A characterization and sum decomposition for operator ideals

Blass, Andreas; Weiss, Gary

doi:10.1090/s0002-9947-1978-0515547-x

Cited by 11 publications

(5 citation statements)

References 10 publications

(3 reference statements)

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“…and this last set is forced to be a subset of k: by ps, hence also by the extension (r, s). Since x = AcG"), we have verified requirement (2). It remains to check (5): r IF "For every n E o, there is an extension t of s in (Po/Gk)/H forcing 'B* n n .x(m) ".…”

Section: B*n(k+l)~b*nmg{i<m(q(i)=l}mentioning

confidence: 87%

“…Indeed, (1) to ( 3) are trivial for k = 0 (as PO is trivial), (4) says p E N which is true by our choice of N, and ( 5) is exactly fact 4.5 above. Third, (1) implies that, for each k, pn 1 k is independent of n once n 2 k. Since P, is an inverse limit, we can define 4 E p, bY then 4 extends every pk, in particular p, and it forces "B* s A" by (2). Thus, CJ will be as desired, so the proof will be complete once we construct the pk's.…”

Section: ) Pk R K If "Pk 1 [K O) E N[gk]" and (5) Pk 1 K It-"for Every N E O There Is An Extension T Of Pk 1 [K O) In P/g Forcing 'B* N Nmentioning

confidence: 89%

“…First, as incdicated by the context, II-refers to forcing with P, in (2) and to forcing with Pk in ( 4) and ( 5); the word 'forcing' in (5) refers to forcing with Pa/G,.…”

Section: ) Pk R K If "Pk 1 [K O) E N[gk]" and (5) Pk 1 K It-"for Every N E O There Is An Extension T Of Pk 1 [K O) In P/g Forcing 'B* N Nmentioning

confidence: 99%

“…The statement (A), or rather its negation, arose in the work of Blass and Weiss [2] on decompositions of ideals of Hilbert space operators. It was shown there that, if one could refute (A) in ZFC, then one could also eliminate the continuum hypothesis from the main theorem of that paper.…”

Section: Introductionmentioning

confidence: 99%

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There may be simple Pℵ1 and Pℵ2-points and the Rudin-Keisler ordering may be downward directed

Blass

Shelah

1987

Annals of Pure and Applied Logic

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122

119

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We prove the consistency, relative to ZFC, of each of the following two (mutually contradictory) statements. (A) Every two non-principal ultrafilters on o have a common image via a finite-to-one function. (B) Simple &,-points and simple &+-points both exist. These results, proved by the second author, answer questions of the first author and P. Nyikos, who had obtained numerous consequences of (A) and (B), respectively. In the models we construct, the bounding number is K,, while the dominating number, the splitting number, and the cardinality of the continuum are HZ.

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Section: B*n(k+l)~b*nmg{i<m(q(i)=l}mentioning

confidence: 87%

Section: ) Pk R K If "Pk 1 [K O) E N[gk]" and (5) Pk 1 K It-"for Every N E O There Is An Extension T Of Pk 1 [K O) In P/g Forcing 'B* N Nmentioning

confidence: 89%

“…First, as incdicated by the context, II-refers to forcing with P, in (2) and to forcing with Pk in ( 4) and ( 5); the word 'forcing' in (5) refers to forcing with Pa/G,.…”

Section: ) Pk R K If "Pk 1 [K O) E N[gk]" and (5) Pk 1 K It-"for Every N E O There Is An Extension T Of Pk 1 [K O) In P/g Forcing 'B* N Nmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

There may be simple Pℵ1 and Pℵ2-points and the Rudin-Keisler ordering may be downward directed

Blass

Shelah

1987

Annals of Pure and Applied Logic

Self Cite

122

119

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“…As a ( ) is am-closed, this yields: Further consequences of Theorem 10 are obtained by exploiting lattice properties of ideals, in particular, of some classes of principal ideals. Blass and Weiss (19) proved that K(H) is the sum of two proper ideals (neither of which can be countably generated) and in general, every ideal that properly contains F is the sum of two proper ideals. Here we obtain that with respect to the inclusion order, the lattice of principal ideals has no ''gaps'', that is, between any two principal ideals lies another one.…”

Section: Dimension Of I͞[i B(h)]mentioning

confidence: 99%

Traces, ideals, and arithmetic means

Kaftal

Weiss²

2002

Proc. Natl. Acad. Sci. U.S.A.

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This article grew out of recent work of Dykema, Figiel, Weiss, and Wodzicki (Commutator structure of operator ideals) which inter alia characterizes commutator ideals in terms of arithmetic means. In this paper we study ideals that are arithmetically mean (am) stable, am-closed, am-open, soft-edged and soft-complemented. We show that many of the ideals in the literature possess such properties. We apply these notions to prove that for all the ideals considered, the linear codimension of their commutator space (the ''number of traces on the ideal'') is either 0, 1, or ؕ. We identify the largest ideal which supports a unique nonsingular trace as the intersection of certain Lorentz ideals. An application to elementary operators is given. We study properties of arithmetic mean operations on ideals, e.g., we prove that the am-closure of a sum of ideals is the sum of their am-closures. We obtain cancellation properties for arithmetic means: for principal ideals, a necessary and sufficient condition for first order cancellations is the regularity of the generator; for second order cancellations, sufficient conditions are that the generator satisfies the exponential ⌬2-condition or is regular. We construct an example where second order cancellation fails, thus settling an open question. We also consider cancellation properties for inclusions. And we find and use lattice properties of ideals associated with the existence of ''gaps.'' T he algebra B(H) of bounded linear operators on a separable, infinite-dimensional, complex Hilbert space has only one nonzero proper closed two-sided ideal, the class of compact operators K(H). There is, however, a rich structure of nonclosed two-sided ideals of B(H) (operator ideals). Their study was initiated by Calkin (1), who established a lattice isomorphism between ideals and characteristic sets, i.e., the hereditary (solid) positive cones ⌺ ʚ c* o (the collection of monotone sequences decreasing to 0) that are invariant under ampliation: 3 ( 1 , 1 , 2 , 2 , 3 , 3 , . . .). Given an ideal I, call ⌺(I) :ϭ {s(X)͉X ʦ I} the characteristic set of I where s(X) :ϭ ͗s n (X)͘ is the sequence of s-numbers of X, i.e., the eigenvalues of ͉X͉ counting multiplicities and arranged in decreasing order with infinitely many zeroes added in case X is finite rank. Conversely, if ⌺ is a characteristic set, the diagonal operators diag with ʦ ⌺ generate the (unique) ideal I such that ⌺(I) ϭ ⌺.For each ideal I, we denote by [I, B(H)] the commutator space for I (also known as the commutator ideal), i.e., the linear span of all the commutators XB-BX where X ʦ I and B ʦ B(H). Commutator spaces are central to the theory of operator ideals. For instance, they play a key role in defining traces (see the next section). Starting with Halmos (2) and Pearcy and Topping (3), a great deal of effort has been devoted over the years to characterizing commutator spaces for various ideals (see ref. This result, along with others in ref. 4, have consequences in the area of operator ideals and traces. Here we explore some of ...

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Classification of certain commutator ideals and the tensor product closure property

Weiss

1989

Integr equ oper theory

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A characterization and sum decomposition for operator ideals

Cited by 11 publications

References 10 publications

There may be simple Pℵ1 and Pℵ2-points and the Rudin-Keisler ordering may be downward directed

There may be simple Pℵ1 and Pℵ2-points and the Rudin-Keisler ordering may be downward directed

Traces, ideals, and arithmetic means

Classification of certain commutator ideals and the tensor product closure property

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