. Somewhat more generally one can consider a
8:
4185:
4171:
3203:
3178:
1908:
1895:
855:
823:
807:
793:
311:
297:
4489:The Theory of Unitary Group Representations
2922:{\displaystyle \varphi :X-E\rightarrow Y-F}
230:measure if and only if there is a null set
4542:
4528:
4520:
4385:. Thus, there is a standard measure μ on
218:is one that differs from a Borel set by a
4261:
4251:
4246:
4225:
4188:
4178:
4169:
4026:
4016:
4011:
3999:
3997:
3924:
3914:
3909:
3897:
3849:
3836:
3835:
3828:
3816:
3747:
3737:
3736:
3729:
3717:
3609:
3607:
3526:
3516:
3511:
3494:
3493:
3471:
3461:
3456:
3446:
3445:
3443:
3354:
3344:
3339:
3333:
3291:
3281:
3276:
3270:
3216:
3185:
3165:
3160:
3158:
3093:
3071:
3065:
2983:
2976:
2954:
2948:
2890:
2845:
2835:
2830:
2801:
2791:
2786:
2780:
2674:
2664:
2659:
2641:
2580:
2570:
2565:
2559:
2442:
2429:
2424:
2412:
2385:
2384:
2360:
2350:
2345:
2335:
2334:
2310:
2305:
2293:
2224:
2211:
2198:
2193:
2180:
2179:
2158:
2145:
2140:
2130:
2129:
2123:
2122:
2101:
2088:
2083:
2073:
2072:
2070:
2043:
2042:
2018:
2008:
2003:
1993:
1992:
1962:
1949:
1944:
1938:
1902:
1880:
1857:
1854:
1842:Measurable families of operators with an
1707:
1636:
1597:
1585:
1583:
1502:
1482:
1467:
1418:
1401:
1389:
1367:
1350:
1338:
1326:
1324:
1297:
1287:
1286:
1279:
1267:
1231:
1221:
1220:
1213:
1189:
1188:
1182:
1172:
1167:
1161:
1082:
1081:
1075:
1065:
1060:
1036:
1035:
1029:
1019:
1014:
1008:
974:
970:
955:
945:
942:
929:
859:
858:
838:
817:
799:
791:
734:
733:
727:
717:
712:
706:
588:
582:
553:
547:
506:
498:
474:
466:
462:
461:
456:
443:
438:
435:
395:
378:
373:
363:
357:
314:
304:
295:
131:
126:
120:
4491:, The University of Chicago Press, 1976.
3808:is a factor. Thus, in this special case
3643:{\displaystyle \mathbf {Z} (A)=A\cap A'}
3591:. The center is the set of operators in
3011:Direct integrals of von Neumann algebras
2733:with the algebra of diagonal operators.
2490:) to be identified with the image of φ.
66:Direct integral theory was also used by
44:. The concept was introduced in 1949 by
4499:On Rings of Operators. Reduction Theory
4416:
4359:, where representations are said to be
2772:) acting on the direct integral spaces
2718:* is the algebra of diagonal operators
2062:acting in a pointwise fashion, that is
1812:if and only if its restriction to each
4085:Measurable families of representations
3958:is a direct integral decomposition of
3778:is a von Neumann algebra on the range
3325:consists of all operators of the form
1571:are defined as the operators that are
179:and the elements of the distinguished
32:is a generalization of the concept of
4862:Spectral theory of normal C*-algebras
4660:Spectral theory of normal C*-algebras
4203:{\displaystyle \{\pi _{x}\}_{x\in X}}
3661:) is an Abelian von Neumann algebra.
2746:. If the Abelian von Neumann algebra
7:
4857:Spectral theory of compact operators
427:-dimensional Hilbert space, that is
4077:is a von Neumann algebra that is a
3671:) is 1-dimensional. In general, if
2275:-valued) measurable functions λ on
262:measurable family of Hilbert spaces
246:. All measures considered here are
108:and the space of square-integrable
48:in one of the papers in the series
5009:Cohen–Hewitt factorization theorem
4446:, Chapter IV, Theorem 7.10, p. 259
4295:
3162:
3080:
2532:, there is a standard Borel space
2328:
2311:
1986:
1914:
1876:
1873:
1870:
1864:
1861:
1858:
1190:
1083:
1037:
956:
946:
860:
735:
158:{\displaystyle L_{\mu }^{2}(X,H).}
86:Direct integrals of Hilbert spaces
14:
5014:Extensions of symmetric operators
1120:The simplest example occurs when
4832:Positive operator-valued measure
4331:with μ measure zero, such that π
4000:
3989:is a standard Borel space. Then
3610:
3595:that commute with all operators
2750:is unitarily equivalent to both
1930:define bounded linear operators
439:
374:
80:locally compact separable groups
5116:Rayleigh–Faber–Krahn inequality
4294:
2982:
2825:
2604:
1316:is given by an infinite matrix
384:
56:classifying semi-simple rings.
4288:
4282:
4273:
4267:
4236:
4230:
4044:
4038:
3942:
3936:
3620:
3614:
3575:is a von Neumann algebra. Let
3547:
3541:
3489:
3483:
3372:
3366:
3309:
3303:
3206:
3175:
3099:
3086:
2964:
2958:
2907:
2863:
2857:
2819:
2813:
2692:
2686:
2652:
2598:
2592:
2460:
2454:
2417:
2381:
2375:
2325:
2322:
2316:
2264:Such operators are said to be
2245:
2239:
2230:
2217:
2176:
2170:
2119:
2113:
2039:
2033:
1980:
1974:
1203:
1197:
1096:
1090:
1053:
1050:
1044:
934:
873:
867:
852:
846:
839:
835:
829:
800:
748:
742:
149:
137:
1:
5024:Limiting absorption principle
4429:Theory of Operator Algebras I
4141:on a separable Hilbert space
2528:on a separable Hilbert space
1140:and μ is counting measure on
4650:Singular value decomposition
4363:if and only if there are no
4323:Moreover, there is a subset
3966:is a von Neumann algebra on
5081:Hearing the shape of a drum
4764:Decomposition of a spectrum
2622:is unitarily equivalent to
2523:Abelian von Neumann algebra
5208:
4669:Special Elements/Operators
600:{\displaystyle \ell ^{2}.}
208:countably additive measure
74:and his general theory of
5141:Superstrong approximation
5004:Banach algebra cohomology
4837:Projection-valued measure
4822:Borel functional calculus
4594:Projection-valued measure
3119:there is a countable set
571:square summable sequences
562:{\displaystyle \ell ^{2}}
345:by measurable subsets of
234:such that its complement
4733:Spectrum of a C*-algebra
4604:Spectrum of a C*-algebra
4399:belongs to the class of
4127:locally compact groups.
2940:are null sets such that
1844:essentially bounded norm
575:separable Hilbert spaces
72:systems of imprimitivity
54:Artin–Wedderburn theorem
40:and direct integrals of
5161:Wiener–Khinchin theorem
5096:Kuznetsov trace formula
5071:Almost Mathieu operator
4889:Banach function algebra
4878:Amenable Banach algebra
4635:Gelfand–Naimark theorem
4589:Noncommutative topology
4110:unitary representations
103:separable Hilbert space
76:induced representations
5136:Sturm–Liouville theory
5034:Sherman–Takeda theorem
4914:Tomita–Takesaki theory
4689:Hermitian/Self-adjoint
4640:Gelfand representation
4365:intertwining operators
4343:are disjoint whenever
4314:
4204:
4051:
3949:
3859:
3757:
3644:
3557:
3379:
3316:
3226:
3106:
2994:
2923:
2870:
2699:
2609:
2467:
2395:
2255:
2053:
1921:
1760:
1569:decomposable operators
1558:
1307:
1251:Decomposable operators
1241:
1106:
1000:is a unitary operator
991:
880:
755:
601:
563:
533:
405:
336:
159:
4630:Gelfand–Mazur theorem
4405:group representations
4315:
4205:
4114:locally compact group
4060:where for almost all
4052:
3950:
3860:
3758:
3645:
3567:Central decomposition
3558:
3380:
3317:
3227:
3107:
2995:
2924:
2871:
2700:
2610:
2468:
2396:
2256:
2054:
1922:
1761:
1559:
1308:
1242:
1144:, then any sequence {
1107:
992:
881:
756:
602:
564:
534:
406:
337:
160:
50:On Rings of Operators
5192:Von Neumann algebras
5106:Proto-value function
5085:Dirichlet eigenvalue
4999:Abstract index group
4884:Approximate identity
4847:Rigged Hilbert space
4723:Krein–Rutman theorem
4569:Involution/*-algebra
4459:Von Neumann algebras
4224:
4168:
3996:
3896:
3815:
3796:. It is easy to see
3716:
3606:
3442:
3332:
3269:
3157:
3064:
3036:von Neumann algebras
2947:
2889:
2779:
2640:
2558:
2411:
2292:
2069:
1937:
1853:
1582:
1323:
1266:
1160:
1007:
928:
790:
705:
581:
546:
434:
356:
294:
244:standard Borel space
222:. The measure μ on
175:is referred to as a
119:
42:von Neumann algebras
5182:Functional analysis
4909:Von Neumann algebra
4645:Polar decomposition
4425:Takesaki, Masamichi
4256:
4107:strongly continuous
4101:. In the case that
4021:
3919:
3534:
3521:
3466:
3349:
3286:
2840:
2796:
2669:
2575:
2536:and a measure μ on
2434:
2355:
2315:
2203:
2150:
2093:
2013:
1954:
1809:strongly measurable
1177:
1070:
1024:
722:
169:Terminological note
136:
70:in his analysis of
22:functional analysis
5039:Unbounded operator
4968:Essential spectrum
4947:Schur–Horn theorem
4937:Bauer–Fike theorem
4932:Alon–Boppana bound
4925:Finite-Dimensional
4899:Nuclear C*-algebra
4743:Spectral asymmetry
4505:Masamichi Takesaki
4310:
4242:
4200:
4047:
4007:
3945:
3905:
3855:
3841:
3753:
3742:
3667:. The center of L(
3640:
3553:
3522:
3507:
3452:
3375:
3335:
3312:
3272:
3222:
3102:
2990:
2988:
2919:
2866:
2826:
2782:
2695:
2655:
2605:
2561:
2463:
2420:
2391:
2341:
2301:
2251:
2189:
2136:
2079:
2049:
1999:
1940:
1917:
1756:
1747:
1554:
1545:
1303:
1292:
1237:
1226:
1163:
1102:
1056:
1010:
987:
876:
751:
708:
597:
577:are isomorphic to
559:
529:
524:
511:
479:
401:
332:
268:, μ) is a family {
155:
122:
112:-valued functions
5169:
5168:
5146:Transfer operator
5121:Spectral geometry
4806:Spectral abscissa
4786:Approximate point
4728:Normal eigenvalue
3824:
3725:
3168:
2987:
2986:almost everywhere
2881:Borel isomorphism
2437:
2368:
2206:
2153:
2096:
2026:
1957:
1869:
1275:
1209:
964:
696:almost everywhere
510:
478:
423:is the canonical
193:topological space
5199:
5151:Transform theory
4871:Special algebras
4852:Spectral theorem
4815:Spectral Theorem
4655:Spectral theorem
4544:
4537:
4530:
4521:
4447:
4445:
4421:
4319:
4317:
4316:
4311:
4266:
4265:
4255:
4250:
4209:
4207:
4206:
4201:
4199:
4198:
4183:
4182:
4056:
4054:
4053:
4048:
4031:
4030:
4020:
4015:
4003:
3954:
3952:
3951:
3946:
3929:
3928:
3918:
3913:
3864:
3862:
3861:
3856:
3854:
3853:
3840:
3839:
3762:
3760:
3759:
3754:
3752:
3751:
3741:
3740:
3649:
3647:
3646:
3641:
3639:
3613:
3562:
3560:
3559:
3554:
3530:
3520:
3515:
3503:
3499:
3498:
3476:
3475:
3465:
3460:
3451:
3450:
3384:
3382:
3381:
3376:
3359:
3358:
3348:
3343:
3321:
3319:
3318:
3313:
3296:
3295:
3285:
3280:
3231:
3229:
3228:
3223:
3221:
3220:
3190:
3189:
3171:
3170:
3169:
3166:
3111:
3109:
3108:
3103:
3098:
3097:
3076:
3075:
2999:
2997:
2996:
2991:
2989:
2985:
2981:
2980:
2968:
2967:
2928:
2926:
2925:
2920:
2875:
2873:
2872:
2867:
2850:
2849:
2839:
2834:
2806:
2805:
2795:
2790:
2704:
2702:
2701:
2696:
2679:
2678:
2668:
2663:
2614:
2612:
2611:
2606:
2585:
2584:
2574:
2569:
2472:
2470:
2469:
2464:
2447:
2446:
2435:
2433:
2428:
2400:
2398:
2397:
2392:
2390:
2389:
2366:
2365:
2364:
2354:
2349:
2340:
2339:
2314:
2309:
2260:
2258:
2257:
2252:
2229:
2228:
2216:
2215:
2204:
2202:
2197:
2185:
2184:
2163:
2162:
2151:
2149:
2144:
2135:
2134:
2128:
2127:
2106:
2105:
2094:
2092:
2087:
2078:
2077:
2058:
2056:
2055:
2050:
2048:
2047:
2024:
2023:
2022:
2012:
2007:
1998:
1997:
1967:
1966:
1955:
1953:
1948:
1926:
1924:
1923:
1918:
1907:
1906:
1891:
1890:
1879:
1867:
1806:) is said to be
1765:
1763:
1762:
1757:
1752:
1751:
1712:
1711:
1641:
1640:
1602:
1601:
1563:
1561:
1560:
1555:
1550:
1549:
1510:
1509:
1490:
1489:
1475:
1474:
1426:
1425:
1406:
1405:
1394:
1393:
1375:
1374:
1355:
1354:
1343:
1342:
1312:
1310:
1309:
1304:
1302:
1301:
1291:
1290:
1246:
1244:
1243:
1238:
1236:
1235:
1225:
1224:
1193:
1187:
1186:
1176:
1171:
1130:discrete measure
1111:
1109:
1108:
1103:
1086:
1080:
1079:
1069:
1064:
1040:
1034:
1033:
1023:
1018:
996:
994:
993:
988:
983:
982:
978:
969:
965:
963:
959:
953:
949:
943:
885:
883:
882:
877:
863:
842:
822:
821:
803:
760:
758:
757:
752:
738:
732:
731:
721:
716:
606:
604:
603:
598:
593:
592:
569:is the space of
568:
566:
565:
560:
558:
557:
538:
536:
535:
530:
528:
525:
512:
508:
503:
502:
480:
476:
471:
470:
465:
448:
447:
442:
410:
408:
407:
402:
400:
399:
383:
382:
377:
368:
367:
341:
339:
338:
333:
331:
330:
309:
308:
164:
162:
161:
156:
135:
130:
96:measurable space
46:John von Neumann
30:Hilbert integral
5207:
5206:
5202:
5201:
5200:
5198:
5197:
5196:
5172:
5171:
5170:
5165:
5126:Spectral method
5111:Ramanujan graph
5059:
5043:
5019:Fredholm theory
4987:
4982:Shilov boundary
4978:Structure space
4956:Generalizations
4951:
4942:Numerical range
4920:
4904:Uniform algebra
4866:
4842:Riesz projector
4827:Min-max theorem
4810:
4796:Direct integral
4752:
4738:Spectral radius
4709:
4664:
4618:
4609:Spectral radius
4557:
4551:Spectral theory
4548:
4451:
4450:
4443:
4433:Springer-Verlag
4423:
4422:
4418:
4413:
4398:
4342:
4336:
4257:
4222:
4221:
4184:
4174:
4166:
4165:
4093:is a separable
4087:
4076:
4022:
3994:
3993:
3980:
3920:
3894:
3893:
3845:
3813:
3812:
3807:
3795:
3786:
3777:
3743:
3714:
3713:
3708:
3698:
3632:
3604:
3603:
3569:
3492:
3467:
3440:
3439:
3434:
3424:
3405:
3396:
3350:
3330:
3329:
3287:
3267:
3266:
3261:
3251:
3212:
3181:
3161:
3155:
3154:
3141:
3131:
3089:
3067:
3062:
3061:
3056:
3046:
3033:
3023:
3013:
2972:
2950:
2945:
2944:
2887:
2886:
2841:
2797:
2777:
2776:
2767:
2756:
2724:
2670:
2638:
2637:
2628:
2576:
2556:
2555:
2546:
2513:
2502:
2485:
2438:
2409:
2408:
2356:
2290:
2289:
2220:
2207:
2154:
2097:
2067:
2066:
2014:
1958:
1935:
1934:
1898:
1856:
1851:
1850:
1838:
1830:is constant on
1829:
1820:
1805:
1796:
1787:
1777:
1746:
1745:
1740:
1735:
1730:
1725:
1719:
1718:
1713:
1703:
1701:
1696:
1691:
1685:
1684:
1679:
1674:
1669:
1664:
1658:
1657:
1652:
1647:
1642:
1632:
1630:
1624:
1623:
1618:
1613:
1608:
1603:
1593:
1586:
1580:
1579:
1544:
1543:
1538:
1533:
1528:
1523:
1517:
1516:
1511:
1498:
1496:
1491:
1478:
1476:
1463:
1460:
1459:
1454:
1449:
1444:
1439:
1433:
1432:
1427:
1414:
1412:
1407:
1397:
1395:
1385:
1382:
1381:
1376:
1363:
1361:
1356:
1346:
1344:
1334:
1327:
1321:
1320:
1293:
1264:
1263:
1253:
1227:
1178:
1158:
1157:
1152:
1118:
1071:
1025:
1005:
1004:
954:
944:
938:
937:
926:
925:
906:
813:
788:
787:
782:
772:
723:
703:
702:
694:that are equal
685:
668:
659:
650:
640:
631:
621:
584:
579:
578:
549:
544:
543:
523:
522:
504:
494:
491:
490:
472:
460:
452:
437:
432:
431:
422:
391:
372:
359:
354:
353:
310:
300:
292:
291:
286:
276:
117:
116:
88:
26:direct integral
12:
11:
5:
5205:
5203:
5195:
5194:
5189:
5187:Measure theory
5184:
5174:
5173:
5167:
5166:
5164:
5163:
5158:
5153:
5148:
5143:
5138:
5133:
5128:
5123:
5118:
5113:
5108:
5103:
5098:
5093:
5088:
5078:
5076:Corona theorem
5073:
5067:
5065:
5061:
5060:
5058:
5057:
5055:Wiener algebra
5051:
5049:
5045:
5044:
5042:
5041:
5036:
5031:
5026:
5021:
5016:
5011:
5006:
5001:
4995:
4993:
4989:
4988:
4986:
4985:
4975:
4973:Pseudospectrum
4970:
4965:
4963:Dirac spectrum
4959:
4957:
4953:
4952:
4950:
4949:
4944:
4939:
4934:
4928:
4926:
4922:
4921:
4919:
4918:
4917:
4916:
4906:
4901:
4896:
4891:
4886:
4880:
4874:
4872:
4868:
4867:
4865:
4864:
4859:
4854:
4849:
4844:
4839:
4834:
4829:
4824:
4818:
4816:
4812:
4811:
4809:
4808:
4803:
4798:
4793:
4788:
4783:
4782:
4781:
4776:
4771:
4760:
4758:
4754:
4753:
4751:
4750:
4745:
4740:
4735:
4730:
4725:
4719:
4717:
4711:
4710:
4708:
4707:
4702:
4694:
4686:
4678:
4672:
4670:
4666:
4665:
4663:
4662:
4657:
4652:
4647:
4642:
4637:
4632:
4626:
4624:
4620:
4619:
4617:
4616:
4614:Operator space
4611:
4606:
4601:
4596:
4591:
4586:
4581:
4576:
4574:Banach algebra
4571:
4565:
4563:
4562:Basic concepts
4559:
4558:
4549:
4547:
4546:
4539:
4532:
4524:
4518:
4517:
4502:
4495:J. von Neumann
4492:
4482:
4469:
4449:
4448:
4441:
4415:
4414:
4412:
4409:
4394:
4372:quasi-spectrum
4367:between them.
4338:
4332:
4321:
4320:
4309:
4306:
4303:
4300:
4297:
4293:
4290:
4287:
4284:
4281:
4278:
4275:
4272:
4269:
4264:
4260:
4254:
4249:
4245:
4241:
4238:
4235:
4232:
4229:
4211:
4210:
4197:
4194:
4191:
4187:
4181:
4177:
4173:
4086:
4083:
4072:
4058:
4057:
4046:
4043:
4040:
4037:
4034:
4029:
4025:
4019:
4014:
4010:
4006:
4002:
3978:
3956:
3955:
3944:
3941:
3938:
3935:
3932:
3927:
3923:
3917:
3912:
3908:
3904:
3901:
3866:
3865:
3852:
3848:
3844:
3838:
3834:
3831:
3827:
3823:
3820:
3803:
3791:
3782:
3773:
3764:
3763:
3750:
3746:
3739:
3735:
3732:
3728:
3724:
3721:
3700:
3694:
3651:
3650:
3638:
3635:
3631:
3628:
3625:
3622:
3619:
3616:
3612:
3568:
3565:
3564:
3563:
3552:
3549:
3546:
3543:
3540:
3537:
3533:
3529:
3525:
3519:
3514:
3510:
3506:
3502:
3497:
3491:
3488:
3485:
3482:
3479:
3474:
3470:
3464:
3459:
3455:
3449:
3426:
3420:
3401:
3392:
3386:
3385:
3374:
3371:
3368:
3365:
3362:
3357:
3353:
3347:
3342:
3338:
3323:
3322:
3311:
3308:
3305:
3302:
3299:
3294:
3290:
3284:
3279:
3275:
3253:
3247:
3233:
3232:
3219:
3215:
3211:
3208:
3205:
3202:
3199:
3196:
3193:
3188:
3184:
3180:
3177:
3174:
3164:
3133:
3127:
3117:if and only if
3115:is measurable
3113:
3112:
3101:
3096:
3092:
3088:
3085:
3082:
3079:
3074:
3070:
3048:
3042:
3025:
3019:
3012:
3009:
3001:
3000:
2979:
2975:
2971:
2966:
2963:
2960:
2957:
2953:
2930:
2929:
2918:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2877:
2876:
2865:
2862:
2859:
2856:
2853:
2848:
2844:
2838:
2833:
2829:
2824:
2821:
2818:
2815:
2812:
2809:
2804:
2800:
2794:
2789:
2785:
2765:
2754:
2722:
2706:
2705:
2694:
2691:
2688:
2685:
2682:
2677:
2673:
2667:
2662:
2658:
2654:
2651:
2648:
2645:
2626:
2616:
2615:
2603:
2600:
2597:
2594:
2591:
2588:
2583:
2579:
2573:
2568:
2564:
2544:
2512:
2509:
2500:
2483:
2474:
2473:
2462:
2459:
2456:
2453:
2450:
2445:
2441:
2432:
2427:
2423:
2419:
2416:
2402:
2401:
2388:
2383:
2380:
2377:
2374:
2371:
2363:
2359:
2353:
2348:
2344:
2338:
2333:
2330:
2327:
2324:
2321:
2318:
2313:
2308:
2304:
2300:
2297:
2285:. The mapping
2262:
2261:
2250:
2247:
2244:
2241:
2238:
2235:
2232:
2227:
2223:
2219:
2214:
2210:
2201:
2196:
2192:
2188:
2183:
2178:
2175:
2172:
2169:
2166:
2161:
2157:
2148:
2143:
2139:
2133:
2126:
2121:
2118:
2115:
2112:
2109:
2104:
2100:
2091:
2086:
2082:
2076:
2060:
2059:
2046:
2041:
2038:
2035:
2032:
2029:
2021:
2017:
2011:
2006:
2002:
1996:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1965:
1961:
1952:
1947:
1943:
1928:
1927:
1916:
1913:
1910:
1905:
1901:
1897:
1894:
1889:
1886:
1883:
1878:
1875:
1872:
1866:
1863:
1860:
1834:
1825:
1816:
1801:
1792:
1779:
1773:
1767:
1766:
1755:
1750:
1744:
1741:
1739:
1736:
1734:
1731:
1729:
1726:
1724:
1721:
1720:
1717:
1714:
1710:
1706:
1702:
1700:
1697:
1695:
1692:
1690:
1687:
1686:
1683:
1680:
1678:
1675:
1673:
1670:
1668:
1665:
1663:
1660:
1659:
1656:
1653:
1651:
1648:
1646:
1643:
1639:
1635:
1631:
1629:
1626:
1625:
1622:
1619:
1617:
1614:
1612:
1609:
1607:
1604:
1600:
1596:
1592:
1591:
1589:
1573:block diagonal
1565:
1564:
1553:
1548:
1542:
1539:
1537:
1534:
1532:
1529:
1527:
1524:
1522:
1519:
1518:
1515:
1512:
1508:
1505:
1501:
1497:
1495:
1492:
1488:
1485:
1481:
1477:
1473:
1470:
1466:
1462:
1461:
1458:
1455:
1453:
1450:
1448:
1445:
1443:
1440:
1438:
1435:
1434:
1431:
1428:
1424:
1421:
1417:
1413:
1411:
1408:
1404:
1400:
1396:
1392:
1388:
1384:
1383:
1380:
1377:
1373:
1370:
1366:
1362:
1360:
1357:
1353:
1349:
1345:
1341:
1337:
1333:
1332:
1330:
1314:
1313:
1300:
1296:
1289:
1285:
1282:
1278:
1274:
1271:
1252:
1249:
1248:
1247:
1234:
1230:
1223:
1219:
1216:
1212:
1208:
1205:
1202:
1199:
1196:
1192:
1185:
1181:
1175:
1170:
1166:
1148:
1117:
1114:
1113:
1112:
1101:
1098:
1095:
1092:
1089:
1085:
1078:
1074:
1068:
1063:
1059:
1055:
1052:
1049:
1046:
1043:
1039:
1032:
1028:
1022:
1017:
1013:
998:
997:
986:
981:
977:
973:
968:
962:
958:
952:
948:
941:
936:
933:
902:
887:
886:
875:
872:
869:
866:
862:
857:
854:
851:
848:
845:
841:
837:
834:
831:
828:
825:
820:
816:
812:
809:
806:
802:
798:
795:
774:
768:
762:
761:
750:
747:
744:
741:
737:
730:
726:
720:
715:
711:
681:
664:
655:
642:
636:
623:
617:
596:
591:
587:
556:
552:
542:In the above,
540:
539:
527:
521:
518:
515:
509: if
505:
501:
497:
493:
492:
489:
486:
483:
477: if
473:
469:
464:
459:
458:
455:
451:
446:
441:
418:
412:
411:
398:
394:
390:
387:
381:
376:
371:
366:
362:
343:
342:
329:
326:
323:
320:
317:
313:
307:
303:
299:
278:
272:
216:measurable set
200:if and only if
166:
165:
154:
151:
148:
145:
142:
139:
134:
129:
125:
87:
84:
38:Hilbert spaces
13:
10:
9:
6:
4:
3:
2:
5204:
5193:
5190:
5188:
5185:
5183:
5180:
5179:
5177:
5162:
5159:
5157:
5154:
5152:
5149:
5147:
5144:
5142:
5139:
5137:
5134:
5132:
5129:
5127:
5124:
5122:
5119:
5117:
5114:
5112:
5109:
5107:
5104:
5102:
5099:
5097:
5094:
5092:
5089:
5086:
5082:
5079:
5077:
5074:
5072:
5069:
5068:
5066:
5062:
5056:
5053:
5052:
5050:
5046:
5040:
5037:
5035:
5032:
5030:
5027:
5025:
5022:
5020:
5017:
5015:
5012:
5010:
5007:
5005:
5002:
5000:
4997:
4996:
4994:
4992:Miscellaneous
4990:
4983:
4979:
4976:
4974:
4971:
4969:
4966:
4964:
4961:
4960:
4958:
4954:
4948:
4945:
4943:
4940:
4938:
4935:
4933:
4930:
4929:
4927:
4923:
4915:
4912:
4911:
4910:
4907:
4905:
4902:
4900:
4897:
4895:
4892:
4890:
4887:
4885:
4881:
4879:
4876:
4875:
4873:
4869:
4863:
4860:
4858:
4855:
4853:
4850:
4848:
4845:
4843:
4840:
4838:
4835:
4833:
4830:
4828:
4825:
4823:
4820:
4819:
4817:
4813:
4807:
4804:
4802:
4799:
4797:
4794:
4792:
4789:
4787:
4784:
4780:
4777:
4775:
4772:
4770:
4767:
4766:
4765:
4762:
4761:
4759:
4757:Decomposition
4755:
4749:
4746:
4744:
4741:
4739:
4736:
4734:
4731:
4729:
4726:
4724:
4721:
4720:
4718:
4716:
4712:
4706:
4703:
4701:
4698:
4695:
4693:
4690:
4687:
4685:
4682:
4679:
4677:
4674:
4673:
4671:
4667:
4661:
4658:
4656:
4653:
4651:
4648:
4646:
4643:
4641:
4638:
4636:
4633:
4631:
4628:
4627:
4625:
4621:
4615:
4612:
4610:
4607:
4605:
4602:
4600:
4597:
4595:
4592:
4590:
4587:
4585:
4582:
4580:
4577:
4575:
4572:
4570:
4567:
4566:
4564:
4560:
4556:
4552:
4545:
4540:
4538:
4533:
4531:
4526:
4525:
4522:
4516:
4515:
4514:3-540-42248-X
4511:
4506:
4503:
4500:
4496:
4493:
4490:
4486:
4483:
4481:
4480:0-7204-0762-1
4477:
4474:
4470:
4468:
4467:0-444-86308-7
4464:
4460:
4456:
4453:
4452:
4444:
4442:3-540-42248-X
4438:
4434:
4430:
4426:
4420:
4417:
4410:
4408:
4406:
4402:
4397:
4392:
4388:
4384:
4380:
4376:
4373:
4368:
4366:
4362:
4358:
4354:
4350:
4346:
4341:
4335:
4330:
4326:
4307:
4304:
4301:
4298:
4291:
4285:
4279:
4276:
4270:
4262:
4258:
4252:
4247:
4243:
4239:
4233:
4227:
4220:
4219:
4218:
4216:
4195:
4192:
4189:
4179:
4175:
4164:
4163:
4162:
4160:
4156:
4152:
4148:
4144:
4140:
4136:
4132:
4128:
4126:
4122:
4118:
4115:
4111:
4108:
4104:
4100:
4096:
4092:
4084:
4082:
4080:
4075:
4071:
4067:
4063:
4041:
4035:
4032:
4027:
4023:
4017:
4012:
4008:
4004:
3992:
3991:
3990:
3988:
3984:
3977:
3973:
3969:
3965:
3961:
3939:
3933:
3930:
3925:
3921:
3915:
3910:
3906:
3902:
3899:
3892:
3891:
3890:
3888:
3884:
3882:
3878:
3873:
3871:
3850:
3846:
3842:
3832:
3829:
3825:
3821:
3818:
3811:
3810:
3809:
3806:
3802:
3799:
3794:
3790:
3785:
3781:
3776:
3772:
3769:
3748:
3744:
3733:
3730:
3726:
3722:
3719:
3712:
3711:
3710:
3707:
3703:
3697:
3693:
3689:
3684:
3682:
3678:
3674:
3670:
3666:
3662:
3660:
3656:
3636:
3633:
3629:
3626:
3623:
3617:
3602:
3601:
3600:
3598:
3594:
3590:
3586:
3582:
3578:
3574:
3566:
3550:
3544:
3538:
3535:
3531:
3527:
3523:
3517:
3512:
3508:
3504:
3500:
3486:
3480:
3477:
3472:
3468:
3462:
3457:
3453:
3438:
3437:
3436:
3433:
3429:
3423:
3419:
3415:
3411:
3407:
3404:
3400:
3395:
3391:
3369:
3363:
3360:
3355:
3351:
3345:
3340:
3336:
3328:
3327:
3326:
3306:
3300:
3297:
3292:
3288:
3282:
3277:
3273:
3265:
3264:
3263:
3260:
3256:
3250:
3246:
3242:
3238:
3217:
3213:
3209:
3200:
3197:
3194:
3191:
3186:
3182:
3172:
3153:
3152:
3151:
3149:
3145:
3140:
3136:
3130:
3126:
3122:
3118:
3094:
3090:
3083:
3077:
3072:
3068:
3060:
3059:
3058:
3055:
3051:
3045:
3041:
3037:
3032:
3028:
3022:
3018:
3010:
3008:
3004:
2977:
2973:
2969:
2961:
2955:
2951:
2943:
2942:
2941:
2939:
2935:
2916:
2913:
2910:
2904:
2901:
2898:
2895:
2892:
2885:
2884:
2883:
2882:
2860:
2854:
2851:
2846:
2842:
2836:
2831:
2827:
2822:
2816:
2810:
2807:
2802:
2798:
2792:
2787:
2783:
2775:
2774:
2773:
2771:
2764:
2760:
2753:
2749:
2745:
2741:
2739:
2734:
2732:
2728:
2721:
2717:
2714:
2711:
2689:
2683:
2680:
2675:
2671:
2665:
2660:
2656:
2649:
2646:
2643:
2636:
2635:
2634:
2632:
2625:
2621:
2601:
2595:
2589:
2586:
2581:
2577:
2571:
2566:
2562:
2554:
2553:
2552:
2550:
2543:
2539:
2535:
2531:
2527:
2524:
2520:
2516:
2510:
2508:
2506:
2499:
2495:
2491:
2489:
2482:
2477:
2457:
2451:
2448:
2443:
2439:
2430:
2425:
2421:
2414:
2407:
2406:
2405:
2378:
2372:
2369:
2361:
2357:
2351:
2346:
2342:
2331:
2319:
2306:
2302:
2298:
2295:
2288:
2287:
2286:
2284:
2280:
2278:
2274:
2269:
2267:
2248:
2242:
2236:
2233:
2225:
2221:
2212:
2208:
2199:
2194:
2190:
2186:
2173:
2167:
2164:
2159:
2155:
2146:
2141:
2137:
2116:
2110:
2107:
2102:
2098:
2089:
2084:
2080:
2065:
2064:
2063:
2036:
2030:
2027:
2019:
2015:
2009:
2004:
2000:
1989:
1983:
1977:
1971:
1968:
1963:
1959:
1950:
1945:
1941:
1933:
1932:
1931:
1911:
1903:
1899:
1892:
1887:
1884:
1881:
1849:
1848:
1847:
1845:
1840:
1837:
1833:
1828:
1824:
1819:
1815:
1811:
1810:
1804:
1800:
1795:
1791:
1786:
1782:
1776:
1772:
1753:
1748:
1742:
1737:
1732:
1727:
1722:
1715:
1708:
1704:
1698:
1693:
1688:
1681:
1676:
1671:
1666:
1661:
1654:
1649:
1644:
1637:
1633:
1627:
1620:
1615:
1610:
1605:
1598:
1594:
1587:
1578:
1577:
1576:
1574:
1570:
1551:
1546:
1540:
1535:
1530:
1525:
1520:
1513:
1506:
1503:
1499:
1493:
1486:
1483:
1479:
1471:
1468:
1464:
1456:
1451:
1446:
1441:
1436:
1429:
1422:
1419:
1415:
1409:
1402:
1398:
1390:
1386:
1378:
1371:
1368:
1364:
1358:
1351:
1347:
1339:
1335:
1328:
1319:
1318:
1317:
1298:
1294:
1283:
1280:
1276:
1272:
1269:
1262:
1261:
1260:
1258:
1250:
1232:
1228:
1217:
1214:
1210:
1206:
1200:
1194:
1183:
1179:
1173:
1168:
1164:
1156:
1155:
1154:
1151:
1147:
1143:
1139:
1135:
1132:. Thus, when
1131:
1127:
1126:countable set
1123:
1115:
1099:
1093:
1087:
1076:
1072:
1066:
1061:
1057:
1047:
1041:
1030:
1026:
1020:
1015:
1011:
1003:
1002:
1001:
984:
979:
975:
971:
966:
960:
950:
939:
931:
924:
923:
922:
920:
916:
912:
909:
905:
901:
898:
894:
890:
870:
864:
849:
843:
832:
826:
818:
814:
810:
804:
796:
786:
785:
784:
781:
777:
771:
767:
745:
739:
728:
724:
718:
713:
709:
701:
700:
699:
697:
693:
689:
684:
680:
676:
672:
667:
663:
658:
654:
649:
645:
639:
635:
632:is a family {
630:
626:
620:
616:
612:
611:cross-section
607:
594:
589:
585:
576:
572:
554:
550:
519:
516:
513:
499:
495:
487:
484:
481:
467:
453:
449:
444:
430:
429:
428:
426:
421:
417:
396:
392:
388:
385:
379:
369:
364:
360:
352:
351:
350:
348:
327:
324:
321:
318:
315:
305:
301:
290:
289:
288:
285:
281:
275:
271:
267:
263:
259:
255:
251:
249:
245:
241:
237:
233:
229:
225:
221:
217:
213:
209:
205:
201:
198:
194:
190:
186:
182:
178:
174:
170:
152:
146:
143:
140:
132:
127:
123:
115:
114:
113:
111:
107:
104:
100:
97:
93:
85:
83:
81:
77:
73:
69:
68:George Mackey
64:
62:
57:
55:
51:
47:
43:
39:
35:
31:
27:
23:
19:
5064:Applications
4894:Disk algebra
4795:
4748:Spectral gap
4623:Main results
4507:
4488:
4485:G. W. Mackey
4472:
4471:J. Dixmier,
4458:
4428:
4419:
4400:
4395:
4390:
4386:
4382:
4378:
4374:
4371:
4369:
4360:
4356:
4352:
4348:
4344:
4339:
4333:
4328:
4324:
4322:
4214:
4212:
4158:
4154:
4150:
4146:
4142:
4138:
4134:
4130:
4129:
4120:
4116:
4102:
4098:
4090:
4088:
4078:
4073:
4069:
4065:
4061:
4059:
3986:
3982:
3975:
3971:
3967:
3963:
3959:
3957:
3886:
3885:
3880:
3876:
3874:
3869:
3867:
3804:
3800:
3797:
3792:
3788:
3783:
3779:
3774:
3770:
3767:
3765:
3705:
3701:
3695:
3691:
3687:
3685:
3680:
3676:
3672:
3668:
3664:
3663:
3658:
3654:
3652:
3596:
3592:
3588:
3580:
3576:
3572:
3570:
3431:
3427:
3421:
3417:
3413:
3412:
3408:
3402:
3398:
3393:
3389:
3387:
3324:
3258:
3254:
3248:
3244:
3240:
3236:
3234:
3147:
3143:
3138:
3134:
3128:
3124:
3120:
3114:
3053:
3049:
3043:
3039:
3030:
3026:
3020:
3016:
3014:
3005:
3002:
2937:
2933:
2931:
2878:
2769:
2762:
2758:
2751:
2747:
2743:
2742:
2737:
2735:
2730:
2726:
2719:
2715:
2712:
2709:
2707:
2630:
2623:
2619:
2617:
2548:
2541:
2537:
2533:
2529:
2525:
2518:
2517:
2514:
2504:
2497:
2493:
2492:
2487:
2480:
2479:This allows
2478:
2475:
2403:
2282:
2281:
2276:
2272:
2270:
2266:decomposable
2265:
2263:
2061:
1929:
1841:
1835:
1831:
1826:
1822:
1817:
1813:
1807:
1802:
1798:
1793:
1789:
1784:
1780:
1774:
1770:
1768:
1568:
1566:
1315:
1256:
1254:
1149:
1145:
1141:
1137:
1133:
1121:
1119:
999:
918:
914:
913:
910:
903:
899:
896:
892:
891:
888:
779:
775:
769:
765:
763:
691:
687:
682:
678:
674:
670:
665:
661:
656:
652:
647:
643:
637:
633:
628:
624:
618:
614:
610:
608:
541:
424:
419:
415:
413:
346:
344:
283:
279:
273:
269:
265:
261:
257:
253:
252:
239:
235:
231:
227:
223:
211:
204:Polish space
196:
184:
176:
172:
168:
167:
109:
105:
98:
91:
89:
65:
58:
49:
29:
25:
15:
5091:Heat kernel
4791:Compression
4676:Isospectral
4473:C* algebras
4393:such that π
3868:represents
2279:. In fact,
1128:and μ is a
177:Borel space
61:C*-algebras
18:mathematics
5176:Categories
4769:Continuous
4584:C*-algebra
4579:B*-algebra
4455:J. Dixmier
4411:References
4217:such that
4095:C*-algebra
3970:so that Z(
3889:. Suppose
3709:such that
2708:such that
2618:To assert
2521:. For any
1846:, that is
651:such that
349:such that
254:Definition
189:Borel sets
34:direct sum
4555:-algebras
4302:∈
4296:∀
4280:μ
4259:π
4253:⊕
4244:∫
4228:π
4193:∈
4176:π
4125:separable
4036:μ
4018:⊕
4009:∫
3934:μ
3916:⊕
3907:∫
3833:∈
3826:⨁
3734:∈
3727:∑
3630:∩
3583:) be the
3539:μ
3518:⊕
3509:∫
3481:μ
3463:⊕
3454:∫
3364:μ
3346:⊕
3337:∫
3301:μ
3283:⊕
3274:∫
3235:where W*(
3198:∈
3173:
3084:
3078:⊆
2956:ϕ
2914:−
2908:→
2902:−
2893:φ
2855:ν
2837:⊕
2828:∫
2811:μ
2793:⊕
2784:∫
2684:μ
2666:⊕
2657:∫
2653:→
2590:μ
2572:⊕
2563:∫
2452:μ
2440:λ
2431:⊕
2422:∫
2418:↦
2415:λ
2404:given by
2373:μ
2352:⊕
2343:∫
2332:
2326:→
2312:∞
2307:μ
2296:ϕ
2237:μ
2200:⊕
2191:∫
2168:μ
2147:⊕
2138:∫
2111:μ
2090:⊕
2081:∫
2031:μ
2010:⊕
2001:∫
1990:
1984:∈
1972:μ
1951:⊕
1942:∫
1915:∞
1909:‖
1896:‖
1893:
1885:∈
1743:⋱
1738:⋮
1733:⋯
1728:⋮
1723:⋮
1716:⋯
1705:λ
1699:⋯
1682:⋯
1677:⋮
1672:⋱
1667:⋮
1662:⋮
1655:⋯
1645:⋯
1634:λ
1621:⋯
1611:⋯
1595:λ
1541:⋱
1536:⋮
1531:⋯
1526:⋮
1521:⋮
1514:⋯
1494:⋯
1457:⋯
1452:⋮
1447:⋱
1442:⋮
1437:⋮
1430:⋯
1410:⋯
1379:⋯
1359:⋯
1284:∈
1277:⨁
1218:∈
1211:⨁
1207:≅
1195:μ
1174:⊕
1165:∫
1088:ν
1067:⊕
1058:∫
1054:→
1042:μ
1021:⊕
1012:∫
961:ν
951:μ
935:↦
865:μ
856:⟩
824:⟨
815:∫
808:⟩
794:⟨
740:μ
719:⊕
710:∫
586:ℓ
551:ℓ
520:ω
496:ℓ
488:ω
389:∈
328:ω
325:≤
319:≤
181:σ-algebra
128:μ
5156:Weyl law
5101:Lax pair
5048:Examples
4882:With an
4801:Discrete
4779:Residual
4715:Spectrum
4700:operator
4692:operator
4684:operator
4599:Spectrum
4427:(2001),
4361:disjoint
3985:) where
3637:′
3571:Suppose
3532:′
3501:′
669:for all
248:σ-finite
228:standard
220:null set
197:standard
4697:Unitary
4131:Theorem
3887:Theorem
3665:Example
3414:Theorem
2744:Theorem
2519:Theorem
2494:Theorem
2283:Theorem
1116:Example
915:Theorem
4681:Normal
4512:
4478:
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4133:. Let
4079:factor
3681:factor
3585:center
3416:. If {
3243:. If {
2932:where
2761:) and
2436:
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897:fibers
893:Remark
573:; all
414:where
256:. Let
4774:Point
4112:of a
3766:then
3686:When
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3653:Then
3057:with
3015:Let {
1788:with
1124:is a
242:is a
226:is a
210:μ on
4705:Unit
4553:and
4510:ISBN
4476:ISBN
4463:ISBN
4437:ISBN
3962:and
3388:for
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613:of {
485:<
264:on (
214:, a
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1259:on
187:as
183:of
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5178::
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3406:.
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3146:∈
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3132:}
3052:∈
3029:∈
2936:,
2268:.
1839:.
1783:∈
1403:22
1391:21
1352:12
1340:11
1136:=
778:∈
690:,
673:∈
660:∈
646:∈
627:∈
609:A
282:∈
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238:−
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5083:(
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4980:(
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4180:x
4172:{
4159:X
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4042:x
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4033:d
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4005:=
4001:A
3987:X
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3979:μ
3976:L
3972:A
3968:H
3964:A
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3943:)
3940:x
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3903:=
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3822:=
3819:A
3805:i
3801:E
3798:A
3793:i
3789:E
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3780:H
3775:i
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3749:i
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3731:i
3723:=
3720:1
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3702:i
3699:}
3696:i
3692:E
3688:A
3677:A
3673:A
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3657:(
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3505:=
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3403:x
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3394:x
3390:T
3373:)
3370:x
3367:(
3361:d
3356:x
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3310:)
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3304:(
3298:d
3293:x
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3278:X
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3255:x
3252:}
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3245:A
3241:S
3237:S
3218:x
3214:A
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3207:)
3204:}
3201:D
3195:S
3192::
3187:x
3183:S
3179:{
3176:(
3167:*
3163:W
3148:X
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3121:D
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3095:x
3091:H
3087:(
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3050:x
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3038:{
3031:X
3027:x
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3021:x
3017:H
2978:x
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2970:=
2965:)
2962:x
2959:(
2952:K
2938:F
2934:E
2917:F
2911:Y
2905:E
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2896::
2864:)
2861:y
2858:(
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2832:Y
2823:,
2820:)
2817:x
2814:(
2808:d
2803:x
2799:H
2788:X
2770:Y
2768:(
2766:ν
2763:L
2759:X
2757:(
2755:μ
2752:L
2748:A
2738:X
2731:A
2727:X
2725:(
2723:μ
2720:L
2716:U
2713:A
2710:U
2693:)
2690:x
2687:(
2681:d
2676:x
2672:H
2661:X
2650:H
2647::
2644:U
2631:X
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2627:μ
2624:L
2620:A
2602:.
2599:)
2596:x
2593:(
2587:d
2582:x
2578:H
2567:X
2549:X
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2545:μ
2542:L
2538:X
2534:X
2530:H
2526:A
2505:X
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2501:μ
2498:L
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2481:L
2461:)
2458:x
2455:(
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2303:L
2299::
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2249:.
2246:)
2243:x
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2234:d
2231:)
2226:x
2222:s
2218:(
2213:x
2209:T
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2187:=
2182:)
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2174:x
2171:(
2165:d
2160:x
2156:s
2142:X
2132:(
2125:]
2120:)
2117:x
2114:(
2108:d
2103:x
2099:T
2085:X
2075:[
2045:)
2040:)
2037:x
2034:(
2028:d
2020:x
2016:H
2005:X
1995:(
1987:L
1981:)
1978:x
1975:(
1969:d
1964:x
1960:T
1946:X
1904:x
1900:T
1888:X
1882:x
1877:p
1874:u
1871:s
1868:-
1865:s
1862:s
1859:e
1836:n
1832:X
1827:x
1823:H
1818:n
1814:X
1803:x
1799:H
1794:x
1790:T
1785:X
1781:x
1778:}
1775:x
1771:T
1754:.
1749:]
1709:n
1694:0
1689:0
1650:0
1638:2
1628:0
1616:0
1606:0
1599:1
1588:[
1552:.
1547:]
1507:n
1504:n
1500:T
1487:2
1484:n
1480:T
1472:1
1469:n
1465:T
1423:n
1420:2
1416:T
1399:T
1387:T
1372:n
1369:1
1365:T
1348:T
1336:T
1329:[
1299:k
1295:H
1288:N
1281:k
1273:=
1270:H
1257:T
1233:k
1229:H
1222:N
1215:k
1204:)
1201:x
1198:(
1191:d
1184:x
1180:H
1169:X
1150:k
1146:H
1142:N
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1134:X
1122:X
1100:.
1097:)
1094:x
1091:(
1084:d
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1048:x
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1038:d
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1016:X
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980:2
976:/
972:1
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957:d
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861:d
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850:x
847:(
844:t
840:|
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805:t
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797:s
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773:}
770:x
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749:)
746:x
743:(
736:d
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595:.
590:2
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514:n
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454:{
450:=
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306:n
302:X
298:{
284:X
280:x
277:}
274:x
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258:X
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150:)
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