5051:
43:
1050:
2130:
321:, 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
1473:
2404:, 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
751:
1356:
2239:
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
2627:
1518:
1409:
2311:
2057:
838:
1242:
1164:
675:
918:
581:
492:
641:
1475:, 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
941:
1692:
3521:
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.
382:
3151:
1966:
3298:
3258:
3194:
3027:
2987:
2931:
2895:
2847:
1838:
243:
2063:
1934:
1778:
1594:
1864:
1752:
1662:
1628:
3094:
1726:
4438:
408:
3338:
3318:
3214:
3114:
3067:
3047:
2951:
1904:
1884:
1802:
3819:. 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.
1539:. '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".
4940:
4540:
1551:
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
4173:
4776:
3727:
John A. Holbrook, David W. Kribs, and
Raymond Laflamme. "Noiseless Subsystems and the Structure of the Commutant in Quantum Error Correction."
4603:
3453:
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
1414:
4195:
3503:)) is a type I von Neumann algebra. In fact it is sufficient to consider only factor representations, i.e. representations Ï for which Ï(
2416:
1547:
C*-algebras have a large number of properties that are technically convenient. Some of these properties can be established by using the
4766:
3648:
2529:
Concrete C*-algebras of compact operators admit a characterization similar to
Wedderburn's theorem for finite dimensional C*-algebras:
4178:
3951:
4200:
4893:
4748:
3430:
3479:
implies that any C*-algebra has a universal enveloping W*-algebra, such that any homomorphism to a W*-algebra factors through it.
694:
4724:
4525:
4188:
1520:. For these reasons, the term B*-algebra is rarely used in current terminology, and has been replaced by the term 'C*-algebra'.
4418:
3793:. This book is widely regarded as a source of new research material, providing much supporting intuition, but it is difficult.
1306:
4271:
4069:
3835:
4266:
107:
2571:
1478:
1369:
79:
4616:
4705:
4596:
4423:
3894:
3854:
3813:
3787:
3757:
2507:
126:
86:
4975:
4241:
2268:
4620:
4210:
3468:, known as W* algebras before the 1960s, are a special kind of C*-algebra. They are required to be closed in the
2362:. This vector uniquely determines the isomorphism class of a finite-dimensional C*-algebra. In the language of
2026:
776:
4124:
4008:
1781:
1548:
1188:
1109:
64:
17:
652:
93:
4771:
4433:
3944:
3875:
1254:
5080:
5054:
4827:
4761:
4589:
4059:
2244:
865:
521:
333:
We begin with the abstract characterization of C*-algebras given in the 1943 paper by
Gelfand and Naimark.
435:
4791:
4570:
4490:
4044:
3870:
2479:
2409:
1976:
587:
310:
yielded an abstract characterisation of C*-algebras making no reference to operators on a
Hilbert space.
291:
attempted to establish a general framework for these algebras, which culminated in a series of papers on
75:
60:
5036:
4990:
4914:
4796:
4545:
4443:
4323:
3601:
3476:
1972:
170:
1045:{\displaystyle \|x\|^{2}=\|x^{*}x\|=\sup\{|\lambda |:x^{*}x-\lambda \,1{\text{ is not invertible}}\}.}
5031:
4847:
4550:
4413:
4246:
4231:
4039:
4003:
922:
which is sometimes called the B*-identity. For history behind the names C*- and B*-algebras, see the
415:
4883:
4781:
4684:
4142:
4132:
4013:
3937:
3454:
3117:
2156:
1980:
152:
1675:
4980:
4756:
4505:
4480:
4298:
4287:
3998:
3653:
3349:
3345:
3217:
53:
31:
3865:
2125:{\displaystyle 0\leq e_{\lambda }\leq e_{\mu }\leq 1\quad {\mbox{ whenever }}\lambda \leq \mu .}
354:
5011:
4955:
4919:
4356:
4346:
4341:
4049:
3469:
3157:
3123:
1939:
314:
202:
3267:
3227:
3163:
2996:
2956:
2900:
2864:
2816:
1807:
1257:, i.e. bounded with norm †1. Furthermore, an injective *-homomorphism between C*-algebras is
212:
4718:
4101:
3423:
2236:
1913:
1757:
1566:
318:
4714:
3692:
3676:
1843:
1731:
1637:
1603:
295:
of operators. These papers considered a special class of C*-algebras that are now known as
5075:
4994:
4515:
4494:
4408:
4293:
4256:
3763:. An excellent introduction to the subject, accessible for those with a knowledge of basic
3365:
3072:
2990:
2670:
2216:
2000:
1711:
1293:
163:
4581:
100:
8:
4960:
4898:
4612:
4318:
4054:
3764:
3605:
3465:
2397:
2232:
2181:
1552:
296:
140:
387:
4985:
4852:
4448:
4377:
4308:
4152:
4114:
3776:
3511:
3389:
3323:
3303:
3199:
3099:
3052:
3032:
2936:
2240:
1889:
1869:
1787:
1058:
322:
292:
2897:
under pointwise multiplication and addition. The involution is pointwise conjugation.
4965:
4555:
4530:
4215:
4137:
3890:
3850:
3831:
3809:
3783:
3753:
3637:
3632:
3530:
3442:
2405:
2401:
2363:
349:
276:
264:
156:
3921:
3120:, which applies to locally compact Hausdorff spaces. Any such sequence of functions
4970:
4888:
4857:
4837:
4822:
4817:
4812:
4560:
4261:
4109:
4064:
3988:
3916:
3622:
2447:
1984:
1297:
288:
280:
268:
4649:
2223:
becomes a C*-algebra if we consider matrices as operators on the
Euclidean space,
1366:
This condition automatically implies that the *-involution is isometric, that is,
4832:
4786:
4734:
4729:
4700:
4535:
4520:
4428:
4391:
4387:
4351:
4313:
4251:
4236:
4205:
4147:
4106:
4093:
4018:
3960:
3929:
3882:
3823:
3797:
3745:
3642:
2858:
2810:
2495:
2432:
2400:, â , is used in the name because physicists typically use the symbol to denote a
933:, it implies that the C*-norm is uniquely determined by the algebraic structure:
930:
257:
254:
4659:
3860:. Mathematically rigorous reference which provides extensive physics background.
3802:
5021:
4873:
4674:
4485:
4464:
4382:
4372:
4183:
4090:
4023:
3983:
3617:
2794:
2259:
345:
341:
303:
284:
174:
160:
148:
30:
This article is about an area of mathematics. For the concept in rocketry, see
929:
The C*-identity is a very strong requirement. For instance, together with the
5069:
5026:
4950:
4679:
4664:
4654:
3341:
3221:
2436:
2367:
2228:
1702:
192:
177:
3517:
However, if a C*-algebra has non-type I representations, then by results of
2187:
Similarly, a closed two-sided ideal of a C*-algebra is itself a C*-algebra.
5016:
4669:
4639:
4303:
4157:
4098:
3904:
3771:
3627:
1705:
1524:
307:
4945:
4935:
4842:
4644:
4500:
4085:
3518:
3261:
2515:
2332:, is isomorphic (in a noncanonical way) to the full matrix algebra M(dim(
1631:
3491:
is of type I if and only if for all non-degenerate representations Ï of
3160:
states that every commutative C*-algebra is *-isomorphic to the algebra
209:
Another important class of non-Hilbert C*-algebras includes the algebra
4710:
3600:
This C*-algebra approach is used in the HaagâKastler axiomatization of
1468:{\displaystyle \lVert xx^{*}\rVert =\lVert x\rVert \lVert x^{*}\rVert }
272:
188:
2415:
An immediate generalization of finite dimensional C*-algebras are the
1672:, even though this terminology conflicts with its use for elements of
3978:
3964:
853:
42:
4565:
4510:
3847:
Algebraic
Methods in Statistical Mechanics and Quantum Field Theory
3510:
A locally compact group is said to be of type I if and only if its
3029:
this is immediate: consider the directed set of compact subsets of
2774:
2177:
1258:
2526:). It is also closed under involution; hence it is a C*-algebra.
2320:
is the set of minimal nonzero self-adjoint central projections of
923:
2393:
2466:, is *-isomorphic to a norm-closed adjoint closed subalgebra of
1535:), namely, the space of bounded operators on some Hilbert space
3828:
Characterizations of C*-algebras: The
Gelfand-Naimark Theorems
3533:, one typically describes a physical system with a C*-algebra
2176:
Using approximate identities, one can show that the algebraic
2773:). For separable Hilbert spaces, it is the unique ideal. The
263:
C*-algebras were first considered primarily for their use in
3907:(1947), "Irreducible representations of operator algebras",
1979:
of a C*-algebra, which in turn can be used to construct the
1287:
746:{\displaystyle (\lambda x)^{*}={\overline {\lambda }}x^{*}.}
4611:
3589:) such that Ï(1) = 1. The expected value of the observable
3096:
be a function of compact support which is identically 1 on
2150:
will have a sequential approximate identity if and only if
1971:
This partially ordered subspace allows the definition of a
3344:. This characterization is one of the motivations for the
2243:, from which fact one can deduce the following theorem of
2685:
is isomorphic to the space of square summable sequences
2427:
The prototypical example of a C*-algebra is the algebra
1351:{\displaystyle \lVert xx^{*}\rVert =\lVert x\rVert ^{2}}
3524:
3429:. This is defined as the enveloping C*-algebra of the
2146:
has a sequential approximate identity. More generally,
1990:
313:
C*-algebras are now an important tool in the theory of
3368:, there is a unique (up to C*-isomorphism) C*-algebra
2104:
1634:. This cone is identical to the elements of the form
390:
357:
3645:, a unital subspace of a C*-algebra that is *-closed.
3561:
of the system is defined as a positive functional on
3326:
3306:
3270:
3230:
3202:
3166:
3126:
3102:
3075:
3055:
3035:
2999:
2959:
2939:
2903:
2867:
2819:
2574:
2271:
2066:
2029:
1975:
on a C*-algebra, which in turn is used to define the
1942:
1916:
1892:
1872:
1846:
1810:
1790:
1760:
1734:
1714:
1678:
1640:
1606:
1569:
1481:
1417:
1372:
1309:
1191:
1112:
944:
868:
779:
697:
655:
590:
524:
438:
215:
3731:. Volume 2, Number 5, pp. 381–419. Oct 2003.
2669:) does not have an identity element, a sequential
2622:{\displaystyle A\cong \bigoplus _{i\in I}K(H_{i}),}
2485:
1513:{\displaystyle \lVert x\rVert =\lVert x^{*}\rVert }
1404:{\displaystyle \lVert x\rVert =\lVert x^{*}\rVert }
67:. Unsourced material may be challenged and removed.
4941:Spectral theory of ordinary differential equations
4541:Spectral theory of ordinary differential equations
3959:
3801:
3775:
3332:
3312:
3292:
3252:
3208:
3188:
3145:
3108:
3088:
3061:
3041:
3021:
2981:
2945:
2925:
2889:
2841:
2621:
2305:
2231:||·|| on matrices. The involution is given by the
2124:
2051:
1960:
1928:
1898:
1878:
1858:
1832:
1796:
1772:
1746:
1720:
1686:
1656:
1622:
1588:
1512:
1467:
1403:
1350:
1236:
1158:
1044:
912:
832:
745:
669:
635:
575:
486:
402:
376:
237:
4439:SchröderâBernstein theorems for operator algebras
2933:has a multiplicative unit element if and only if
2195:
1697:The set of self-adjoint elements of a C*-algebra
283:and in a more mathematically developed form with
5067:
3537:with unit element; the self-adjoint elements of
3422:Of particular importance is the C*-algebra of a
2632:where the (C*-)direct sum consists of elements (
2498:infinite-dimensional Hilbert space. The algebra
2289:
986:
1527:in 1947 to describe norm-closed subalgebras of
1249:In the case of C*-algebras, any *-homomorphism
3557:, the measurable quantities, of the system. A
4597:
3945:
3909:Bulletin of the American Mathematical Society
3392:, that is, every other continuous *-morphism
2180:of a C*-algebra by a closed proper two-sided
1261:. These are consequences of the C*-identity.
3140:
3127:
2417:approximately finite dimensional C*-algebras
2306:{\displaystyle A=\bigoplus _{e\in \min A}Ae}
1563:Self-adjoint elements are those of the form
1507:
1494:
1488:
1482:
1462:
1449:
1446:
1440:
1434:
1418:
1398:
1385:
1379:
1373:
1339:
1332:
1326:
1310:
1036:
989:
980:
964:
952:
945:
898:
891:
885:
869:
824:
811:
808:
802:
796:
780:
328:
3863:
2422:
2235:. More generally, one can consider finite
1728:. In this ordering, a self-adjoint element
1542:
4604:
4590:
3952:
3938:
3822:
3688:
3672:
3472:, which is weaker than the norm topology.
2849:of complex-valued continuous functions on
2800:
2396:for a finite-dimensional C*-algebra. The
2184:, with the natural norm, is a C*-algebra.
2052:{\displaystyle xe_{\lambda }\rightarrow x}
833:{\displaystyle \|x^{*}x\|=\|x\|\|x^{*}\|.}
245:of complex-valued continuous functions on
3920:
3593:, if the system is in state Ï, is then Ï(
3403:factors uniquely through Ï. The algebra
1680:
1288:Some history: B*-algebras and C*-algebras
1237:{\displaystyle \pi (x^{*})=\pi (x)^{*}\,}
1233:
1159:{\displaystyle \pi (xy)=\pi (x)\pi (y)\,}
1155:
1027:
663:
127:Learn how and when to remove this message
4894:Group algebra of a locally compact group
3804:Les C*-algÚbres et leurs représentations
3355:
2003:. In fact, there is a directed family {
1558:
670:{\displaystyle \lambda \in \mathbb {C} }
201:is closed under the operation of taking
3796:
3744:
3482:
3460:
1362:in the given B*-algebra. (B*-condition)
14:
5068:
3770:
2953:is compact. As does any C*-algebra,
1596:. The set of elements of a C*-algebra
1523:The term C*-algebra was introduced by
1292:The term B*-algebra was introduced by
4585:
4272:Spectral theory of normal C*-algebras
4070:Spectral theory of normal C*-algebras
3933:
3903:
3881:
3715:
3704:
2548:), then there exists Hilbert spaces {
2431:of bounded (equivalently continuous)
913:{\displaystyle \|xx^{*}\|=\|x\|^{2},}
576:{\displaystyle (x+y)^{*}=x^{*}+y^{*}}
4267:Spectral theory of compact operators
3844:
3525:C*-algebras and quantum field theory
2681:) can be developed. To be specific,
2340:). The finite family indexed on min
1991:Quotients and approximate identities
487:{\displaystyle x^{**}=(x^{*})^{*}=x}
275:. This line of research began with
65:adding citations to reliable sources
36:
2392:) is the name occasionally used in
1664:. Elements of this cone are called
848:The first four identities say that
636:{\displaystyle (xy)^{*}=y^{*}x^{*}}
24:
4419:CohenâHewitt factorization theorem
3649:GelfandâNaimarkâSegal construction
3499:)âł (that is, the bicommutant of Ï(
2727:be the orthogonal projection onto
2262:isomorphic to a finite direct sum
1708:; the ordering is usually denoted
856:. The last identity is called the
25:
5092:
4424:Extensions of symmetric operators
3608:is associated with a C*-algebra.
2861:) forms a commutative C*-algebra
2765:) is a two-sided closed ideal of
2254:A finite-dimensional C*-algebra,
1701:naturally has the structure of a
159:. A particular case is that of a
155:satisfying the properties of the
27:Topological complex vector space
5050:
5049:
4976:Topological quantum field theory
4242:Positive operator-valued measure
3300:as C*-algebras, it follows that
2708:be the subspace of sequences of
2486:C*-algebras of compact operators
2474:) for a suitable Hilbert space,
1804:is non-negative, if and only if
180:with two additional properties:
139:In mathematics, specifically in
41:
4526:RayleighâFaberâKrahn inequality
3922:10.1090/S0002-9904-1947-08742-5
3116:. Such functions exist by the
2747:is an approximate identity for
2102:
410:with the following properties:
249:that vanish at infinity, where
52:needs additional citations for
3729:Quantum Information Processing
3721:
3709:
3698:
3682:
3666:
3287:
3281:
3247:
3241:
3183:
3177:
3016:
3010:
2976:
2970:
2920:
2914:
2884:
2878:
2836:
2830:
2613:
2600:
2196:Finite-dimensional C*-algebras
2043:
1549:continuous functional calculus
1224:
1217:
1208:
1195:
1152:
1146:
1140:
1134:
1125:
1116:
1001:
993:
708:
698:
601:
591:
538:
525:
469:
455:
361:
232:
226:
13:
1:
4772:Uniform boundedness principle
4434:Limiting absorption principle
3738:
3441:provides context for general
2639:) of the Cartesian product Î
2478:; this is the content of the
2462:. In fact, every C*-algebra,
4060:Singular value decomposition
3153:is an approximate identity.
2813:Hausdorff space. The space
2014:of self-adjoint elements of
1866:. Two self-adjoint elements
1687:{\displaystyle \mathbb {R} }
1300:that satisfy the condition:
725:
287:around 1933. Subsequently,
7:
4491:Hearing the shape of a drum
4174:Decomposition of a spectrum
3887:C*-algebras and W*-algebras
3871:Encyclopedia of Mathematics
3826:; Belfi, Victor A. (1986),
3750:An Invitation to C*-Algebra
3611:
2857:(defined in the article on
2697:. For each natural number
2410:quantum information science
2190:
1264:A bijective *-homomorphism
377:{\textstyle x\mapsto x^{*}}
171:continuous linear operators
147:(pronounced "C-star") is a
10:
5097:
4915:Invariant subspace problem
4079:Special Elements/Operators
3604:, where every open set of
3602:local quantum field theory
3495:the von Neumann algebra Ï(
2160:, i.e. a positive element
1973:positive linear functional
29:
5045:
5004:
4928:
4907:
4866:
4805:
4747:
4693:
4635:
4628:
4551:Superstrong approximation
4473:
4457:
4414:Banach algebra cohomology
4401:
4365:
4334:
4280:
4247:Projection-valued measure
4232:Borel functional calculus
4224:
4166:
4123:
4078:
4032:
4004:Projection-valued measure
3971:
3360:Given a Banach *-algebra
3146:{\displaystyle \{f_{K}\}}
2712:which vanish for indices
2157:strictly positive element
1961:{\displaystyle x-y\geq 0}
649:For every complex number
329:Abstract characterization
302:Around 1943, the work of
4884:Spectrum of a C*-algebra
4143:Spectrum of a C*-algebra
4014:Spectrum of a C*-algebra
3778:Non-commutative geometry
3659:
3553:) are thought of as the
3455:spectrum of a C*-algebra
3415:of the Banach *-algebra
3376:) and *-morphism Ï from
3293:{\displaystyle C_{0}(Y)}
3253:{\displaystyle C_{0}(X)}
3189:{\displaystyle C_{0}(X)}
3118:Tietze extension theorem
3022:{\displaystyle C_{0}(X)}
2982:{\displaystyle C_{0}(X)}
2926:{\displaystyle C_{0}(X)}
2890:{\displaystyle C_{0}(X)}
2842:{\displaystyle C_{0}(X)}
2423:C*-algebras of operators
2388:(or, more explicitly, a
1981:spectrum of a C*-algebra
1833:{\displaystyle x=s^{*}s}
1543:Structure of C*-algebras
238:{\displaystyle C_{0}(X)}
4981:Noncommutative geometry
4571:WienerâKhinchin theorem
4506:Kuznetsov trace formula
4481:Almost Mathieu operator
4299:Banach function algebra
4288:Amenable Banach algebra
4045:GelfandâNaimark theorem
3999:Noncommutative topology
3654:Jordan operator algebra
3350:noncommutative geometry
3346:noncommutative topology
3049:, and for each compact
2801:Commutative C*-algebras
2480:GelfandâNaimark theorem
1929:{\displaystyle x\geq y}
1773:{\displaystyle x\geq 0}
1589:{\displaystyle x=x^{*}}
1253:between C*-algebras is
1033: is not invertible
931:spectral radius formula
315:unitary representations
32:characteristic velocity
5037:TomitaâTakesaki theory
5012:Approximation property
4956:Calculus of variations
4546:SturmâLiouville theory
4444:ShermanâTakeda theorem
4324:TomitaâTakesaki theory
4099:Hermitian/Self-adjoint
4050:Gelfand representation
3849:, Wiley-Interscience,
3689:Doran & Belfi 1986
3673:Doran & Belfi 1986
3477:ShermanâTakeda theorem
3470:weak operator topology
3437:. The C*-algebra of
3334:
3314:
3294:
3254:
3210:
3190:
3158:Gelfand representation
3147:
3110:
3090:
3063:
3043:
3023:
2983:
2947:
2927:
2891:
2843:
2659:
2623:
2540:is a C*-subalgebra of
2326:
2307:
2126:
2053:
1962:
1930:
1900:
1880:
1860:
1859:{\displaystyle s\in A}
1834:
1798:
1774:
1748:
1747:{\displaystyle x\in A}
1722:
1688:
1658:
1657:{\displaystyle xx^{*}}
1624:
1623:{\displaystyle x^{*}x}
1590:
1514:
1469:
1405:
1352:
1238:
1160:
1073:, between C*-algebras
1046:
914:
860:and is equivalent to:
834:
747:
671:
637:
577:
488:
404:
378:
319:locally compact groups
239:
5032:BanachâMazur distance
4995:Generalized functions
4040:GelfandâMazur theorem
3864:A.I. Shtern (2001) ,
3569:-linear map Ï :
3424:locally compact group
3413:C*-enveloping algebra
3356:C*-enveloping algebra
3335:
3315:
3295:
3255:
3211:
3191:
3148:
3111:
3091:
3089:{\displaystyle f_{K}}
3064:
3044:
3024:
2984:
2948:
2928:
2892:
2844:
2689:; we may assume that
2624:
2531:
2435:defined on a complex
2366:, this vector is the
2308:
2249:
2127:
2054:
1963:
1931:
1901:
1881:
1861:
1835:
1799:
1775:
1749:
1723:
1721:{\displaystyle \geq }
1689:
1659:
1625:
1591:
1559:Self-adjoint elements
1515:
1470:
1406:
1353:
1239:
1161:
1047:
915:
835:
748:
672:
638:
578:
489:
405:
379:
271:algebras of physical
240:
4777:Kakutani fixed-point
4762:Riesz representation
4516:Proto-value function
4495:Dirichlet eigenvalue
4409:Abstract index group
4294:Approximate identity
4257:Rigged Hilbert space
4133:KreinâRutman theorem
3979:Involution/*-algebra
3808:, Gauthier-Villars,
3483:Type for C*-algebras
3466:Von Neumann algebras
3461:Von Neumann algebras
3366:approximate identity
3324:
3304:
3268:
3228:
3200:
3164:
3124:
3100:
3073:
3053:
3033:
2997:
2991:approximate identity
2957:
2937:
2901:
2865:
2817:
2671:approximate identity
2572:
2269:
2106: whenever
2064:
2027:
2001:approximate identity
1940:
1914:
1890:
1870:
1844:
1808:
1788:
1758:
1732:
1712:
1676:
1638:
1604:
1567:
1479:
1415:
1370:
1307:
1296:in 1946 to describe
1189:
1110:
942:
866:
777:
695:
653:
588:
522:
436:
388:
355:
297:von Neumann algebras
213:
61:improve this article
5081:Functional analysis
4961:Functional calculus
4920:Mahler's conjecture
4899:Von Neumann algebra
4613:Functional analysis
4319:Von Neumann algebra
4055:Polar decomposition
3765:functional analysis
3752:, Springer-Verlag,
3606:Minkowski spacetime
2233:conjugate transpose
1780:if and only if the
1553:Gelfand isomorphism
403:{\textstyle x\in A}
323:nuclear C*-algebras
187:is a topologically
141:functional analysis
4986:Riemann hypothesis
4685:Topological vector
4449:Unbounded operator
4378:Essential spectrum
4357:SchurâHorn theorem
4347:BauerâFike theorem
4342:AlonâBoppana bound
4335:Finite-Dimensional
4309:Nuclear C*-algebra
4153:Spectral asymmetry
3330:
3310:
3290:
3250:
3224:. Furthermore, if
3220:equipped with the
3206:
3186:
3143:
3106:
3086:
3059:
3039:
3019:
2979:
2943:
2923:
2887:
2855:vanish at infinity
2839:
2619:
2596:
2303:
2296:
2122:
2108:
2049:
1958:
1926:
1896:
1876:
1856:
1830:
1794:
1770:
1744:
1718:
1684:
1654:
1620:
1586:
1510:
1465:
1401:
1348:
1234:
1156:
1059:bounded linear map
1042:
910:
830:
743:
667:
633:
573:
484:
400:
374:
348:, together with a
344:over the field of
235:
5063:
5062:
4966:Integral operator
4743:
4742:
4579:
4578:
4556:Transfer operator
4531:Spectral geometry
4216:Spectral abscissa
4196:Approximate point
4138:Normal eigenvalue
3845:Emch, G. (1972),
3837:978-0-8247-7569-8
3638:Operator K-theory
3633:Hilbert C*-module
3531:quantum mechanics
3443:harmonic analysis
3333:{\displaystyle Y}
3313:{\displaystyle X}
3209:{\displaystyle X}
3109:{\displaystyle K}
3062:{\displaystyle K}
3042:{\displaystyle X}
2993:. In the case of
2946:{\displaystyle X}
2859:local compactness
2581:
2508:compact operators
2408:, and especially
2406:quantum mechanics
2402:Hermitian adjoint
2328:Each C*-algebra,
2278:
2107:
1899:{\displaystyle y}
1879:{\displaystyle x}
1797:{\displaystyle x}
1703:partially ordered
1298:Banach *-algebras
1034:
728:
277:Werner Heisenberg
265:quantum mechanics
151:together with an
137:
136:
129:
111:
16:(Redirected from
5088:
5053:
5052:
4971:Jones polynomial
4889:Operator algebra
4633:
4632:
4606:
4599:
4592:
4583:
4582:
4561:Transform theory
4281:Special algebras
4262:Spectral theorem
4225:Spectral Theorem
4065:Spectral theorem
3954:
3947:
3940:
3931:
3930:
3925:
3924:
3899:
3878:
3859:
3840:
3824:Doran, Robert S.
3818:
3807:
3798:Dixmier, Jacques
3792:
3781:
3762:
3732:
3725:
3719:
3713:
3707:
3702:
3696:
3686:
3680:
3675:, pp. 5â6,
3670:
3623:Banach *-algebra
3512:group C*-algebra
3507:)âł is a factor.
3411:) is called the
3402:
3339:
3337:
3336:
3331:
3319:
3317:
3316:
3311:
3299:
3297:
3296:
3291:
3280:
3279:
3259:
3257:
3256:
3251:
3240:
3239:
3216:is the space of
3215:
3213:
3212:
3207:
3195:
3193:
3192:
3187:
3176:
3175:
3152:
3150:
3149:
3144:
3139:
3138:
3115:
3113:
3112:
3107:
3095:
3093:
3092:
3087:
3085:
3084:
3068:
3066:
3065:
3060:
3048:
3046:
3045:
3040:
3028:
3026:
3025:
3020:
3009:
3008:
2988:
2986:
2985:
2980:
2969:
2968:
2952:
2950:
2949:
2944:
2932:
2930:
2929:
2924:
2913:
2912:
2896:
2894:
2893:
2888:
2877:
2876:
2848:
2846:
2845:
2840:
2829:
2828:
2734:. The sequence {
2628:
2626:
2625:
2620:
2612:
2611:
2595:
2450:of the operator
2448:adjoint operator
2433:linear operators
2390:â -closed algebra
2356:dimension vector
2312:
2310:
2309:
2304:
2295:
2245:ArtinâWedderburn
2131:
2129:
2128:
2123:
2109:
2105:
2095:
2094:
2082:
2081:
2058:
2056:
2055:
2050:
2042:
2041:
1985:GNS construction
1967:
1965:
1964:
1959:
1935:
1933:
1932:
1927:
1905:
1903:
1902:
1897:
1885:
1883:
1882:
1877:
1865:
1863:
1862:
1857:
1839:
1837:
1836:
1831:
1826:
1825:
1803:
1801:
1800:
1795:
1779:
1777:
1776:
1771:
1753:
1751:
1750:
1745:
1727:
1725:
1724:
1719:
1693:
1691:
1690:
1685:
1683:
1663:
1661:
1660:
1655:
1653:
1652:
1629:
1627:
1626:
1621:
1616:
1615:
1595:
1593:
1592:
1587:
1585:
1584:
1519:
1517:
1516:
1511:
1506:
1505:
1474:
1472:
1471:
1466:
1461:
1460:
1433:
1432:
1410:
1408:
1407:
1402:
1397:
1396:
1357:
1355:
1354:
1349:
1347:
1346:
1325:
1324:
1272:, in which case
1243:
1241:
1240:
1235:
1232:
1231:
1207:
1206:
1165:
1163:
1162:
1157:
1051:
1049:
1048:
1043:
1035:
1032:
1017:
1016:
1004:
996:
976:
975:
960:
959:
919:
917:
916:
911:
906:
905:
884:
883:
839:
837:
836:
831:
823:
822:
792:
791:
752:
750:
749:
744:
739:
738:
729:
721:
716:
715:
676:
674:
673:
668:
666:
642:
640:
639:
634:
632:
631:
622:
621:
609:
608:
582:
580:
579:
574:
572:
571:
559:
558:
546:
545:
493:
491:
490:
485:
477:
476:
467:
466:
451:
450:
409:
407:
406:
401:
383:
381:
380:
375:
373:
372:
289:John von Neumann
281:matrix mechanics
244:
242:
241:
236:
225:
224:
132:
125:
121:
118:
112:
110:
69:
45:
37:
21:
5096:
5095:
5091:
5090:
5089:
5087:
5086:
5085:
5066:
5065:
5064:
5059:
5041:
5005:Advanced topics
5000:
4924:
4903:
4862:
4828:HilbertâSchmidt
4801:
4792:GelfandâNaimark
4739:
4689:
4624:
4610:
4580:
4575:
4536:Spectral method
4521:Ramanujan graph
4469:
4453:
4429:Fredholm theory
4397:
4392:Shilov boundary
4388:Structure space
4366:Generalizations
4361:
4352:Numerical range
4330:
4314:Uniform algebra
4276:
4252:Riesz projector
4237:Min-max theorem
4220:
4206:Direct integral
4162:
4148:Spectral radius
4119:
4074:
4028:
4019:Spectral radius
3967:
3961:Spectral theory
3958:
3897:
3857:
3838:
3816:
3790:
3760:
3741:
3736:
3735:
3726:
3722:
3714:
3710:
3703:
3699:
3687:
3683:
3671:
3667:
3662:
3643:Operator system
3614:
3527:
3485:
3463:
3393:
3358:
3325:
3322:
3321:
3305:
3302:
3301:
3275:
3271:
3269:
3266:
3265:
3235:
3231:
3229:
3226:
3225:
3201:
3198:
3197:
3171:
3167:
3165:
3162:
3161:
3134:
3130:
3125:
3122:
3121:
3101:
3098:
3097:
3080:
3076:
3074:
3071:
3070:
3054:
3051:
3050:
3034:
3031:
3030:
3004:
3000:
2998:
2995:
2994:
2964:
2960:
2958:
2955:
2954:
2938:
2935:
2934:
2908:
2904:
2902:
2899:
2898:
2872:
2868:
2866:
2863:
2862:
2824:
2820:
2818:
2815:
2814:
2811:locally compact
2803:
2746:
2739:
2732:
2725:
2706:
2655:
2648:
2637:
2607:
2603:
2585:
2573:
2570:
2569:
2564:
2553:
2488:
2425:
2376:
2353:
2282:
2270:
2267:
2266:
2227:, and use the
2198:
2193:
2103:
2090:
2086:
2077:
2073:
2065:
2062:
2061:
2037:
2033:
2028:
2025:
2024:
2013:
2009:
1995:Any C*-algebra
1993:
1941:
1938:
1937:
1915:
1912:
1911:
1891:
1888:
1887:
1871:
1868:
1867:
1845:
1842:
1841:
1821:
1817:
1809:
1806:
1805:
1789:
1786:
1785:
1759:
1756:
1755:
1733:
1730:
1729:
1713:
1710:
1709:
1679:
1677:
1674:
1673:
1648:
1644:
1639:
1636:
1635:
1630:forms a closed
1611:
1607:
1605:
1602:
1601:
1580:
1576:
1568:
1565:
1564:
1561:
1545:
1501:
1497:
1480:
1477:
1476:
1456:
1452:
1428:
1424:
1416:
1413:
1412:
1392:
1388:
1371:
1368:
1367:
1342:
1338:
1320:
1316:
1308:
1305:
1304:
1290:
1280:are said to be
1227:
1223:
1202:
1198:
1190:
1187:
1186:
1111:
1108:
1107:
1031:
1012:
1008:
1000:
992:
971:
967:
955:
951:
943:
940:
939:
926:section below.
901:
897:
879:
875:
867:
864:
863:
818:
814:
787:
783:
778:
775:
774:
734:
730:
720:
711:
707:
696:
693:
692:
662:
654:
651:
650:
627:
623:
617:
613:
604:
600:
589:
586:
585:
567:
563:
554:
550:
541:
537:
523:
520:
519:
472:
468:
462:
458:
443:
439:
437:
434:
433:
389:
386:
385:
368:
364:
356:
353:
352:
346:complex numbers
331:
255:locally compact
220:
216:
214:
211:
210:
133:
122:
116:
113:
70:
68:
58:
46:
35:
28:
23:
22:
15:
12:
11:
5:
5094:
5084:
5083:
5078:
5061:
5060:
5058:
5057:
5046:
5043:
5042:
5040:
5039:
5034:
5029:
5024:
5022:Choquet theory
5019:
5014:
5008:
5006:
5002:
5001:
4999:
4998:
4988:
4983:
4978:
4973:
4968:
4963:
4958:
4953:
4948:
4943:
4938:
4932:
4930:
4926:
4925:
4923:
4922:
4917:
4911:
4909:
4905:
4904:
4902:
4901:
4896:
4891:
4886:
4881:
4876:
4874:Banach algebra
4870:
4868:
4864:
4863:
4861:
4860:
4855:
4850:
4845:
4840:
4835:
4830:
4825:
4820:
4815:
4809:
4807:
4803:
4802:
4800:
4799:
4797:BanachâAlaoglu
4794:
4789:
4784:
4779:
4774:
4769:
4764:
4759:
4753:
4751:
4745:
4744:
4741:
4740:
4738:
4737:
4732:
4727:
4725:Locally convex
4722:
4708:
4703:
4697:
4695:
4691:
4690:
4688:
4687:
4682:
4677:
4672:
4667:
4662:
4657:
4652:
4647:
4642:
4636:
4630:
4626:
4625:
4609:
4608:
4601:
4594:
4586:
4577:
4576:
4574:
4573:
4568:
4563:
4558:
4553:
4548:
4543:
4538:
4533:
4528:
4523:
4518:
4513:
4508:
4503:
4498:
4488:
4486:Corona theorem
4483:
4477:
4475:
4471:
4470:
4468:
4467:
4465:Wiener algebra
4461:
4459:
4455:
4454:
4452:
4451:
4446:
4441:
4436:
4431:
4426:
4421:
4416:
4411:
4405:
4403:
4399:
4398:
4396:
4395:
4385:
4383:Pseudospectrum
4380:
4375:
4373:Dirac spectrum
4369:
4367:
4363:
4362:
4360:
4359:
4354:
4349:
4344:
4338:
4336:
4332:
4331:
4329:
4328:
4327:
4326:
4316:
4311:
4306:
4301:
4296:
4290:
4284:
4282:
4278:
4277:
4275:
4274:
4269:
4264:
4259:
4254:
4249:
4244:
4239:
4234:
4228:
4226:
4222:
4221:
4219:
4218:
4213:
4208:
4203:
4198:
4193:
4192:
4191:
4186:
4181:
4170:
4168:
4164:
4163:
4161:
4160:
4155:
4150:
4145:
4140:
4135:
4129:
4127:
4121:
4120:
4118:
4117:
4112:
4104:
4096:
4088:
4082:
4080:
4076:
4075:
4073:
4072:
4067:
4062:
4057:
4052:
4047:
4042:
4036:
4034:
4030:
4029:
4027:
4026:
4024:Operator space
4021:
4016:
4011:
4006:
4001:
3996:
3991:
3986:
3984:Banach algebra
3981:
3975:
3973:
3972:Basic concepts
3969:
3968:
3957:
3956:
3949:
3942:
3934:
3928:
3927:
3901:
3895:
3879:
3861:
3855:
3842:
3836:
3820:
3814:
3794:
3788:
3768:
3758:
3740:
3737:
3734:
3733:
3720:
3708:
3697:
3681:
3664:
3663:
3661:
3658:
3657:
3656:
3651:
3646:
3640:
3635:
3630:
3625:
3620:
3618:Banach algebra
3613:
3610:
3581:) â„ 0 for all
3526:
3523:
3484:
3481:
3462:
3459:
3357:
3354:
3329:
3309:
3289:
3286:
3283:
3278:
3274:
3249:
3246:
3243:
3238:
3234:
3222:weak* topology
3205:
3185:
3182:
3179:
3174:
3170:
3142:
3137:
3133:
3129:
3105:
3083:
3079:
3058:
3038:
3018:
3015:
3012:
3007:
3003:
2978:
2975:
2972:
2967:
2963:
2942:
2922:
2919:
2916:
2911:
2907:
2886:
2883:
2880:
2875:
2871:
2838:
2835:
2832:
2827:
2823:
2802:
2799:
2795:Calkin algebra
2742:
2737:
2730:
2723:
2704:
2653:
2646:
2635:
2630:
2629:
2618:
2615:
2610:
2606:
2602:
2599:
2594:
2591:
2588:
2584:
2580:
2577:
2556:
2551:
2518:subalgebra of
2487:
2484:
2424:
2421:
2374:
2354:is called the
2349:
2344:given by {dim(
2314:
2313:
2302:
2299:
2294:
2291:
2288:
2285:
2281:
2277:
2274:
2200:The algebra M(
2197:
2194:
2192:
2189:
2174:
2173:
2142:is separable,
2135:
2134:
2133:
2132:
2121:
2118:
2115:
2112:
2101:
2098:
2093:
2089:
2085:
2080:
2076:
2072:
2069:
2059:
2048:
2045:
2040:
2036:
2032:
2011:
2007:
1992:
1989:
1957:
1954:
1951:
1948:
1945:
1925:
1922:
1919:
1895:
1875:
1855:
1852:
1849:
1829:
1824:
1820:
1816:
1813:
1793:
1769:
1766:
1763:
1743:
1740:
1737:
1717:
1682:
1668:(or sometimes
1651:
1647:
1643:
1619:
1614:
1610:
1583:
1579:
1575:
1572:
1560:
1557:
1544:
1541:
1509:
1504:
1500:
1496:
1493:
1490:
1487:
1484:
1464:
1459:
1455:
1451:
1448:
1445:
1442:
1439:
1436:
1431:
1427:
1423:
1420:
1400:
1395:
1391:
1387:
1384:
1381:
1378:
1375:
1364:
1363:
1345:
1341:
1337:
1334:
1331:
1328:
1323:
1319:
1315:
1312:
1289:
1286:
1270:C*-isomorphism
1247:
1246:
1245:
1244:
1230:
1226:
1222:
1219:
1216:
1213:
1210:
1205:
1201:
1197:
1194:
1181:
1180:
1169:
1168:
1167:
1166:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1102:
1101:
1083:*-homomorphism
1055:
1054:
1053:
1052:
1041:
1038:
1030:
1026:
1023:
1020:
1015:
1011:
1007:
1003:
999:
995:
991:
988:
985:
982:
979:
974:
970:
966:
963:
958:
954:
950:
947:
909:
904:
900:
896:
893:
890:
887:
882:
878:
874:
871:
843:
842:
841:
840:
829:
826:
821:
817:
813:
810:
807:
804:
801:
798:
795:
790:
786:
782:
769:
768:
756:
755:
754:
753:
742:
737:
733:
727:
724:
719:
714:
710:
706:
703:
700:
687:
686:
665:
661:
658:
646:
645:
644:
643:
630:
626:
620:
616:
612:
607:
603:
599:
596:
593:
583:
570:
566:
562:
557:
553:
549:
544:
540:
536:
533:
530:
527:
514:
513:
497:
496:
495:
494:
483:
480:
475:
471:
465:
461:
457:
454:
449:
446:
442:
428:
427:
399:
396:
393:
371:
367:
363:
360:
342:Banach algebra
336:A C*-algebra,
330:
327:
304:Israel Gelfand
285:Pascual Jordan
234:
231:
228:
223:
219:
207:
206:
196:
149:Banach algebra
135:
134:
49:
47:
40:
26:
18:C star algebra
9:
6:
4:
3:
2:
5093:
5082:
5079:
5077:
5074:
5073:
5071:
5056:
5048:
5047:
5044:
5038:
5035:
5033:
5030:
5028:
5027:Weak topology
5025:
5023:
5020:
5018:
5015:
5013:
5010:
5009:
5007:
5003:
4996:
4992:
4989:
4987:
4984:
4982:
4979:
4977:
4974:
4972:
4969:
4967:
4964:
4962:
4959:
4957:
4954:
4952:
4951:Index theorem
4949:
4947:
4944:
4942:
4939:
4937:
4934:
4933:
4931:
4927:
4921:
4918:
4916:
4913:
4912:
4910:
4908:Open problems
4906:
4900:
4897:
4895:
4892:
4890:
4887:
4885:
4882:
4880:
4877:
4875:
4872:
4871:
4869:
4865:
4859:
4856:
4854:
4851:
4849:
4846:
4844:
4841:
4839:
4836:
4834:
4831:
4829:
4826:
4824:
4821:
4819:
4816:
4814:
4811:
4810:
4808:
4804:
4798:
4795:
4793:
4790:
4788:
4785:
4783:
4780:
4778:
4775:
4773:
4770:
4768:
4765:
4763:
4760:
4758:
4755:
4754:
4752:
4750:
4746:
4736:
4733:
4731:
4728:
4726:
4723:
4720:
4716:
4712:
4709:
4707:
4704:
4702:
4699:
4698:
4696:
4692:
4686:
4683:
4681:
4678:
4676:
4673:
4671:
4668:
4666:
4663:
4661:
4658:
4656:
4653:
4651:
4648:
4646:
4643:
4641:
4638:
4637:
4634:
4631:
4627:
4622:
4618:
4614:
4607:
4602:
4600:
4595:
4593:
4588:
4587:
4584:
4572:
4569:
4567:
4564:
4562:
4559:
4557:
4554:
4552:
4549:
4547:
4544:
4542:
4539:
4537:
4534:
4532:
4529:
4527:
4524:
4522:
4519:
4517:
4514:
4512:
4509:
4507:
4504:
4502:
4499:
4496:
4492:
4489:
4487:
4484:
4482:
4479:
4478:
4476:
4472:
4466:
4463:
4462:
4460:
4456:
4450:
4447:
4445:
4442:
4440:
4437:
4435:
4432:
4430:
4427:
4425:
4422:
4420:
4417:
4415:
4412:
4410:
4407:
4406:
4404:
4402:Miscellaneous
4400:
4393:
4389:
4386:
4384:
4381:
4379:
4376:
4374:
4371:
4370:
4368:
4364:
4358:
4355:
4353:
4350:
4348:
4345:
4343:
4340:
4339:
4337:
4333:
4325:
4322:
4321:
4320:
4317:
4315:
4312:
4310:
4307:
4305:
4302:
4300:
4297:
4295:
4291:
4289:
4286:
4285:
4283:
4279:
4273:
4270:
4268:
4265:
4263:
4260:
4258:
4255:
4253:
4250:
4248:
4245:
4243:
4240:
4238:
4235:
4233:
4230:
4229:
4227:
4223:
4217:
4214:
4212:
4209:
4207:
4204:
4202:
4199:
4197:
4194:
4190:
4187:
4185:
4182:
4180:
4177:
4176:
4175:
4172:
4171:
4169:
4167:Decomposition
4165:
4159:
4156:
4154:
4151:
4149:
4146:
4144:
4141:
4139:
4136:
4134:
4131:
4130:
4128:
4126:
4122:
4116:
4113:
4111:
4108:
4105:
4103:
4100:
4097:
4095:
4092:
4089:
4087:
4084:
4083:
4081:
4077:
4071:
4068:
4066:
4063:
4061:
4058:
4056:
4053:
4051:
4048:
4046:
4043:
4041:
4038:
4037:
4035:
4031:
4025:
4022:
4020:
4017:
4015:
4012:
4010:
4007:
4005:
4002:
4000:
3997:
3995:
3992:
3990:
3987:
3985:
3982:
3980:
3977:
3976:
3974:
3970:
3966:
3962:
3955:
3950:
3948:
3943:
3941:
3936:
3935:
3932:
3923:
3918:
3914:
3910:
3906:
3905:Segal, Irving
3902:
3898:
3896:3-540-63633-1
3892:
3888:
3884:
3880:
3877:
3873:
3872:
3867:
3862:
3858:
3856:0-471-23900-3
3852:
3848:
3843:
3839:
3833:
3830:, CRC Press,
3829:
3825:
3821:
3817:
3815:0-7204-0762-1
3811:
3806:
3805:
3799:
3795:
3791:
3789:0-12-185860-X
3785:
3780:
3779:
3773:
3772:Connes, Alain
3769:
3766:
3761:
3759:0-387-90176-0
3755:
3751:
3747:
3743:
3742:
3730:
3724:
3717:
3712:
3706:
3701:
3694:
3691:, p. 6,
3690:
3685:
3678:
3674:
3669:
3665:
3655:
3652:
3650:
3647:
3644:
3641:
3639:
3636:
3634:
3631:
3629:
3626:
3624:
3621:
3619:
3616:
3615:
3609:
3607:
3603:
3598:
3596:
3592:
3588:
3584:
3580:
3576:
3572:
3568:
3564:
3560:
3556:
3552:
3548:
3544:
3540:
3536:
3532:
3522:
3520:
3515:
3513:
3508:
3506:
3502:
3498:
3494:
3490:
3487:A C*-algebra
3480:
3478:
3473:
3471:
3467:
3458:
3456:
3452:
3448:
3444:
3440:
3436:
3432:
3431:group algebra
3428:
3425:
3420:
3418:
3414:
3410:
3406:
3401:
3397:
3391:
3387:
3383:
3379:
3375:
3371:
3367:
3363:
3353:
3351:
3347:
3343:
3327:
3307:
3284:
3276:
3272:
3263:
3244:
3236:
3232:
3223:
3219:
3203:
3180:
3172:
3168:
3159:
3154:
3135:
3131:
3119:
3103:
3081:
3077:
3056:
3036:
3013:
3005:
3001:
2992:
2973:
2965:
2961:
2940:
2917:
2909:
2905:
2881:
2873:
2869:
2860:
2856:
2852:
2833:
2825:
2821:
2812:
2808:
2798:
2796:
2792:
2788:
2784:
2780:
2776:
2772:
2768:
2764:
2760:
2756:
2754:
2750:
2745:
2740:
2733:
2726:
2719:
2715:
2711:
2707:
2700:
2696:
2692:
2688:
2684:
2680:
2676:
2672:
2668:
2664:
2658:
2656:
2649:
2642:
2638:
2616:
2608:
2604:
2597:
2592:
2589:
2586:
2582:
2578:
2575:
2568:
2567:
2566:
2563:
2559:
2554:
2547:
2543:
2539:
2535:
2530:
2527:
2525:
2521:
2517:
2513:
2509:
2505:
2501:
2497:
2493:
2483:
2481:
2477:
2473:
2469:
2465:
2461:
2457:
2453:
2449:
2445:
2441:
2438:
2437:Hilbert space
2434:
2430:
2420:
2418:
2413:
2411:
2407:
2403:
2399:
2395:
2391:
2387:
2382:
2380:
2373:
2369:
2368:positive cone
2365:
2361:
2357:
2352:
2347:
2343:
2339:
2335:
2331:
2325:
2323:
2319:
2300:
2297:
2292:
2286:
2283:
2279:
2275:
2272:
2265:
2264:
2263:
2261:
2257:
2253:
2248:
2246:
2242:
2238:
2234:
2230:
2229:operator norm
2226:
2222:
2218:
2215:
2211:
2207:
2203:
2188:
2185:
2183:
2179:
2171:
2167:
2163:
2159:
2158:
2153:
2149:
2145:
2141:
2137:
2136:
2119:
2116:
2113:
2110:
2099:
2096:
2091:
2087:
2083:
2078:
2074:
2070:
2067:
2060:
2046:
2038:
2034:
2030:
2023:
2022:
2021:
2020:
2019:
2017:
2006:
2002:
1998:
1988:
1986:
1982:
1978:
1974:
1969:
1955:
1952:
1949:
1946:
1943:
1923:
1920:
1917:
1909:
1893:
1873:
1853:
1850:
1847:
1827:
1822:
1818:
1814:
1811:
1791:
1783:
1767:
1764:
1761:
1741:
1738:
1735:
1715:
1707:
1704:
1700:
1695:
1671:
1667:
1649:
1645:
1641:
1633:
1617:
1612:
1608:
1599:
1581:
1577:
1573:
1570:
1556:
1554:
1550:
1540:
1538:
1534:
1530:
1526:
1521:
1502:
1498:
1491:
1485:
1457:
1453:
1443:
1437:
1429:
1425:
1421:
1393:
1389:
1382:
1376:
1361:
1343:
1335:
1329:
1321:
1317:
1313:
1303:
1302:
1301:
1299:
1295:
1294:C. E. Rickart
1285:
1283:
1279:
1275:
1271:
1267:
1262:
1260:
1256:
1252:
1228:
1220:
1214:
1211:
1203:
1199:
1192:
1185:
1184:
1183:
1182:
1179:
1175:
1171:
1170:
1149:
1143:
1137:
1131:
1128:
1122:
1119:
1113:
1106:
1105:
1104:
1103:
1100:
1096:
1092:
1088:
1087:
1086:
1084:
1080:
1076:
1072:
1068:
1064:
1060:
1039:
1028:
1024:
1021:
1018:
1013:
1009:
1005:
997:
983:
977:
972:
968:
961:
956:
948:
938:
937:
936:
935:
934:
932:
927:
925:
920:
907:
902:
894:
888:
880:
876:
872:
861:
859:
855:
851:
847:
827:
819:
815:
805:
799:
793:
788:
784:
773:
772:
771:
770:
766:
762:
758:
757:
740:
735:
731:
722:
717:
712:
704:
701:
691:
690:
689:
688:
684:
680:
659:
656:
648:
647:
628:
624:
618:
614:
610:
605:
597:
594:
584:
568:
564:
560:
555:
551:
547:
542:
534:
531:
528:
518:
517:
516:
515:
511:
507:
503:
499:
498:
481:
478:
473:
463:
459:
452:
447:
444:
440:
432:
431:
430:
429:
425:
421:
417:
413:
412:
411:
397:
394:
391:
369:
365:
358:
351:
347:
343:
339:
334:
326:
324:
320:
316:
311:
309:
305:
300:
298:
294:
290:
286:
282:
278:
274:
270:
266:
261:
259:
256:
252:
248:
229:
221:
217:
205:of operators.
204:
200:
197:
195:of operators.
194:
193:norm topology
190:
186:
183:
182:
181:
179:
178:Hilbert space
176:
172:
168:
165:
162:
158:
154:
150:
146:
142:
131:
128:
120:
117:February 2013
109:
106:
102:
99:
95:
92:
88:
85:
81:
78: â
77:
73:
72:Find sources:
66:
62:
56:
55:
50:This article
48:
44:
39:
38:
33:
19:
5017:Balanced set
4991:Distribution
4929:Applications
4878:
4782:KreinâMilman
4767:Closed graph
4474:Applications
4304:Disk algebra
4158:Spectral gap
4033:Main results
3993:
3915:(2): 73â88,
3912:
3908:
3889:, Springer,
3886:
3869:
3866:"C*-algebra"
3846:
3827:
3803:
3777:
3749:
3728:
3723:
3718:, p. 75
3711:
3700:
3693:Google Books
3684:
3677:Google Books
3668:
3599:
3594:
3590:
3586:
3582:
3578:
3574:
3570:
3566:
3562:
3558:
3554:
3550:
3546:
3542:
3538:
3534:
3528:
3516:
3509:
3504:
3500:
3496:
3492:
3488:
3486:
3474:
3464:
3450:
3449:in the case
3446:
3438:
3434:
3426:
3421:
3416:
3412:
3408:
3404:
3399:
3395:
3385:
3381:
3377:
3373:
3369:
3361:
3359:
3342:homeomorphic
3155:
2854:
2850:
2806:
2804:
2790:
2786:
2782:
2778:
2770:
2766:
2762:
2758:
2757:
2752:
2748:
2743:
2735:
2728:
2721:
2717:
2713:
2709:
2702:
2698:
2694:
2690:
2686:
2682:
2678:
2674:
2666:
2662:
2660:
2651:
2644:
2640:
2633:
2631:
2561:
2557:
2549:
2545:
2541:
2537:
2533:
2532:
2528:
2523:
2519:
2511:
2503:
2499:
2491:
2489:
2475:
2471:
2467:
2463:
2459:
2455:
2451:
2446:denotes the
2443:
2439:
2428:
2426:
2414:
2389:
2385:
2383:
2378:
2371:
2359:
2355:
2350:
2345:
2341:
2337:
2333:
2329:
2327:
2321:
2317:
2315:
2255:
2251:
2250:
2224:
2220:
2213:
2209:
2205:
2201:
2199:
2186:
2175:
2169:
2168:is dense in
2165:
2161:
2155:
2151:
2147:
2143:
2139:
2015:
2004:
1996:
1994:
1970:
1907:
1706:vector space
1698:
1696:
1669:
1666:non-negative
1665:
1600:of the form
1597:
1562:
1546:
1536:
1532:
1528:
1522:
1365:
1359:
1291:
1281:
1277:
1273:
1269:
1268:is called a
1265:
1263:
1250:
1248:
1177:
1173:
1098:
1094:
1090:
1082:
1081:is called a
1078:
1074:
1070:
1066:
1062:
1056:
928:
921:
862:
857:
849:
845:
844:
764:
760:
682:
678:
509:
505:
501:
423:
419:
418:, for every
337:
335:
332:
312:
308:Mark Naimark
301:
262:
250:
246:
208:
198:
184:
166:
144:
138:
123:
114:
104:
97:
90:
83:
76:"C*-algebra"
71:
59:Please help
54:verification
51:
5076:C*-algebras
4946:Heat kernel
4936:Hardy space
4843:Trace class
4757:HahnâBanach
4719:Topological
4501:Heat kernel
4201:Compression
4086:Isospectral
3746:Arveson, W.
3555:observables
3519:James Glimm
3514:is type I.
2516:norm closed
2260:canonically
2237:direct sums
2154:contains a
1632:convex cone
1525:I. E. Segal
1255:contractive
858:C* identity
273:observables
5070:Categories
4879:C*-algebra
4694:Properties
4179:Continuous
3994:C*-algebra
3989:B*-algebra
3739:References
3716:Segal 1947
3705:Segal 1947
3541:(elements
3388:) that is
3352:programs.
3262:isomorphic
3218:characters
2565:such that
2316:where min
2241:semisimple
2164:such that
2018:such that
1983:using the
1754:satisfies
1282:isomorphic
677:and every
416:involution
189:closed set
153:involution
87:newspapers
4853:Unbounded
4848:Transpose
4806:Operators
4735:Separable
4730:Reflexive
4715:Algebraic
4701:Barrelled
3965:-algebras
3883:Sakai, S.
3876:EMS Press
3628:*-algebra
3390:universal
2793:) is the
2650:) with ||
2590:∈
2583:⨁
2579:≅
2496:separable
2386:â -algebra
2377:group of
2287:∈
2280:⨁
2117:μ
2114:≤
2111:λ
2097:≤
2092:μ
2084:≤
2079:λ
2071:≤
2044:→
2039:λ
1953:≥
1947:−
1921:≥
1851:∈
1840:for some
1823:∗
1765:≥
1739:∈
1716:≥
1650:∗
1613:∗
1582:∗
1508:‖
1503:∗
1495:‖
1489:‖
1483:‖
1463:‖
1458:∗
1450:‖
1447:‖
1441:‖
1435:‖
1430:∗
1419:‖
1411:. Hence,
1399:‖
1394:∗
1386:‖
1380:‖
1374:‖
1340:‖
1333:‖
1327:‖
1322:∗
1311:‖
1259:isometric
1229:∗
1215:π
1204:∗
1193:π
1144:π
1132:π
1114:π
1025:λ
1022:−
1014:∗
998:λ
981:‖
973:∗
965:‖
953:‖
946:‖
899:‖
892:‖
886:‖
881:∗
870:‖
854:*-algebra
825:‖
820:∗
812:‖
809:‖
803:‖
797:‖
789:∗
781:‖
736:∗
726:¯
723:λ
713:∗
702:λ
660:∈
657:λ
629:∗
619:∗
606:∗
569:∗
556:∗
543:∗
474:∗
464:∗
448:∗
445:∗
414:It is an
395:∈
370:∗
362:↦
258:Hausdorff
145:C-algebra
5055:Category
4867:Algebras
4749:Theorems
4706:Complete
4675:Schwartz
4621:glossary
4566:Weyl law
4511:Lax pair
4458:Examples
4292:With an
4211:Discrete
4189:Residual
4125:Spectrum
4110:operator
4102:operator
4094:operator
4009:Spectrum
3885:(1971),
3800:(1969),
3774:(1994),
3748:(1976),
3612:See also
3364:with an
3196:, where
2775:quotient
2720:and let
2534:Theorem.
2454: :
2364:K-theory
2252:Theorem.
2217:matrices
2191:Examples
2178:quotient
2138:In case
1910:satisfy
1782:spectrum
1670:positive
1358:for all
1065: :
759:For all
500:For all
203:adjoints
4858:Unitary
4838:Nuclear
4823:Compact
4818:Bounded
4813:Adjoint
4787:Minâmax
4680:Sobolev
4665:Nuclear
4655:Hilbert
4650:Fréchet
4615: (
4107:Unitary
3577:with Ï(
2989:has an
2661:Though
2657:|| â 0.
2442:; here
2394:physics
2370:of the
1999:has an
924:history
846:Remark.
340:, is a
260:space.
191:in the
175:complex
164:algebra
161:complex
157:adjoint
101:scholar
4833:Normal
4670:Orlicz
4660:Hölder
4640:Banach
4629:Spaces
4617:topics
4091:Normal
3893:
3853:
3834:
3812:
3786:
3756:
3394:Ïâ'â:
2398:dagger
2247:type:
1977:states
103:
96:
89:
82:
74:
4645:Besov
4184:Point
3660:Notes
3559:state
3545:with
3380:into
2853:that
2809:be a
2785:) by
2514:is a
2506:) of
2494:be a
2258:, is
2219:over
2208:) of
2182:ideal
852:is a
293:rings
269:model
253:is a
173:on a
108:JSTOR
94:books
4993:(or
4711:Dual
4115:Unit
3963:and
3891:ISBN
3851:ISBN
3832:ISBN
3810:ISBN
3784:ISBN
3754:ISBN
3475:The
3348:and
3340:are
3320:and
3156:The
3069:let
2805:Let
2701:let
2673:for
2490:Let
2429:B(H)
1886:and
1276:and
1172:For
1093:and
1089:For
1077:and
384:for
306:and
143:, a
80:news
3917:doi
3597:).
3579:u*u
3565:(a
3529:In
3445:of
3433:of
3264:to
3260:is
2777:of
2755:).
2536:If
2510:on
2358:of
2336:),
2290:min
2166:hAh
2012:λâI
1936:if
1906:of
1784:of
1176:in
1097:in
1085:if
987:sup
763:in
681:in
508:in
422:in
350:map
317:of
279:'s
267:to
169:of
63:by
5072::
4619:â
3913:53
3911:,
3874:,
3868:,
3782:,
3585:â
3573:â
3549:=
3547:x*
3457:.
3419:.
3398:â
2797:.
2716:â„
2693:=
2482:.
2458:â
2444:x*
2419:.
2412:.
2384:A
2381:.
2348:)}
2330:Ae
2212:Ă
2204:,
1987:.
1968:.
1694:)
1555:.
1284:.
1069:â
1061:,
1057:A
504:,
325:.
299:.
4997:)
4721:)
4717:/
4713:(
4623:)
4605:e
4598:t
4591:v
4497:)
4493:(
4394:)
4390:(
3953:e
3946:t
3939:v
3926:.
3919::
3900:.
3841:.
3767:.
3695:.
3679:.
3595:x
3591:x
3587:A
3583:u
3575:C
3571:A
3567:C
3563:A
3551:x
3543:x
3539:A
3535:A
3505:A
3501:A
3497:A
3493:A
3489:A
3451:G
3447:G
3439:G
3435:G
3427:G
3417:A
3409:A
3407:(
3405:E
3400:B
3396:A
3386:A
3384:(
3382:E
3378:A
3374:A
3372:(
3370:E
3362:A
3328:Y
3308:X
3288:)
3285:Y
3282:(
3277:0
3273:C
3248:)
3245:X
3242:(
3237:0
3233:C
3204:X
3184:)
3181:X
3178:(
3173:0
3169:C
3141:}
3136:K
3132:f
3128:{
3104:K
3082:K
3078:f
3057:K
3037:X
3017:)
3014:X
3011:(
3006:0
3002:C
2977:)
2974:X
2971:(
2966:0
2962:C
2941:X
2921:)
2918:X
2915:(
2910:0
2906:C
2885:)
2882:X
2879:(
2874:0
2870:C
2851:X
2837:)
2834:X
2831:(
2826:0
2822:C
2807:X
2791:H
2789:(
2787:K
2783:H
2781:(
2779:B
2771:H
2769:(
2767:B
2763:H
2761:(
2759:K
2753:H
2751:(
2749:K
2744:n
2741:}
2738:n
2736:e
2731:n
2729:H
2724:n
2722:e
2718:n
2714:k
2710:l
2705:n
2703:H
2699:n
2695:l
2691:H
2687:l
2683:H
2679:H
2677:(
2675:K
2667:H
2665:(
2663:K
2654:i
2652:T
2647:i
2645:H
2643:(
2641:K
2636:i
2634:T
2617:,
2614:)
2609:i
2605:H
2601:(
2598:K
2593:I
2587:i
2576:A
2562:I
2560:â
2558:i
2555:}
2552:i
2550:H
2546:H
2544:(
2542:K
2538:A
2524:H
2522:(
2520:B
2512:H
2504:H
2502:(
2500:K
2492:H
2476:H
2472:H
2470:(
2468:B
2464:A
2460:H
2456:H
2452:x
2440:H
2379:A
2375:0
2372:K
2360:A
2351:e
2346:e
2342:A
2338:C
2334:e
2324:.
2322:A
2318:A
2301:e
2298:A
2293:A
2284:e
2276:=
2273:A
2256:A
2225:C
2221:C
2214:n
2210:n
2206:C
2202:n
2172:.
2170:A
2162:h
2152:A
2148:A
2144:A
2140:A
2120:.
2100:1
2088:e
2075:e
2068:0
2047:x
2035:e
2031:x
2016:A
2010:}
2008:λ
2005:e
1997:A
1956:0
1950:y
1944:x
1924:y
1918:x
1908:A
1894:y
1874:x
1854:A
1848:s
1828:s
1819:s
1815:=
1812:x
1792:x
1768:0
1762:x
1742:A
1736:x
1699:A
1681:R
1646:x
1642:x
1618:x
1609:x
1598:A
1578:x
1574:=
1571:x
1537:H
1533:H
1531:(
1529:B
1499:x
1492:=
1486:x
1454:x
1444:x
1438:=
1426:x
1422:x
1390:x
1383:=
1377:x
1360:x
1344:2
1336:x
1330:=
1318:x
1314:x
1278:B
1274:A
1266:Ï
1251:Ï
1225:)
1221:x
1218:(
1212:=
1209:)
1200:x
1196:(
1178:A
1174:x
1153:)
1150:y
1147:(
1141:)
1138:x
1135:(
1129:=
1126:)
1123:y
1120:x
1117:(
1099:A
1095:y
1091:x
1079:B
1075:A
1071:B
1067:A
1063:Ï
1040:.
1037:}
1029:1
1019:x
1010:x
1006::
1002:|
994:|
990:{
984:=
978:x
969:x
962:=
957:2
949:x
908:,
903:2
895:x
889:=
877:x
873:x
850:A
828:.
816:x
806:x
800:=
794:x
785:x
767::
765:A
761:x
741:.
732:x
718:=
709:)
705:x
699:(
685::
683:A
679:x
664:C
625:x
615:y
611:=
602:)
598:y
595:x
592:(
565:y
561:+
552:x
548:=
539:)
535:y
532:+
529:x
526:(
512::
510:A
506:y
502:x
482:x
479:=
470:)
460:x
456:(
453:=
441:x
426::
424:A
420:x
398:A
392:x
366:x
359:x
338:A
251:X
247:X
233:)
230:X
227:(
222:0
218:C
199:A
185:A
167:A
130:)
124:(
119:)
115:(
105:·
98:·
91:·
84:·
57:.
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
