# locally compact quantum groups from von Neumann/$C^{*}$- algebras with Haar measures

## 0.1 Hilbert spaces, Von Neumann algebras and Quantum Groups

John von Neumann introduced a mathematical foundation for Quantum Mechanics in the form of $W^{*}$-algebras (http://planetmath.org/WeakHopfCAlgebra2) of (quantum) bounded operators in a (quantum:= presumed separable, i.e. with a countable basis) Hilbert space $H_{S}$. Recently, such von Neumann algebras , $W^{*}$ (http://planetmath.org/WeakHopfCAlgebra2) and/or (more generally) C*-algebras are, for example, employed to define locally compact quantum groups $CQG_{lc}$ (http://planetmath.org/LocallyCompactQuantumGroup) by equipping such algebras with a co-associative multiplication (http://planetmath.org/WeakHopfCAlgebra2) and also with associated, both left– and right– Haar measures, defined by two semi-finite normal weights .

### 0.1.1 Remark on Jordan-Banach-von Neumann (JBW) algebras, $JBWA$

A JLB–algebra is a $JB$–algebra $\mathfrak{A}_{\mathbb{R}}$ together with a Poisson bracket for which it becomes a Jordan–Lie algebra $JL$ for some $\hslash^{2}\geq 0$ . Such JLB–algebras often constitute the real part of several widely studied complex associative algebras. For the purpose of quantization, there are fundamental relations between $\mathfrak{A}^{sa}$, JLB and Poisson algebras (http://planetmath.org/JordanBanachAndJordanLieAlgebras).

These appeared in the work of von Neumann who developed an orthomodular lattice theory of projections on $\mathcal{L}(H)$ on which to study quantum logic . BW-algebras have the following property: whereas $\mathfrak{A}^{sa}$ is a J(L)B–algebra, the self-adjoint part of a von Neumann algebra is a JBW–algebra.

## References

Title locally compact quantum groups from von Neumann/$C^{*}$- algebras with Haar measures LocallyCompactQuantumGroupsFromVonNeumannCAlgebrasWithHaarMeasures 2013-03-22 18:24:28 2013-03-22 18:24:28 bci1 (20947) bci1 (20947) 16 bci1 (20947) Topic msc 47A70 msc 46N50 msc 47L30 msc 47N50 msc 81P15 msc 46C05 locally compact quantum groups quantum groupoids Hopf and weak Hopf algebras HilbertSpace QuantumSpaceTimes VonNeumannAlgebra WeakHopfCAlgebra2 ClassificationOfHilbertSpaces QuantumSpaceTimes VonNeumannAlgebra WeakHopfCAlgebra2 JordanBanachAndJordanLieAlgebras QuantumLogic Distribution4 JBW-algebras $JBW$ algebras $JB$–algebra $JBWA$ $JL$ Jordan-Banach-von Neumann algebras $CQG_{lc}$
{"ft_lang_label":"__label__en","ft_lang_prob":0.64077044,"math_prob":0.9673394,"size":4878,"snap":"2019-35-2019-39","text_gpt3_token_len":1388,"char_repetition_ratio":0.14464505,"word_repetition_ratio":0.041800644,"special_character_ratio":0.24456744,"punctuation_ratio":0.18,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9965441,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-09-19T18:49:53Z\",\"WARC-Record-ID\":\"<urn:uuid:a96e3cf5-61c7-43e8-9569-7c32d64a5fbc>\",\"Content-Length\":\"31949\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:97e5e195-bf44-489f-a207-893c61f32a60>\",\"WARC-Concurrent-To\":\"<urn:uuid:a29080b2-b805-44b4-923d-e138c8063713>\",\"WARC-IP-Address\":\"129.97.206.129\",\"WARC-Target-URI\":\"https://planetmath.org/locallycompactquantumgroupsfromvonneumanncalgebraswithhaarmeasures\",\"WARC-Payload-Digest\":\"sha1:ROPRH4SXFOBXH5N6NC3T7PBH746LIKWO\",\"WARC-Block-Digest\":\"sha1:RT5RKMWCRYXPXFU3DFYL6HVP7X4GBGHC\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-39/CC-MAIN-2019-39_segments_1568514573570.6_warc_CC-MAIN-20190919183843-20190919205843-00241.warc.gz\"}"}