Timeline for The monotone closure of a $C^*$-algebra
Current License: CC BY-SA 3.0
16 events
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| Jul 24, 2016 at 11:52 | vote | accept | Masayoshi Kaneda | ||
| Aug 5, 2012 at 16:17 | comment | added | Nik Weaver | Huh. Well, I'm not sure where to look, then. | |
| Aug 5, 2012 at 16:03 | comment | added | Masayoshi Kaneda | Yes, of course. Notes and remarks are provided at the end of each section in his book. The relevant section (Section 2.4. The up-down theorem) refers to [Kadison, Ann. of Math. (1956)], [Pedersen, Amer. J. Math. (1972)] as well as [Pedersen, The `Up-Down' problem for operator algebras, Proc. Nat. Acad. Sci. USA, 68 (1971), 1896-1897]. The last article is not available at MathSciNet, but it seems that the contents are repeated in [Pedersen, Amer. J. Math. (1972)] with detailed proofs. | |
| Aug 5, 2012 at 15:40 | comment | added | Nik Weaver | Specifically, did you check the notes at the end of the relevant chapter in Gert's book? | |
| Aug 5, 2012 at 13:19 | vote | accept | Masayoshi Kaneda | ||
| Jul 24, 2016 at 11:52 | |||||
| Aug 5, 2012 at 13:15 | comment | added | Masayoshi Kaneda | (Continuation) In any case, Question 2 was more important to me at this time, so thank you again for your help. | |
| Aug 5, 2012 at 13:14 | comment | added | Masayoshi Kaneda | (Continuation) The result is also rephrased as ``for any $C^*$-algebra $A$ in $B(H)$ one can characterize $\bar{A_{s.a.}}$ as being the smallest class $M$ in $B(H)$, containing $A_{s.a.}$, such that $M^m=M_m=M$.’’ on p. 956 of [Pedersen, Monotone closures in operator algebras, Amer. J. Math. 94 (1972), 955-962]. Here $M^m$ and $M_m$ are in the usual sense. Although $M$ can be any “class”, this does not answer my question since both “increasing” and “decreasing” are involved. However, I feel that the statement of Question 1 is false as you think. | |
| Aug 5, 2012 at 13:11 | comment | added | Masayoshi Kaneda | (Continuation) Kadison’s result cited in Hamana’s paper can be found on pp. 316-317 of [Kadison, Unitary invariants for representations of operator algebras, Ann. of Math. (2) 66 (1957), 304-379], though it is not presented as a theorem or lemma. (Note that in this paper, $A_1^m$ is used in a different meaning from the usual one. It is the union of the monotone closures of $A_1$ taken “transfinitely recursively”. Also the monotone closures are taken in both directions, i.e., both increasing and decreasing.) | |
| Aug 5, 2012 at 13:07 | comment | added | Masayoshi Kaneda | (Continuation) In Pedersen’s book (Theorem 2.4.4), Kadison’s other (but related) result “A $C^*$-subalgebra $M$ of $B(H)$ is a von Neumann algebra if and only if $(M_{sa})^m= M_{sa}$” (Lemma 1 of [Kadison, Operator algebras with a faithful weakly-closed representation, Ann. of Math. (2) 64 (1956), 175-181]) is presented. But the proof in Pedersen’s book is not Kadison’s original one, but one using his own lemma which was used to prove his “up-down” theorem. | |
| Aug 5, 2012 at 13:05 | comment | added | Masayoshi Kaneda | Nik, thank you again for your response. For Question 2, I see, the point is to consider $l^{\infty}[0,1]$ instead of $L^{\infty}[0,1]$. Thank you for a nice example. For Question 1, of course, I looked over Pedersen’s book, but I couldn’t find the answer. This is why I am placing the question on this board. | |
| Aug 4, 2012 at 15:20 | history | edited | Nik Weaver | CC BY-SA 3.0 |
added 196 characters in body
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| Aug 4, 2012 at 14:47 | history | undeleted | Nik Weaver | ||
| Aug 4, 2012 at 14:46 | history | deleted | Nik Weaver | ||
| Aug 4, 2012 at 11:14 | comment | added | Masayoshi Kaneda | For Question 2: If you don't mind, could you please provide me with your counterexample in the abelian case? Thank you. P.S. My Question 1 seems exactly what Jon is asking. | |
| Aug 4, 2012 at 10:45 | comment | added | Masayoshi Kaneda |
Thank you Nik for your answer. Please allow me to ask further questions following your answer. For Question 1: Yes, I know that paper by Hamana. In fact, I previously highlighted the sentence $B_{sa}$ is itself a unique real linear subspace $S$ of $B_{sa}$ such that $A_{sa}\subset S$ and $S^m=S$'' in Introduction. I think that you are referring this part. In this statement, $S$ is assumed to be a real linear subspace of $B_{sa}$. However, in my question, $(A_{sa})^m$ is just a selfadjoint subset'' of $B(H)_{sa}$ and not a linear subspace in general. Could you please clarify this point?
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| Aug 3, 2012 at 15:36 | history | answered | Nik Weaver | CC BY-SA 3.0 |