5040:
32:
1039:
2119:
310:, and are also used in algebraic formulations of quantum mechanics. Another active area of research is the program to obtain classification, or to determine the extent of which classification is possible, for separable simple
1462:
2393:, and are often not worried about the subtleties associated with an infinite number of dimensions. (Mathematicians usually use the asterisk, *, to denote the Hermitian adjoint.) â -algebras feature prominently in
740:
1345:
2228:
of matrix algebras. In fact, all C*-algebras that are finite dimensional as vector spaces are of this form, up to isomorphism. The self-adjoint requirement means finite-dimensional C*-algebras are
2616:
1507:
1398:
2300:
4427:
2046:
827:
1231:
1153:
664:
907:
570:
481:
630:
1464:, and therefore, a B*-algebra is also a C*-algebra. Conversely, the C*-condition implies the B*-condition. This is nontrivial, and can be proved without using the condition
930:
4929:
4529:
1681:
3510:
it also has representations of type II and type III. Thus for C*-algebras and locally compact groups, it is only meaningful to speak of type I and non type I properties.
371:
3140:
1955:
3287:
3247:
3183:
3016:
2976:
2920:
2884:
2836:
1827:
232:
2052:
1923:
1767:
1583:
1853:
1741:
1651:
1617:
4162:
3083:
1715:
397:
4592:
3327:
3307:
3203:
3103:
3056:
3036:
2940:
1893:
1873:
1791:
3637:
2405:
3808:. This is a somewhat dated reference, but is still considered as a high-quality technical exposition. It is available in English from North Holland press.
4755:
1528:. 'C' stood for 'closed'. In his paper Segal defines a C*-algebra as a "uniformly closed, self-adjoint algebra of bounded operators on a Hilbert space".
4167:
3940:
4882:
4737:
4514:
3419:
4713:
1540:
or by reduction to commutative C*-algebras. In the latter case, we can use the fact that the structure of these is completely determined by the
4407:
3716:
John A. Holbrook, David W. Kribs, and
Raymond Laflamme. "Noiseless Subsystems and the Structure of the Commutant in Quantum Error Correction."
4260:
4058:
3824:
3442:
is non-abelian. In particular, the dual of a locally compact group is defined to be the primitive ideal space of the group C*-algebra. See
1403:
4255:
3492:)) is a type I von Neumann algebra. In fact it is sufficient to consider only factor representations, i.e. representations Ï for which Ï(
1536:
C*-algebras have a large number of properties that are technically convenient. Some of these properties can be established by using the
2518:
Concrete C*-algebras of compact operators admit a characterization similar to
Wedderburn's theorem for finite dimensional C*-algebras:
4605:
4694:
4585:
4412:
3883:
3843:
3802:
3776:
3746:
2496:
115:
3468:
implies that any C*-algebra has a universal enveloping W*-algebra, such that any homomorphism to a W*-algebra factors through it.
683:
4964:
4230:
1509:. For these reasons, the term B*-algebra is rarely used in current terminology, and has been replaced by the term 'C*-algebra'.
4609:
4199:
3782:. This book is widely regarded as a source of new research material, providing much supporting intuition, but it is difficult.
1295:
4189:
4184:
4177:
4113:
3997:
1770:
1537:
53:
96:
4760:
4422:
3933:
3864:
1243:
4816:
2560:
2233:
1467:
1358:
68:
5069:
5043:
4765:
4750:
4578:
4048:
4780:
4559:
4033:
2468:
4479:
3859:
2398:
1965:
49:
5025:
4785:
4534:
4432:
4312:
3465:
75:
4979:
4903:
3590:
1961:
159:
5020:
4028:
2257:
42:
4836:
4539:
4402:
4235:
4220:
3992:
3457:, known as W* algebras before the 1960s, are a special kind of C*-algebra. They are required to be closed in the
404:
4770:
4121:
2351:. This vector uniquely determines the isomorphism class of a finite-dimensional C*-algebra. In the language of
2015:
765:
4872:
4673:
4131:
4002:
3926:
3443:
3106:
2145:
1969:
1177:
1098:
141:
4745:
641:
82:
4969:
4494:
4469:
4287:
4276:
3987:
3642:
3338:
3334:
3206:
20:
4345:
4335:
4330:
854:
510:
322:
We begin with the abstract characterization of C*-algebras given in the 1943 paper by
Gelfand and Naimark.
5000:
4944:
4908:
4038:
3458:
3146:
424:
303:
191:
576:
299:
yielded an abstract characterisation of C*-algebras making no reference to operators on a
Hilbert space.
280:
attempted to establish a general framework for these algebras, which culminated in a series of papers on
64:
4090:
3412:
2225:
307:
1034:{\displaystyle \|x\|^{2}=\|x^{*}x\|=\sup\{|\lambda |:x^{*}x-\lambda \,1{\text{ is not invertible}}\}.}
5064:
4983:
4504:
4483:
4397:
4282:
4245:
3354:
2979:
2659:
2205:
1989:
1282:
911:
which is sometimes called the B*-identity. For history behind the names C*- and B*-algebras, see the
152:
4949:
4887:
4601:
4307:
4043:
3753:
3594:
3454:
2386:
2221:
2170:
1541:
285:
129:
1664:
4974:
4841:
4437:
4366:
4297:
4141:
4103:
3500:
3378:
2229:
1047:
311:
281:
3854:
2114:{\displaystyle 0\leq e_{\lambda }\leq e_{\mu }\leq 1\quad {\mbox{ whenever }}\lambda \leq \mu .}
343:
3112:
1928:
4954:
4544:
4519:
4204:
4126:
3879:
3839:
3820:
3798:
3772:
3742:
3626:
3621:
3519:
3431:
3256:
3216:
3152:
2985:
2945:
2889:
2853:
2805:
2394:
2390:
2352:
1796:
1246:, i.e. bounded with norm †1. Furthermore, an injective *-homomorphism between C*-algebras is
338:
265:
253:
201:
145:
1902:
1746:
1555:
4959:
4877:
4846:
4826:
4811:
4806:
4801:
4638:
4549:
4250:
4098:
4053:
3977:
3905:
3681:
3665:
3611:
2436:
1973:
1832:
1720:
1626:
1592:
1286:
284:
of operators. These papers considered a special class of C*-algebras that are now known as
277:
269:
257:
3752:. An excellent introduction to the subject, accessible for those with a knowledge of basic
3061:
1700:
4821:
4775:
4723:
4718:
4689:
4570:
4524:
4509:
4417:
4380:
4376:
4340:
4302:
4240:
4225:
4194:
4136:
4095:
4082:
4007:
3949:
3871:
3812:
3786:
3734:
3631:
2847:
2799:
2484:
2421:
919:
246:
243:
89:
4648:
376:
5010:
4862:
4663:
4474:
4453:
4371:
4361:
4172:
4079:
4012:
3972:
3765:
3606:
3312:
3292:
3188:
3088:
3041:
3021:
2925:
2783:
2248:
1878:
1858:
1776:
334:
330:
292:
273:
163:
149:
137:
2886:
under pointwise multiplication and addition. The involution is pointwise conjugation.
5058:
5015:
4939:
4668:
4653:
4643:
3330:
3210:
2425:
2356:
2217:
1691:
181:
166:
3910:
3109:, which applies to locally compact Hausdorff spaces. Any such sequence of functions
5005:
4658:
4628:
4292:
4146:
4087:
3893:
3760:
3616:
1694:
1513:
296:
2212:
becomes a C*-algebra if we consider matrices as operators on the
Euclidean space,
1355:
This condition automatically implies that the *-involution is isometric, that is,
3918:
2389:, â , is used in the name because physicists typically use the symbol to denote a
922:, it implies that the C*-norm is uniquely determined by the algebraic structure:
4934:
4924:
4831:
4633:
4489:
4074:
3849:. Mathematically rigorous reference which provides extensive physics background.
3791:
3507:
3250:
2504:
1620:
31:
19:
This article is about an area of mathematics. For the concept in rocketry, see
4707:
4703:
4699:
918:
The C*-identity is a very strong requirement. For instance, together with the
261:
177:
3967:
3953:
3506:
However, if a C*-algebra has non-type I representations, then by results of
2176:
Similarly, a closed two-sided ideal of a C*-algebra is itself a C*-algebra.
842:
4554:
4499:
2763:
2321:, is isomorphic (in a noncanonical way) to the full matrix algebra M(dim(
2166:
1247:
3480:
is of type I if and only if for all non-degenerate representations Ï of
3149:
states that every commutative C*-algebra is *-isomorphic to the algebra
198:
Another important class of non-Hilbert C*-algebras includes the algebra
3589:
This C*-algebra approach is used in the HaagâKastler axiomatization of
2382:
1457:{\displaystyle \lVert xx^{*}\rVert =\lVert x\rVert \lVert x^{*}\rVert }
2404:
An immediate generalization of finite dimensional C*-algebras are the
1661:, even though this terminology conflicts with its use for elements of
3836:
Algebraic
Methods in Statistical Mechanics and Quantum Field Theory
3499:
A locally compact group is said to be of type I if and only if its
3018:
this is immediate: consider the directed set of compact subsets of
2515:). It is also closed under involution; hence it is a C*-algebra.
2309:
is the set of minimal nonzero self-adjoint central projections of
912:
2455:, is *-isomorphic to a norm-closed adjoint closed subalgebra of
1524:), namely, the space of bounded operators on some Hilbert space
4574:
3922:
3817:
Characterizations of C*-algebras: The
Gelfand-Naimark Theorems
3522:, one typically describes a physical system with a C*-algebra
2165:
Using approximate identities, one can show that the algebraic
25:
2762:). For separable Hilbert spaces, it is the unique ideal. The
252:
C*-algebras were first considered primarily for their use in
3896:(1947), "Irreducible representations of operator algebras",
1968:
of a C*-algebra, which in turn can be used to construct the
735:{\displaystyle (\lambda x)^{*}={\overline {\lambda }}x^{*}.}
3578:) such that Ï(1) = 1. The expected value of the observable
3085:
be a function of compact support which is identically 1 on
2139:
will have a sequential approximate identity if and only if
1960:
This partially ordered subspace allows the definition of a
3333:. This characterization is one of the motivations for the
2232:, from which fact one can deduce the following theorem of
2674:
is isomorphic to the space of square summable sequences
2416:
The prototypical example of a C*-algebra is the algebra
1340:{\displaystyle \lVert xx^{*}\rVert =\lVert x\rVert ^{2}}
3418:. This is defined as the enveloping C*-algebra of the
2135:
has a sequential approximate identity. More generally,
302:
C*-algebras are now an important tool in the theory of
3357:, there is a unique (up to C*-isomorphism) C*-algebra
2093:
1623:. This cone is identical to the elements of the form
379:
346:
3634:, a unital subspace of a C*-algebra that is *-closed.
3550:
of the system is defined as a positive functional on
3315:
3295:
3259:
3219:
3191:
3155:
3115:
3091:
3064:
3044:
3024:
2988:
2948:
2928:
2892:
2856:
2808:
2563:
2260:
2055:
2018:
1964:
on a C*-algebra, which in turn is used to define the
1931:
1905:
1881:
1861:
1835:
1799:
1779:
1749:
1723:
1703:
1667:
1629:
1595:
1558:
1470:
1406:
1361:
1298:
1180:
1101:
933:
857:
768:
686:
644:
579:
513:
427:
204:
4993:
4917:
4896:
4855:
4794:
4736:
4682:
4617:
4462:
4446:
4390:
4354:
4323:
4269:
4213:
4155:
4112:
4067:
4021:
3960:
3720:. Volume 2, Number 5, pp. 381–419. Oct 2003.
2658:) does not have an identity element, a sequential
2611:{\displaystyle A\cong \bigoplus _{i\in I}K(H_{i}),}
1502:{\displaystyle \lVert x\rVert =\lVert x^{*}\rVert }
1393:{\displaystyle \lVert x\rVert =\lVert x^{*}\rVert }
56:. Unsourced material may be challenged and removed.
4930:Spectral theory of ordinary differential equations
4530:Spectral theory of ordinary differential equations
3790:
3764:
3321:
3301:
3281:
3241:
3197:
3177:
3134:
3097:
3077:
3050:
3030:
3010:
2970:
2934:
2914:
2878:
2830:
2610:
2294:
2220:||·|| on matrices. The involution is given by the
2113:
2040:
1949:
1917:
1887:
1867:
1847:
1821:
1785:
1761:
1735:
1709:
1675:
1645:
1611:
1577:
1501:
1456:
1392:
1339:
1225:
1147:
1033:
901:
821:
734:
658:
624:
564:
475:
391:
365:
226:
4428:SchröderâBernstein theorems for operator algebras
2922:has a multiplicative unit element if and only if
1686:The set of self-adjoint elements of a C*-algebra
272:and in a more mathematically developed form with
3526:with unit element; the self-adjoint elements of
3411:Of particular importance is the C*-algebra of a
2621:where the (C*-)direct sum consists of elements (
2487:infinite-dimensional Hilbert space. The algebra
2278:
975:
1516:in 1947 to describe norm-closed subalgebras of
1238:In the case of C*-algebras, any *-homomorphism
3546:, the measurable quantities, of the system. A
4586:
3934:
3898:Bulletin of the American Mathematical Society
3381:, that is, every other continuous *-morphism
2169:of a C*-algebra by a closed proper two-sided
1250:. These are consequences of the C*-identity.
8:
3129:
3116:
2406:approximately finite dimensional C*-algebras
2295:{\displaystyle A=\bigoplus _{e\in \min A}Ae}
1552:Self-adjoint elements are those of the form
1496:
1483:
1477:
1471:
1451:
1438:
1435:
1429:
1423:
1407:
1387:
1374:
1368:
1362:
1328:
1321:
1315:
1299:
1025:
978:
969:
953:
941:
934:
887:
880:
874:
858:
813:
800:
797:
791:
785:
769:
2224:. More generally, one can consider finite
1717:. In this ordering, a self-adjoint element
4621:
4593:
4579:
4571:
3941:
3927:
3919:
3677:
3661:
3461:, which is weaker than the norm topology.
2838:of complex-valued continuous functions on
2385:for a finite-dimensional C*-algebra. The
2173:, with the natural norm, is a C*-algebra.
2041:{\displaystyle xe_{\lambda }\rightarrow x}
822:{\displaystyle \|x^{*}x\|=\|x\|\|x^{*}\|.}
234:of complex-valued continuous functions on
3909:
3582:, if the system is in state Ï, is then Ï(
3392:factors uniquely through Ï. The algebra
3314:
3294:
3264:
3258:
3224:
3218:
3190:
3160:
3154:
3123:
3114:
3090:
3069:
3063:
3043:
3023:
2993:
2987:
2953:
2947:
2927:
2897:
2891:
2861:
2855:
2813:
2807:
2596:
2574:
2562:
2271:
2259:
2092:
2079:
2066:
2054:
2026:
2017:
1930:
1904:
1880:
1860:
1834:
1810:
1798:
1778:
1748:
1722:
1702:
1669:
1668:
1666:
1637:
1628:
1600:
1594:
1569:
1557:
1490:
1469:
1445:
1417:
1405:
1381:
1360:
1331:
1309:
1297:
1277:Some history: B*-algebras and C*-algebras
1226:{\displaystyle \pi (x^{*})=\pi (x)^{*}\,}
1222:
1216:
1191:
1179:
1148:{\displaystyle \pi (xy)=\pi (x)\pi (y)\,}
1144:
1100:
1020:
1016:
1001:
989:
981:
960:
944:
932:
890:
868:
856:
807:
776:
767:
723:
709:
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685:
652:
651:
643:
616:
606:
593:
578:
556:
543:
530:
512:
461:
451:
432:
426:
378:
357:
345:
209:
203:
116:Learn how and when to remove this message
4883:Group algebra of a locally compact group
3793:Les C*-algÚbres et leurs représentations
1992:. In fact, there is a directed family {
659:{\displaystyle \lambda \in \mathbb {C} }
190:is closed under the operation of taking
3654:
1351:in the given B*-algebra. (B*-condition)
2942:is compact. As does any C*-algebra,
1585:. The set of elements of a C*-algebra
1512:The term C*-algebra was introduced by
1281:The term B*-algebra was introduced by
4261:Spectral theory of normal C*-algebras
4059:Spectral theory of normal C*-algebras
3704:
3693:
2537:), then there exists Hilbert spaces {
2420:of bounded (equivalently continuous)
902:{\displaystyle \|xx^{*}\|=\|x\|^{2},}
565:{\displaystyle (x+y)^{*}=x^{*}+y^{*}}
7:
4256:Spectral theory of compact operators
3514:C*-algebras and quantum field theory
2670:) can be developed. To be specific,
2329:). The finite family indexed on min
1980:Quotients and approximate identities
476:{\displaystyle x^{**}=(x^{*})^{*}=x}
264:. This line of research began with
54:adding citations to reliable sources
2381:) is the name occasionally used in
1653:. Elements of this cone are called
837:The first four identities say that
625:{\displaystyle (xy)^{*}=y^{*}x^{*}}
4408:CohenâHewitt factorization theorem
3638:GelfandâNaimarkâSegal construction
3488:)âł (that is, the bicommutant of Ï(
2716:be the orthogonal projection onto
2251:isomorphic to a finite direct sum
1697:; the ordering is usually denoted
845:. The last identity is called the
14:
4413:Extensions of symmetric operators
3597:is associated with a C*-algebra.
2850:) forms a commutative C*-algebra
2754:) is a two-sided closed ideal of
2243:A finite-dimensional C*-algebra,
1690:naturally has the structure of a
148:. A particular case is that of a
144:satisfying the properties of the
16:Topological complex vector space
5039:
5038:
4965:Topological quantum field theory
4231:Positive operator-valued measure
3289:as C*-algebras, it follows that
2697:be the subspace of sequences of
2475:C*-algebras of compact operators
2463:) for a suitable Hilbert space,
1793:is non-negative, if and only if
169:with two additional properties:
128:In mathematics, specifically in
30:
4515:RayleighâFaberâKrahn inequality
3911:10.1090/S0002-9904-1947-08742-5
3105:. Such functions exist by the
2736:is an approximate identity for
2091:
399:with the following properties:
238:that vanish at infinity, where
41:needs additional citations for
3718:Quantum Information Processing
3276:
3270:
3236:
3230:
3172:
3166:
3005:
2999:
2965:
2959:
2909:
2903:
2873:
2867:
2825:
2819:
2602:
2589:
2185:Finite-dimensional C*-algebras
2032:
1538:continuous functional calculus
1213:
1206:
1197:
1184:
1141:
1135:
1129:
1123:
1114:
1105:
990:
982:
697:
687:
590:
580:
527:
514:
458:
444:
350:
221:
215:
1:
4761:Uniform boundedness principle
4423:Limiting absorption principle
3430:provides context for general
2628:) of the Cartesian product Î
2467:; this is the content of the
2451:. In fact, every C*-algebra,
4049:Singular value decomposition
3142:is an approximate identity.
2802:Hausdorff space. The space
2003:of self-adjoint elements of
1855:. Two self-adjoint elements
1676:{\displaystyle \mathbb {R} }
1289:that satisfy the condition:
714:
276:around 1933. Subsequently,
4480:Hearing the shape of a drum
4163:Decomposition of a spectrum
3876:C*-algebras and W*-algebras
3860:Encyclopedia of Mathematics
3815:; Belfi, Victor A. (1986),
3739:An Invitation to C*-Algebra
2846:(defined in the article on
2686:. For each natural number
2399:quantum information science
1253:A bijective *-homomorphism
366:{\textstyle x\mapsto x^{*}}
160:continuous linear operators
136:(pronounced "C-star") is a
5086:
4904:Invariant subspace problem
4068:Special Elements/Operators
3593:, where every open set of
3591:local quantum field theory
3484:the von Neumann algebra Ï(
2149:, i.e. a positive element
1962:positive linear functional
18:
5034:
4624:
4540:Superstrong approximation
4403:Banach algebra cohomology
4236:Projection-valued measure
4221:Borel functional calculus
3993:Projection-valued measure
3349:Given a Banach *-algebra
3135:{\displaystyle \{f_{K}\}}
2701:which vanish for indices
2146:strictly positive element
1950:{\displaystyle x-y\geq 0}
638:For every complex number
318:Abstract characterization
291:Around 1943, the work of
4873:Spectrum of a C*-algebra
4132:Spectrum of a C*-algebra
4003:Spectrum of a C*-algebra
3767:Non-commutative geometry
3542:) are thought of as the
3444:spectrum of a C*-algebra
3404:of the Banach *-algebra
3365:) and *-morphism Ï from
3282:{\displaystyle C_{0}(Y)}
3242:{\displaystyle C_{0}(X)}
3178:{\displaystyle C_{0}(X)}
3107:Tietze extension theorem
3011:{\displaystyle C_{0}(X)}
2971:{\displaystyle C_{0}(X)}
2915:{\displaystyle C_{0}(X)}
2879:{\displaystyle C_{0}(X)}
2831:{\displaystyle C_{0}(X)}
2412:C*-algebras of operators
2377:(or, more explicitly, a
1970:spectrum of a C*-algebra
1822:{\displaystyle x=s^{*}s}
1532:Structure of C*-algebras
227:{\displaystyle C_{0}(X)}
4970:Noncommutative geometry
4560:WienerâKhinchin theorem
4495:Kuznetsov trace formula
4470:Almost Mathieu operator
4288:Banach function algebra
4277:Amenable Banach algebra
4034:GelfandâNaimark theorem
3988:Noncommutative topology
3643:Jordan operator algebra
3339:noncommutative geometry
3335:noncommutative topology
3038:, and for each compact
2790:Commutative C*-algebras
2469:GelfandâNaimark theorem
1918:{\displaystyle x\geq y}
1762:{\displaystyle x\geq 0}
1578:{\displaystyle x=x^{*}}
1242:between C*-algebras is
1022: is not invertible
920:spectral radius formula
304:unitary representations
21:characteristic velocity
5026:TomitaâTakesaki theory
5001:Approximation property
4945:Calculus of variations
4535:SturmâLiouville theory
4433:ShermanâTakeda theorem
4313:TomitaâTakesaki theory
4088:Hermitian/Self-adjoint
4039:Gelfand representation
3838:, Wiley-Interscience,
3678:Doran & Belfi 1986
3662:Doran & Belfi 1986
3466:ShermanâTakeda theorem
3459:weak operator topology
3426:. The C*-algebra of
3323:
3303:
3283:
3243:
3199:
3179:
3147:Gelfand representation
3136:
3099:
3079:
3052:
3032:
3012:
2972:
2936:
2916:
2880:
2832:
2648:
2612:
2529:is a C*-subalgebra of
2315:
2296:
2115:
2042:
1951:
1919:
1889:
1869:
1849:
1848:{\displaystyle s\in A}
1823:
1787:
1763:
1737:
1736:{\displaystyle x\in A}
1711:
1677:
1647:
1646:{\displaystyle xx^{*}}
1613:
1612:{\displaystyle x^{*}x}
1579:
1503:
1458:
1394:
1341:
1227:
1149:
1062:, between C*-algebras
1035:
903:
849:and is equivalent to:
823:
736:
660:
626:
566:
477:
393:
367:
308:locally compact groups
228:
5021:BanachâMazur distance
4984:Generalized functions
4029:GelfandâMazur theorem
3853:A.I. Shtern (2001) ,
3558:-linear map Ï :
3413:locally compact group
3402:C*-enveloping algebra
3345:C*-enveloping algebra
3324:
3304:
3284:
3244:
3200:
3180:
3137:
3100:
3080:
3078:{\displaystyle f_{K}}
3053:
3033:
3013:
2973:
2937:
2917:
2881:
2833:
2678:; we may assume that
2613:
2520:
2424:defined on a complex
2355:, this vector is the
2297:
2238:
2116:
2043:
1952:
1920:
1890:
1870:
1850:
1824:
1788:
1764:
1738:
1712:
1710:{\displaystyle \geq }
1678:
1648:
1614:
1580:
1548:Self-adjoint elements
1504:
1459:
1395:
1342:
1228:
1150:
1036:
904:
824:
737:
661:
627:
567:
478:
394:
368:
260:algebras of physical
229:
4766:Kakutani fixed-point
4751:Riesz representation
4505:Proto-value function
4484:Dirichlet eigenvalue
4398:Abstract index group
4283:Approximate identity
4246:Rigged Hilbert space
4122:KreinâRutman theorem
3968:Involution/*-algebra
3797:, Gauthier-Villars,
3472:Type for C*-algebras
3455:Von Neumann algebras
3450:Von Neumann algebras
3355:approximate identity
3313:
3293:
3257:
3217:
3189:
3153:
3113:
3089:
3062:
3042:
3022:
2986:
2980:approximate identity
2946:
2926:
2890:
2854:
2806:
2660:approximate identity
2561:
2258:
2095: whenever
2053:
2016:
1990:approximate identity
1929:
1903:
1879:
1859:
1833:
1797:
1777:
1747:
1721:
1701:
1665:
1627:
1593:
1556:
1468:
1404:
1359:
1296:
1285:in 1946 to describe
1178:
1099:
931:
855:
766:
684:
642:
577:
511:
425:
377:
344:
286:von Neumann algebras
202:
50:improve this article
5070:Functional analysis
4950:Functional calculus
4909:Mahler's conjecture
4888:Von Neumann algebra
4602:Functional analysis
4308:Von Neumann algebra
4044:Polar decomposition
3754:functional analysis
3741:, Springer-Verlag,
3595:Minkowski spacetime
2222:conjugate transpose
1769:if and only if the
1542:Gelfand isomorphism
392:{\textstyle x\in A}
312:nuclear C*-algebras
176:is a topologically
130:functional analysis
4975:Riemann hypothesis
4674:Topological vector
4438:Unbounded operator
4367:Essential spectrum
4346:SchurâHorn theorem
4336:BauerâFike theorem
4331:AlonâBoppana bound
4324:Finite-Dimensional
4298:Nuclear C*-algebra
4142:Spectral asymmetry
3319:
3299:
3279:
3239:
3213:. Furthermore, if
3209:equipped with the
3195:
3175:
3132:
3095:
3075:
3048:
3028:
3008:
2968:
2932:
2912:
2876:
2844:vanish at infinity
2828:
2608:
2585:
2292:
2285:
2111:
2097:
2038:
1947:
1915:
1885:
1865:
1845:
1819:
1783:
1759:
1733:
1707:
1673:
1643:
1609:
1575:
1499:
1454:
1390:
1337:
1223:
1145:
1048:bounded linear map
1031:
899:
819:
732:
656:
622:
562:
473:
389:
363:
337:, together with a
333:over the field of
224:
5052:
5051:
4955:Integral operator
4732:
4731:
4568:
4567:
4545:Transfer operator
4520:Spectral geometry
4205:Spectral abscissa
4185:Approximate point
4127:Normal eigenvalue
3834:Emch, G. (1972),
3826:978-0-8247-7569-8
3627:Operator K-theory
3622:Hilbert C*-module
3520:quantum mechanics
3432:harmonic analysis
3322:{\displaystyle Y}
3302:{\displaystyle X}
3198:{\displaystyle X}
3098:{\displaystyle K}
3051:{\displaystyle K}
3031:{\displaystyle X}
2982:. In the case of
2935:{\displaystyle X}
2848:local compactness
2570:
2497:compact operators
2397:, and especially
2395:quantum mechanics
2391:Hermitian adjoint
2317:Each C*-algebra,
2267:
2096:
1888:{\displaystyle y}
1868:{\displaystyle x}
1786:{\displaystyle x}
1692:partially ordered
1287:Banach *-algebras
1023:
717:
266:Werner Heisenberg
254:quantum mechanics
140:together with an
126:
125:
118:
100:
5077:
5042:
5041:
4960:Jones polynomial
4878:Operator algebra
4622:
4595:
4588:
4581:
4572:
4550:Transform theory
4270:Special algebras
4251:Spectral theorem
4214:Spectral Theorem
4054:Spectral theorem
3943:
3936:
3929:
3920:
3914:
3913:
3888:
3867:
3848:
3829:
3813:Doran, Robert S.
3807:
3796:
3787:Dixmier, Jacques
3781:
3770:
3751:
3721:
3714:
3708:
3702:
3696:
3691:
3685:
3675:
3669:
3664:, pp. 5â6,
3659:
3612:Banach *-algebra
3501:group C*-algebra
3496:)âł is a factor.
3400:) is called the
3391:
3328:
3326:
3325:
3320:
3308:
3306:
3305:
3300:
3288:
3286:
3285:
3280:
3269:
3268:
3248:
3246:
3245:
3240:
3229:
3228:
3205:is the space of
3204:
3202:
3201:
3196:
3184:
3182:
3181:
3176:
3165:
3164:
3141:
3139:
3138:
3133:
3128:
3127:
3104:
3102:
3101:
3096:
3084:
3082:
3081:
3076:
3074:
3073:
3057:
3055:
3054:
3049:
3037:
3035:
3034:
3029:
3017:
3015:
3014:
3009:
2998:
2997:
2977:
2975:
2974:
2969:
2958:
2957:
2941:
2939:
2938:
2933:
2921:
2919:
2918:
2913:
2902:
2901:
2885:
2883:
2882:
2877:
2866:
2865:
2837:
2835:
2834:
2829:
2818:
2817:
2723:. The sequence {
2617:
2615:
2614:
2609:
2601:
2600:
2584:
2439:of the operator
2437:adjoint operator
2422:linear operators
2379:â -closed algebra
2345:dimension vector
2301:
2299:
2298:
2293:
2284:
2234:ArtinâWedderburn
2120:
2118:
2117:
2112:
2098:
2094:
2084:
2083:
2071:
2070:
2047:
2045:
2044:
2039:
2031:
2030:
1974:GNS construction
1956:
1954:
1953:
1948:
1924:
1922:
1921:
1916:
1894:
1892:
1891:
1886:
1874:
1872:
1871:
1866:
1854:
1852:
1851:
1846:
1828:
1826:
1825:
1820:
1815:
1814:
1792:
1790:
1789:
1784:
1768:
1766:
1765:
1760:
1742:
1740:
1739:
1734:
1716:
1714:
1713:
1708:
1682:
1680:
1679:
1674:
1672:
1652:
1650:
1649:
1644:
1642:
1641:
1618:
1616:
1615:
1610:
1605:
1604:
1584:
1582:
1581:
1576:
1574:
1573:
1508:
1506:
1505:
1500:
1495:
1494:
1463:
1461:
1460:
1455:
1450:
1449:
1422:
1421:
1399:
1397:
1396:
1391:
1386:
1385:
1346:
1344:
1343:
1338:
1336:
1335:
1314:
1313:
1261:, in which case
1232:
1230:
1229:
1224:
1221:
1220:
1196:
1195:
1154:
1152:
1151:
1146:
1040:
1038:
1037:
1032:
1024:
1021:
1006:
1005:
993:
985:
965:
964:
949:
948:
908:
906:
905:
900:
895:
894:
873:
872:
828:
826:
825:
820:
812:
811:
781:
780:
741:
739:
738:
733:
728:
727:
718:
710:
705:
704:
665:
663:
662:
657:
655:
631:
629:
628:
623:
621:
620:
611:
610:
598:
597:
571:
569:
568:
563:
561:
560:
548:
547:
535:
534:
482:
480:
479:
474:
466:
465:
456:
455:
440:
439:
398:
396:
395:
390:
372:
370:
369:
364:
362:
361:
278:John von Neumann
270:matrix mechanics
233:
231:
230:
225:
214:
213:
121:
114:
110:
107:
101:
99:
58:
34:
26:
5085:
5084:
5080:
5079:
5078:
5076:
5075:
5074:
5055:
5054:
5053:
5048:
5030:
4994:Advanced topics
4989:
4913:
4892:
4851:
4817:HilbertâSchmidt
4790:
4781:GelfandâNaimark
4728:
4678:
4613:
4599:
4569:
4564:
4525:Spectral method
4510:Ramanujan graph
4458:
4442:
4418:Fredholm theory
4386:
4381:Shilov boundary
4377:Structure space
4355:Generalizations
4350:
4341:Numerical range
4319:
4303:Uniform algebra
4265:
4241:Riesz projector
4226:Min-max theorem
4209:
4195:Direct integral
4151:
4137:Spectral radius
4108:
4063:
4017:
4008:Spectral radius
3956:
3950:Spectral theory
3947:
3892:
3886:
3870:
3852:
3846:
3833:
3827:
3811:
3805:
3785:
3779:
3759:
3749:
3733:
3730:
3725:
3724:
3715:
3711:
3703:
3699:
3692:
3688:
3676:
3672:
3660:
3656:
3651:
3632:Operator system
3603:
3516:
3474:
3452:
3382:
3347:
3311:
3310:
3291:
3290:
3260:
3255:
3254:
3220:
3215:
3214:
3187:
3186:
3156:
3151:
3150:
3119:
3111:
3110:
3087:
3086:
3065:
3060:
3059:
3040:
3039:
3020:
3019:
2989:
2984:
2983:
2949:
2944:
2943:
2924:
2923:
2893:
2888:
2887:
2857:
2852:
2851:
2809:
2804:
2803:
2800:locally compact
2792:
2735:
2728:
2721:
2714:
2695:
2644:
2637:
2626:
2592:
2559:
2558:
2553:
2542:
2477:
2414:
2365:
2342:
2256:
2255:
2216:, and use the
2187:
2182:
2075:
2062:
2051:
2050:
2022:
2014:
2013:
2002:
1998:
1984:Any C*-algebra
1982:
1927:
1926:
1901:
1900:
1877:
1876:
1857:
1856:
1831:
1830:
1806:
1795:
1794:
1775:
1774:
1745:
1744:
1719:
1718:
1699:
1698:
1663:
1662:
1633:
1625:
1624:
1619:forms a closed
1596:
1591:
1590:
1565:
1554:
1553:
1550:
1534:
1486:
1466:
1465:
1441:
1413:
1402:
1401:
1377:
1357:
1356:
1327:
1305:
1294:
1293:
1279:
1269:are said to be
1212:
1187:
1176:
1175:
1097:
1096:
997:
956:
940:
929:
928:
915:section below.
886:
864:
853:
852:
803:
772:
764:
763:
719:
696:
682:
681:
640:
639:
612:
602:
589:
575:
574:
552:
539:
526:
509:
508:
457:
447:
428:
423:
422:
375:
374:
353:
342:
341:
335:complex numbers
320:
244:locally compact
205:
200:
199:
122:
111:
105:
102:
59:
57:
47:
35:
24:
17:
12:
11:
5:
5083:
5081:
5073:
5072:
5067:
5057:
5056:
5050:
5049:
5047:
5046:
5035:
5032:
5031:
5029:
5028:
5023:
5018:
5013:
5011:Choquet theory
5008:
5003:
4997:
4995:
4991:
4990:
4988:
4987:
4977:
4972:
4967:
4962:
4957:
4952:
4947:
4942:
4937:
4932:
4927:
4921:
4919:
4915:
4914:
4912:
4911:
4906:
4900:
4898:
4894:
4893:
4891:
4890:
4885:
4880:
4875:
4870:
4865:
4863:Banach algebra
4859:
4857:
4853:
4852:
4850:
4849:
4844:
4839:
4834:
4829:
4824:
4819:
4814:
4809:
4804:
4798:
4796:
4792:
4791:
4789:
4788:
4786:BanachâAlaoglu
4783:
4778:
4773:
4768:
4763:
4758:
4753:
4748:
4742:
4740:
4734:
4733:
4730:
4729:
4727:
4726:
4721:
4716:
4714:Locally convex
4711:
4697:
4692:
4686:
4684:
4680:
4679:
4677:
4676:
4671:
4666:
4661:
4656:
4651:
4646:
4641:
4636:
4631:
4625:
4619:
4615:
4614:
4600:
4598:
4597:
4590:
4583:
4575:
4566:
4565:
4563:
4562:
4557:
4552:
4547:
4542:
4537:
4532:
4527:
4522:
4517:
4512:
4507:
4502:
4497:
4492:
4487:
4477:
4475:Corona theorem
4472:
4466:
4464:
4460:
4459:
4457:
4456:
4454:Wiener algebra
4450:
4448:
4444:
4443:
4441:
4440:
4435:
4430:
4425:
4420:
4415:
4410:
4405:
4400:
4394:
4392:
4388:
4387:
4385:
4384:
4374:
4372:Pseudospectrum
4369:
4364:
4362:Dirac spectrum
4358:
4356:
4352:
4351:
4349:
4348:
4343:
4338:
4333:
4327:
4325:
4321:
4320:
4318:
4317:
4316:
4315:
4305:
4300:
4295:
4290:
4285:
4279:
4273:
4271:
4267:
4266:
4264:
4263:
4258:
4253:
4248:
4243:
4238:
4233:
4228:
4223:
4217:
4215:
4211:
4210:
4208:
4207:
4202:
4197:
4192:
4187:
4182:
4181:
4180:
4175:
4170:
4159:
4157:
4153:
4152:
4150:
4149:
4144:
4139:
4134:
4129:
4124:
4118:
4116:
4110:
4109:
4107:
4106:
4101:
4093:
4085:
4077:
4071:
4069:
4065:
4064:
4062:
4061:
4056:
4051:
4046:
4041:
4036:
4031:
4025:
4023:
4019:
4018:
4016:
4015:
4013:Operator space
4010:
4005:
4000:
3995:
3990:
3985:
3980:
3975:
3973:Banach algebra
3970:
3964:
3962:
3961:Basic concepts
3958:
3957:
3948:
3946:
3945:
3938:
3931:
3923:
3917:
3916:
3890:
3884:
3868:
3850:
3844:
3831:
3825:
3809:
3803:
3783:
3777:
3757:
3747:
3729:
3726:
3723:
3722:
3709:
3697:
3686:
3670:
3653:
3652:
3650:
3647:
3646:
3645:
3640:
3635:
3629:
3624:
3619:
3614:
3609:
3607:Banach algebra
3602:
3599:
3570:) â„ 0 for all
3515:
3512:
3473:
3470:
3451:
3448:
3346:
3343:
3318:
3298:
3278:
3275:
3272:
3267:
3263:
3238:
3235:
3232:
3227:
3223:
3211:weak* topology
3194:
3174:
3171:
3168:
3163:
3159:
3131:
3126:
3122:
3118:
3094:
3072:
3068:
3047:
3027:
3007:
3004:
3001:
2996:
2992:
2967:
2964:
2961:
2956:
2952:
2931:
2911:
2908:
2905:
2900:
2896:
2875:
2872:
2869:
2864:
2860:
2827:
2824:
2821:
2816:
2812:
2791:
2788:
2784:Calkin algebra
2731:
2726:
2719:
2712:
2693:
2642:
2635:
2624:
2619:
2618:
2607:
2604:
2599:
2595:
2591:
2588:
2583:
2580:
2577:
2573:
2569:
2566:
2545:
2540:
2507:subalgebra of
2476:
2473:
2413:
2410:
2363:
2343:is called the
2338:
2333:given by {dim(
2303:
2302:
2291:
2288:
2283:
2280:
2277:
2274:
2270:
2266:
2263:
2189:The algebra M(
2186:
2183:
2181:
2178:
2163:
2162:
2131:is separable,
2124:
2123:
2122:
2121:
2110:
2107:
2104:
2101:
2090:
2087:
2082:
2078:
2074:
2069:
2065:
2061:
2058:
2048:
2037:
2034:
2029:
2025:
2021:
2000:
1996:
1981:
1978:
1946:
1943:
1940:
1937:
1934:
1914:
1911:
1908:
1884:
1864:
1844:
1841:
1838:
1818:
1813:
1809:
1805:
1802:
1782:
1758:
1755:
1752:
1732:
1729:
1726:
1706:
1671:
1657:(or sometimes
1640:
1636:
1632:
1608:
1603:
1599:
1572:
1568:
1564:
1561:
1549:
1546:
1533:
1530:
1498:
1493:
1489:
1485:
1482:
1479:
1476:
1473:
1453:
1448:
1444:
1440:
1437:
1434:
1431:
1428:
1425:
1420:
1416:
1412:
1409:
1389:
1384:
1380:
1376:
1373:
1370:
1367:
1364:
1353:
1352:
1334:
1330:
1326:
1323:
1320:
1317:
1312:
1308:
1304:
1301:
1278:
1275:
1259:C*-isomorphism
1236:
1235:
1234:
1233:
1219:
1215:
1211:
1208:
1205:
1202:
1199:
1194:
1190:
1186:
1183:
1170:
1169:
1158:
1157:
1156:
1155:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1122:
1119:
1116:
1113:
1110:
1107:
1104:
1091:
1090:
1072:*-homomorphism
1044:
1043:
1042:
1041:
1030:
1027:
1019:
1015:
1012:
1009:
1004:
1000:
996:
992:
988:
984:
980:
977:
974:
971:
968:
963:
959:
955:
952:
947:
943:
939:
936:
898:
893:
889:
885:
882:
879:
876:
871:
867:
863:
860:
832:
831:
830:
829:
818:
815:
810:
806:
802:
799:
796:
793:
790:
787:
784:
779:
775:
771:
758:
757:
745:
744:
743:
742:
731:
726:
722:
716:
713:
708:
703:
699:
695:
692:
689:
676:
675:
654:
650:
647:
635:
634:
633:
632:
619:
615:
609:
605:
601:
596:
592:
588:
585:
582:
572:
559:
555:
551:
546:
542:
538:
533:
529:
525:
522:
519:
516:
503:
502:
486:
485:
484:
483:
472:
469:
464:
460:
454:
450:
446:
443:
438:
435:
431:
417:
416:
388:
385:
382:
360:
356:
352:
349:
331:Banach algebra
325:A C*-algebra,
319:
316:
293:Israel Gelfand
274:Pascual Jordan
223:
220:
217:
212:
208:
196:
195:
185:
138:Banach algebra
124:
123:
38:
36:
29:
15:
13:
10:
9:
6:
4:
3:
2:
5082:
5071:
5068:
5066:
5063:
5062:
5060:
5045:
5037:
5036:
5033:
5027:
5024:
5022:
5019:
5017:
5016:Weak topology
5014:
5012:
5009:
5007:
5004:
5002:
4999:
4998:
4996:
4992:
4985:
4981:
4978:
4976:
4973:
4971:
4968:
4966:
4963:
4961:
4958:
4956:
4953:
4951:
4948:
4946:
4943:
4941:
4940:Index theorem
4938:
4936:
4933:
4931:
4928:
4926:
4923:
4922:
4920:
4916:
4910:
4907:
4905:
4902:
4901:
4899:
4897:Open problems
4895:
4889:
4886:
4884:
4881:
4879:
4876:
4874:
4871:
4869:
4866:
4864:
4861:
4860:
4858:
4854:
4848:
4845:
4843:
4840:
4838:
4835:
4833:
4830:
4828:
4825:
4823:
4820:
4818:
4815:
4813:
4810:
4808:
4805:
4803:
4800:
4799:
4797:
4793:
4787:
4784:
4782:
4779:
4777:
4774:
4772:
4769:
4767:
4764:
4762:
4759:
4757:
4754:
4752:
4749:
4747:
4744:
4743:
4741:
4739:
4735:
4725:
4722:
4720:
4717:
4715:
4712:
4709:
4705:
4701:
4698:
4696:
4693:
4691:
4688:
4687:
4685:
4681:
4675:
4672:
4670:
4667:
4665:
4662:
4660:
4657:
4655:
4652:
4650:
4647:
4645:
4642:
4640:
4637:
4635:
4632:
4630:
4627:
4626:
4623:
4620:
4616:
4611:
4607:
4603:
4596:
4591:
4589:
4584:
4582:
4577:
4576:
4573:
4561:
4558:
4556:
4553:
4551:
4548:
4546:
4543:
4541:
4538:
4536:
4533:
4531:
4528:
4526:
4523:
4521:
4518:
4516:
4513:
4511:
4508:
4506:
4503:
4501:
4498:
4496:
4493:
4491:
4488:
4485:
4481:
4478:
4476:
4473:
4471:
4468:
4467:
4465:
4461:
4455:
4452:
4451:
4449:
4445:
4439:
4436:
4434:
4431:
4429:
4426:
4424:
4421:
4419:
4416:
4414:
4411:
4409:
4406:
4404:
4401:
4399:
4396:
4395:
4393:
4391:Miscellaneous
4389:
4382:
4378:
4375:
4373:
4370:
4368:
4365:
4363:
4360:
4359:
4357:
4353:
4347:
4344:
4342:
4339:
4337:
4334:
4332:
4329:
4328:
4326:
4322:
4314:
4311:
4310:
4309:
4306:
4304:
4301:
4299:
4296:
4294:
4291:
4289:
4286:
4284:
4280:
4278:
4275:
4274:
4272:
4268:
4262:
4259:
4257:
4254:
4252:
4249:
4247:
4244:
4242:
4239:
4237:
4234:
4232:
4229:
4227:
4224:
4222:
4219:
4218:
4216:
4212:
4206:
4203:
4201:
4198:
4196:
4193:
4191:
4188:
4186:
4183:
4179:
4176:
4174:
4171:
4169:
4166:
4165:
4164:
4161:
4160:
4158:
4156:Decomposition
4154:
4148:
4145:
4143:
4140:
4138:
4135:
4133:
4130:
4128:
4125:
4123:
4120:
4119:
4117:
4115:
4111:
4105:
4102:
4100:
4097:
4094:
4092:
4089:
4086:
4084:
4081:
4078:
4076:
4073:
4072:
4070:
4066:
4060:
4057:
4055:
4052:
4050:
4047:
4045:
4042:
4040:
4037:
4035:
4032:
4030:
4027:
4026:
4024:
4020:
4014:
4011:
4009:
4006:
4004:
4001:
3999:
3996:
3994:
3991:
3989:
3986:
3984:
3981:
3979:
3976:
3974:
3971:
3969:
3966:
3965:
3963:
3959:
3955:
3951:
3944:
3939:
3937:
3932:
3930:
3925:
3924:
3921:
3912:
3907:
3903:
3899:
3895:
3894:Segal, Irving
3891:
3887:
3885:3-540-63633-1
3881:
3877:
3873:
3869:
3866:
3862:
3861:
3856:
3851:
3847:
3845:0-471-23900-3
3841:
3837:
3832:
3828:
3822:
3819:, CRC Press,
3818:
3814:
3810:
3806:
3804:0-7204-0762-1
3800:
3795:
3794:
3788:
3784:
3780:
3778:0-12-185860-X
3774:
3769:
3768:
3762:
3761:Connes, Alain
3758:
3755:
3750:
3748:0-387-90176-0
3744:
3740:
3736:
3732:
3731:
3727:
3719:
3713:
3710:
3706:
3701:
3698:
3695:
3690:
3687:
3683:
3680:, p. 6,
3679:
3674:
3671:
3667:
3663:
3658:
3655:
3648:
3644:
3641:
3639:
3636:
3633:
3630:
3628:
3625:
3623:
3620:
3618:
3615:
3613:
3610:
3608:
3605:
3604:
3600:
3598:
3596:
3592:
3587:
3585:
3581:
3577:
3573:
3569:
3565:
3561:
3557:
3553:
3549:
3545:
3541:
3537:
3533:
3529:
3525:
3521:
3513:
3511:
3509:
3504:
3502:
3497:
3495:
3491:
3487:
3483:
3479:
3476:A C*-algebra
3471:
3469:
3467:
3462:
3460:
3456:
3449:
3447:
3445:
3441:
3437:
3433:
3429:
3425:
3421:
3420:group algebra
3417:
3414:
3409:
3407:
3403:
3399:
3395:
3390:
3386:
3380:
3376:
3372:
3368:
3364:
3360:
3356:
3352:
3344:
3342:
3340:
3336:
3332:
3316:
3296:
3273:
3265:
3261:
3252:
3233:
3225:
3221:
3212:
3208:
3192:
3169:
3161:
3157:
3148:
3143:
3124:
3120:
3108:
3092:
3070:
3066:
3045:
3025:
3002:
2994:
2990:
2981:
2962:
2954:
2950:
2929:
2906:
2898:
2894:
2870:
2862:
2858:
2849:
2845:
2841:
2822:
2814:
2810:
2801:
2797:
2789:
2787:
2785:
2781:
2777:
2773:
2769:
2765:
2761:
2757:
2753:
2749:
2745:
2743:
2739:
2734:
2729:
2722:
2715:
2708:
2704:
2700:
2696:
2689:
2685:
2681:
2677:
2673:
2669:
2665:
2661:
2657:
2653:
2647:
2645:
2638:
2631:
2627:
2605:
2597:
2593:
2586:
2581:
2578:
2575:
2571:
2567:
2564:
2557:
2556:
2555:
2552:
2548:
2543:
2536:
2532:
2528:
2524:
2519:
2516:
2514:
2510:
2506:
2502:
2498:
2494:
2490:
2486:
2482:
2474:
2472:
2470:
2466:
2462:
2458:
2454:
2450:
2446:
2442:
2438:
2434:
2430:
2427:
2426:Hilbert space
2423:
2419:
2411:
2409:
2407:
2402:
2400:
2396:
2392:
2388:
2384:
2380:
2376:
2371:
2369:
2362:
2358:
2357:positive cone
2354:
2350:
2346:
2341:
2336:
2332:
2328:
2324:
2320:
2314:
2312:
2308:
2289:
2286:
2281:
2275:
2272:
2268:
2264:
2261:
2254:
2253:
2252:
2250:
2246:
2242:
2237:
2235:
2231:
2227:
2223:
2219:
2218:operator norm
2215:
2211:
2207:
2204:
2200:
2196:
2192:
2184:
2179:
2177:
2174:
2172:
2168:
2160:
2156:
2152:
2148:
2147:
2142:
2138:
2134:
2130:
2126:
2125:
2108:
2105:
2102:
2099:
2088:
2085:
2080:
2076:
2072:
2067:
2063:
2059:
2056:
2049:
2035:
2027:
2023:
2019:
2012:
2011:
2010:
2009:
2008:
2006:
1995:
1991:
1987:
1979:
1977:
1975:
1971:
1967:
1963:
1958:
1944:
1941:
1938:
1935:
1932:
1912:
1909:
1906:
1898:
1882:
1862:
1842:
1839:
1836:
1816:
1811:
1807:
1803:
1800:
1780:
1772:
1756:
1753:
1750:
1730:
1727:
1724:
1704:
1696:
1693:
1689:
1684:
1660:
1656:
1638:
1634:
1630:
1622:
1606:
1601:
1597:
1588:
1570:
1566:
1562:
1559:
1547:
1545:
1543:
1539:
1531:
1529:
1527:
1523:
1519:
1515:
1510:
1491:
1487:
1480:
1474:
1446:
1442:
1432:
1426:
1418:
1414:
1410:
1382:
1378:
1371:
1365:
1350:
1332:
1324:
1318:
1310:
1306:
1302:
1292:
1291:
1290:
1288:
1284:
1283:C. E. Rickart
1276:
1274:
1272:
1268:
1264:
1260:
1256:
1251:
1249:
1245:
1241:
1217:
1209:
1203:
1200:
1192:
1188:
1181:
1174:
1173:
1172:
1171:
1168:
1164:
1160:
1159:
1138:
1132:
1126:
1120:
1117:
1111:
1108:
1102:
1095:
1094:
1093:
1092:
1089:
1085:
1081:
1077:
1076:
1075:
1073:
1069:
1065:
1061:
1057:
1053:
1049:
1028:
1017:
1013:
1010:
1007:
1002:
998:
994:
986:
972:
966:
961:
957:
950:
945:
937:
927:
926:
925:
924:
923:
921:
916:
914:
909:
896:
891:
883:
877:
869:
865:
861:
850:
848:
844:
840:
836:
816:
808:
804:
794:
788:
782:
777:
773:
762:
761:
760:
759:
755:
751:
747:
746:
729:
724:
720:
711:
706:
701:
693:
690:
680:
679:
678:
677:
673:
669:
648:
645:
637:
636:
617:
613:
607:
603:
599:
594:
586:
583:
573:
557:
553:
549:
544:
540:
536:
531:
523:
520:
517:
507:
506:
505:
504:
500:
496:
492:
488:
487:
470:
467:
462:
452:
448:
441:
436:
433:
429:
421:
420:
419:
418:
414:
410:
406:
402:
401:
400:
386:
383:
380:
358:
354:
347:
340:
336:
332:
328:
323:
317:
315:
313:
309:
305:
300:
298:
294:
289:
287:
283:
279:
275:
271:
267:
263:
259:
255:
250:
248:
245:
241:
237:
218:
210:
206:
194:of operators.
193:
189:
186:
184:of operators.
183:
182:norm topology
179:
175:
172:
171:
170:
168:
167:Hilbert space
165:
161:
157:
154:
151:
147:
143:
139:
135:
131:
120:
117:
109:
106:February 2013
98:
95:
91:
88:
84:
81:
77:
74:
70:
67: â
66:
62:
61:Find sources:
55:
51:
45:
44:
39:This article
37:
33:
28:
27:
22:
5006:Balanced set
4980:Distribution
4918:Applications
4867:
4771:KreinâMilman
4756:Closed graph
4463:Applications
4293:Disk algebra
4147:Spectral gap
4022:Main results
3982:
3904:(2): 73â88,
3901:
3897:
3878:, Springer,
3875:
3858:
3855:"C*-algebra"
3835:
3816:
3792:
3766:
3738:
3717:
3712:
3707:, p. 75
3700:
3689:
3682:Google Books
3673:
3666:Google Books
3657:
3588:
3583:
3579:
3575:
3571:
3567:
3563:
3559:
3555:
3551:
3547:
3543:
3539:
3535:
3531:
3527:
3523:
3517:
3505:
3498:
3493:
3489:
3485:
3481:
3477:
3475:
3463:
3453:
3439:
3438:in the case
3435:
3427:
3423:
3415:
3410:
3405:
3401:
3397:
3393:
3388:
3384:
3374:
3370:
3366:
3362:
3358:
3350:
3348:
3331:homeomorphic
3144:
2843:
2839:
2795:
2793:
2779:
2775:
2771:
2767:
2759:
2755:
2751:
2747:
2746:
2741:
2737:
2732:
2724:
2717:
2710:
2706:
2702:
2698:
2691:
2687:
2683:
2679:
2675:
2671:
2667:
2663:
2655:
2651:
2649:
2640:
2633:
2629:
2622:
2620:
2550:
2546:
2538:
2534:
2530:
2526:
2522:
2521:
2517:
2512:
2508:
2500:
2492:
2488:
2480:
2478:
2464:
2460:
2456:
2452:
2448:
2444:
2440:
2435:denotes the
2432:
2428:
2417:
2415:
2403:
2378:
2374:
2372:
2367:
2360:
2348:
2344:
2339:
2334:
2330:
2326:
2322:
2318:
2316:
2310:
2306:
2304:
2244:
2240:
2239:
2213:
2209:
2202:
2198:
2194:
2190:
2188:
2175:
2164:
2158:
2157:is dense in
2154:
2150:
2144:
2140:
2136:
2132:
2128:
2004:
1993:
1985:
1983:
1959:
1896:
1695:vector space
1687:
1685:
1658:
1655:non-negative
1654:
1589:of the form
1586:
1551:
1535:
1525:
1521:
1517:
1511:
1354:
1348:
1280:
1270:
1266:
1262:
1258:
1257:is called a
1254:
1252:
1239:
1237:
1166:
1162:
1087:
1083:
1079:
1071:
1070:is called a
1067:
1063:
1059:
1055:
1051:
1045:
917:
910:
851:
846:
838:
834:
833:
753:
749:
671:
667:
498:
494:
490:
412:
408:
407:, for every
326:
324:
321:
301:
297:Mark Naimark
290:
251:
239:
235:
197:
187:
173:
155:
133:
127:
112:
103:
93:
86:
79:
72:
65:"C*-algebra"
60:
48:Please help
43:verification
40:
5065:C*-algebras
4935:Heat kernel
4925:Hardy space
4832:Trace class
4746:HahnâBanach
4708:Topological
4490:Heat kernel
4190:Compression
4075:Isospectral
3735:Arveson, W.
3544:observables
3508:James Glimm
3503:is type I.
2505:norm closed
2249:canonically
2226:direct sums
2143:contains a
1621:convex cone
1514:I. E. Segal
1244:contractive
847:C* identity
262:observables
5059:Categories
4868:C*-algebra
4683:Properties
4168:Continuous
3983:C*-algebra
3978:B*-algebra
3728:References
3705:Segal 1947
3694:Segal 1947
3530:(elements
3377:) that is
3341:programs.
3251:isomorphic
3207:characters
2554:such that
2305:where min
2230:semisimple
2153:such that
2007:such that
1972:using the
1743:satisfies
1271:isomorphic
666:and every
405:involution
178:closed set
142:involution
76:newspapers
4842:Unbounded
4837:Transpose
4795:Operators
4724:Separable
4719:Reflexive
4704:Algebraic
4690:Barrelled
3954:-algebras
3872:Sakai, S.
3865:EMS Press
3617:*-algebra
3379:universal
2782:) is the
2639:) with ||
2579:∈
2572:⨁
2568:≅
2485:separable
2375:â -algebra
2366:group of
2276:∈
2269:⨁
2106:μ
2103:≤
2100:λ
2086:≤
2081:μ
2073:≤
2068:λ
2060:≤
2033:→
2028:λ
1942:≥
1936:−
1910:≥
1840:∈
1829:for some
1812:∗
1754:≥
1728:∈
1705:≥
1639:∗
1602:∗
1571:∗
1497:‖
1492:∗
1484:‖
1478:‖
1472:‖
1452:‖
1447:∗
1439:‖
1436:‖
1430:‖
1424:‖
1419:∗
1408:‖
1400:. Hence,
1388:‖
1383:∗
1375:‖
1369:‖
1363:‖
1329:‖
1322:‖
1316:‖
1311:∗
1300:‖
1248:isometric
1218:∗
1204:π
1193:∗
1182:π
1133:π
1121:π
1103:π
1014:λ
1011:−
1003:∗
987:λ
970:‖
962:∗
954:‖
942:‖
935:‖
888:‖
881:‖
875:‖
870:∗
859:‖
843:*-algebra
814:‖
809:∗
801:‖
798:‖
792:‖
786:‖
778:∗
770:‖
725:∗
715:¯
712:λ
702:∗
691:λ
649:∈
646:λ
618:∗
608:∗
595:∗
558:∗
545:∗
532:∗
463:∗
453:∗
437:∗
434:∗
403:It is an
384:∈
359:∗
351:↦
247:Hausdorff
134:C-algebra
5044:Category
4856:Algebras
4738:Theorems
4695:Complete
4664:Schwartz
4610:glossary
4555:Weyl law
4500:Lax pair
4447:Examples
4281:With an
4200:Discrete
4178:Residual
4114:Spectrum
4099:operator
4091:operator
4083:operator
3998:Spectrum
3874:(1971),
3789:(1969),
3763:(1994),
3737:(1976),
3601:See also
3353:with an
3185:, where
2764:quotient
2709:and let
2523:Theorem.
2443: :
2353:K-theory
2241:Theorem.
2206:matrices
2180:Examples
2167:quotient
2127:In case
1899:satisfy
1771:spectrum
1659:positive
1347:for all
1054: :
748:For all
489:For all
192:adjoints
4847:Unitary
4827:Nuclear
4812:Compact
4807:Bounded
4802:Adjoint
4776:Minâmax
4669:Sobolev
4654:Nuclear
4644:Hilbert
4639:Fréchet
4604: (
4096:Unitary
3566:with Ï(
2978:has an
2650:Though
2646:|| â 0.
2431:; here
2383:physics
2359:of the
1988:has an
913:history
835:Remark.
329:, is a
249:space.
180:in the
164:complex
153:algebra
150:complex
146:adjoint
90:scholar
4822:Normal
4659:Orlicz
4649:Hölder
4629:Banach
4618:Spaces
4606:topics
4080:Normal
3882:
3842:
3823:
3801:
3775:
3745:
3383:Ïâ'â:
2387:dagger
2236:type:
1966:states
92:
85:
78:
71:
63:
4634:Besov
4173:Point
3649:Notes
3548:state
3534:with
3369:into
2842:that
2798:be a
2774:) by
2503:is a
2495:) of
2483:be a
2247:, is
2208:over
2197:) of
2171:ideal
841:is a
282:rings
258:model
242:is a
162:on a
97:JSTOR
83:books
4982:(or
4700:Dual
4104:Unit
3952:and
3880:ISBN
3840:ISBN
3821:ISBN
3799:ISBN
3773:ISBN
3743:ISBN
3464:The
3337:and
3329:are
3309:and
3145:The
3058:let
2794:Let
2690:let
2662:for
2479:Let
2418:B(H)
1875:and
1265:and
1161:For
1082:and
1078:For
1066:and
373:for
295:and
132:, a
69:news
3906:doi
3586:).
3568:u*u
3554:(a
3518:In
3434:of
3422:of
3253:to
3249:is
2766:of
2744:).
2525:If
2499:on
2347:of
2325:),
2279:min
2155:hAh
2001:λâI
1925:if
1895:of
1773:of
1165:in
1086:in
1074:if
976:sup
752:in
670:in
497:in
411:in
339:map
306:of
268:'s
256:to
158:of
52:by
5061::
4608:â
3902:53
3900:,
3863:,
3857:,
3771:,
3574:â
3562:â
3538:=
3536:x*
3446:.
3408:.
3387:â
2786:.
2705:â„
2682:=
2471:.
2447:â
2433:x*
2408:.
2401:.
2373:A
2370:.
2337:)}
2319:Ae
2201:Ă
2193:,
1976:.
1957:.
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1273:.
1058:â
1050:,
1046:A
493:,
314:.
288:.
4986:)
4710:)
4706:/
4702:(
4612:)
4594:e
4587:t
4580:v
4486:)
4482:(
4383:)
4379:(
3942:e
3935:t
3928:v
3915:.
3908::
3889:.
3830:.
3756:.
3684:.
3668:.
3584:x
3580:x
3576:A
3572:u
3564:C
3560:A
3556:C
3552:A
3540:x
3532:x
3528:A
3524:A
3494:A
3490:A
3486:A
3482:A
3478:A
3440:G
3436:G
3428:G
3424:G
3416:G
3406:A
3398:A
3396:(
3394:E
3389:B
3385:A
3375:A
3373:(
3371:E
3367:A
3363:A
3361:(
3359:E
3351:A
3317:Y
3297:X
3277:)
3274:Y
3271:(
3266:0
3262:C
3237:)
3234:X
3231:(
3226:0
3222:C
3193:X
3173:)
3170:X
3167:(
3162:0
3158:C
3130:}
3125:K
3121:f
3117:{
3093:K
3071:K
3067:f
3046:K
3026:X
3006:)
3003:X
3000:(
2995:0
2991:C
2966:)
2963:X
2960:(
2955:0
2951:C
2930:X
2910:)
2907:X
2904:(
2899:0
2895:C
2874:)
2871:X
2868:(
2863:0
2859:C
2840:X
2826:)
2823:X
2820:(
2815:0
2811:C
2796:X
2780:H
2778:(
2776:K
2772:H
2770:(
2768:B
2760:H
2758:(
2756:B
2752:H
2750:(
2748:K
2742:H
2740:(
2738:K
2733:n
2730:}
2727:n
2725:e
2720:n
2718:H
2713:n
2711:e
2707:n
2703:k
2699:l
2694:n
2692:H
2688:n
2684:l
2680:H
2676:l
2672:H
2668:H
2666:(
2664:K
2656:H
2654:(
2652:K
2643:i
2641:T
2636:i
2634:H
2632:(
2630:K
2625:i
2623:T
2606:,
2603:)
2598:i
2594:H
2590:(
2587:K
2582:I
2576:i
2565:A
2551:I
2549:â
2547:i
2544:}
2541:i
2539:H
2535:H
2533:(
2531:K
2527:A
2513:H
2511:(
2509:B
2501:H
2493:H
2491:(
2489:K
2481:H
2465:H
2461:H
2459:(
2457:B
2453:A
2449:H
2445:H
2441:x
2429:H
2368:A
2364:0
2361:K
2349:A
2340:e
2335:e
2331:A
2327:C
2323:e
2313:.
2311:A
2307:A
2290:e
2287:A
2282:A
2273:e
2265:=
2262:A
2245:A
2214:C
2210:C
2203:n
2199:n
2195:C
2191:n
2161:.
2159:A
2151:h
2141:A
2137:A
2133:A
2129:A
2109:.
2089:1
2077:e
2064:e
2057:0
2036:x
2024:e
2020:x
2005:A
1999:}
1997:λ
1994:e
1986:A
1945:0
1939:y
1933:x
1913:y
1907:x
1897:A
1883:y
1863:x
1843:A
1837:s
1817:s
1808:s
1804:=
1801:x
1781:x
1757:0
1751:x
1731:A
1725:x
1688:A
1670:R
1635:x
1631:x
1607:x
1598:x
1587:A
1567:x
1563:=
1560:x
1526:H
1522:H
1520:(
1518:B
1488:x
1481:=
1475:x
1443:x
1433:x
1427:=
1415:x
1411:x
1379:x
1372:=
1366:x
1349:x
1333:2
1325:x
1319:=
1307:x
1303:x
1267:B
1263:A
1255:Ï
1240:Ï
1214:)
1210:x
1207:(
1201:=
1198:)
1189:x
1185:(
1167:A
1163:x
1142:)
1139:y
1136:(
1130:)
1127:x
1124:(
1118:=
1115:)
1112:y
1109:x
1106:(
1088:A
1084:y
1080:x
1068:B
1064:A
1060:B
1056:A
1052:Ï
1029:.
1026:}
1018:1
1008:x
999:x
995::
991:|
983:|
979:{
973:=
967:x
958:x
951:=
946:2
938:x
897:,
892:2
884:x
878:=
866:x
862:x
839:A
817:.
805:x
795:x
789:=
783:x
774:x
756::
754:A
750:x
730:.
721:x
707:=
698:)
694:x
688:(
674::
672:A
668:x
653:C
614:x
604:y
600:=
591:)
587:y
584:x
581:(
554:y
550:+
541:x
537:=
528:)
524:y
521:+
518:x
515:(
501::
499:A
495:y
491:x
471:x
468:=
459:)
449:x
445:(
442:=
430:x
415::
413:A
409:x
387:A
381:x
355:x
348:x
327:A
240:X
236:X
222:)
219:X
216:(
211:0
207:C
188:A
174:A
156:A
119:)
113:(
108:)
104:(
94:·
87:·
80:·
73:·
46:.
23:.
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