200:
is a bijection. It follows from this correspondence that every closed ideal is a hereditary C*-subalgebra. Another corollary is that a hereditary C*-subalgebra of a simple C*-algebra is also simple.
132:. C*-algebras are stably isomorphic to their full hereditary C*-subalgebras. Hence, two C*-algebras are stably isomorphic if they contain stably isomorphic full hereditary C*-subalgebras.
303:
78:
424:
28:
is a particular type of C*-subalgebra whose structure is closely related to that of the larger C*-algebra. A C*-subalgebra
125:
267:
These hereditary C*-subalgebras can bring some insight into the notion of Cuntz subequivalence. In particular, if
136:
392:. More generally, a C*-algebra contains a strictly positive element if and only if the algebra has a
385:
282:
148:
There is a bijective correspondence between closed left ideals and hereditary C*-subalgebras of
393:
489:
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470:
81:
is also AF. This is not true for subalgebras that are not hereditary. For instance, every
8:
135:
Also hereditary C*-subalgebras are those C*-subalgebras in which the restriction of any
217:
440:
420:
466:
456:
384:. A commutative C*-algebra contains a strictly positive element if and only if the
261:
380:, a compact operator is strictly positive if and only if its range is dense in
92:
if it is not contained in any proper (two-sided) closed ideal. Two C*-algebras
483:
129:
82:
461:
444:
389:
17:
25:
417:
Operator
Algebras: Theory of C*-Algebras and von Neumann Algebras
348:. This suggests the following notion for the non-unital case:
445:"Stable Isomorphism of Hereditary Subalgebras of C*-algebras"
143:
285:
264:, then every hereditary C*-subalgebra has this form.
203:
248:
is the smallest hereditary C*-subalgebra containing
297:
85:C*-algebra can be embedded into an AF C*-algebra.
481:
376:) of compact operators acting on Hilbert space
79:approximately finite-dimensional C*-algebra
460:
414:
228:is a hereditary C*-subalgebra known as a
36:is a hereditary C*-subalgebra if for all
482:
275:are positive elements of a C*-algebra
144:Correspondence with closed left ideals
439:
128:on a separable infinite-dimensional
336:is unital and the positive element
236:. More generally, given a positive
13:
204:Connections with positive elements
14:
501:
368:. For example, in the C*-algebra
77:A hereditary C*-subalgebra of an
168:under the *-operation. The set
449:Pacific Journal of Mathematics
433:
408:
180:is a C*-subalgebra. In fact,
1:
402:
71:
419:. Springer. pp. 75–79.
160:is a closed left ideal, let
7:
298:{\displaystyle a\precsim b}
10:
506:
188:is hereditary and the map
137:irreducible representation
88:A C*-subalgebra is called
415:Blackadar, Bruce (2006).
244:, the closure of the set
340:is invertible, then Her(
216:(or a projection of the
22:hereditary C*-subalgebra
462:10.2140/pjm.1977.71.335
309: ∈ Her(
172:* is a right ideal and
299:
164:* denote the image of
300:
124:is the C*-algebra of
397:approximate identity
283:
139:is also irreducible.
212:is a projection of
116: ⊗
112: ≅
108: ⊗
441:Brown, Lawrence G.
388:of the algebra is
325:) = Her(
295:
218:multiplier algebra
426:978-3-540-28517-5
358:strictly positive
252:, denoted by Her(
126:compact operators
102:stably isomorphic
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304:
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11:
5:
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477:
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455:(2): 335–348.
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390:σ-compact
356:is said to be
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205:
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145:
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133:
86:
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52:such that 0 ≤
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328:
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317: ~
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240: ∈
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130:Hilbert space
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43:
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27:
23:
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349:
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121:
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37:
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21:
15:
490:C*-algebras
100:are called
18:mathematics
471:0362.46042
403:References
394:sequential
313:). Hence,
72:Properties
60:, we have
26:C*-algebra
290:≾
262:separable
484:Category
443:(1977).
386:spectrum
224:), then
120:, where
360:if Her(
321:if Her(
279:, then
83:abelian
469:
423:
256:). If
230:corner
152:. If
24:of a
421:ISBN
364:) =
344:) =
271:and
196:* ∩
184:* ∩
176:* ∩
96:and
90:full
44:and
20:, a
467:Zbl
457:doi
332:If
329:).
305:if
260:is
246:aAa
232:of
226:pAp
220:of
208:If
104:if
32:of
16:In
486::
465:.
453:71
451:.
447:.
399:.
352:∈
192:↦
156:⊂
68:.
64:∈
56:≤
48:∈
40:∈
473:.
459::
429:.
382:H
378:H
374:H
372:(
370:K
366:A
362:a
354:A
350:a
346:A
342:a
338:a
334:A
327:b
323:a
319:b
315:a
311:a
307:b
293:b
287:a
277:A
273:b
269:a
258:A
254:a
250:a
242:A
238:a
234:A
222:A
214:A
210:p
198:L
194:L
190:L
186:L
182:L
178:L
174:L
170:L
166:L
162:L
158:A
154:L
150:A
122:K
118:K
114:B
110:K
106:A
98:B
94:A
66:B
62:a
58:b
54:a
50:B
46:b
42:A
38:a
34:A
30:B
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