Lie groups were introduced in 1870 by the Norwegian mathematician Sophus Lie. A century later Jean Dieudonnāe quipped that Lie groups had moved to the center of mathematics and that one cannot undertake anything without them.

A *pro-Lie group* is a complete topological group *G* in which every identity neighborhood *U* of *G* contains a normal subgroup *N* such that the quotient *G/N* is a Lie group. Every locally compact connected topological group and every compact group is a pro-Lie group. While the class of locally compact groups is not closed under the formation of arbitrary products, the class of pro-Lie groups is.

For half a century, locally compact pro-Lie groups have drifted through the literature, yet this is the first book which systematically treats the Lie theory and the structure theory of pro-Lie groups irrespective of local compactness. So it fits very well into that current trend which addresses infinite dimensional Lie groups. The results of this text are based on a theory of pro-Lie algebras which parallels the structure theory of finite dimensional real Lie algebras to an astonishing degree even though it has to overcome technical obstacles.

A topological group is said to be almost connected if the quotient group of its connected components is compact. This book exposes a Lie theory of almost connected pro-Lie groups (and hence of almost connected locally compact groups) and illuminates the variety of ways in which their structure theory reduces to that of compact groups on the one hand and of finite dimensional Lie groups on the other. It is therefore a continuation of the authors' monograph on the structure of compact groups (1998, 2006, 2014, 2020, 2023) and is an invaluable tool for researchers in topological groups, Lie theory, harmonic analysis and representation theory. It is written to be accessible to advanced graduate students wishing to study this fascinating and important area of research, which has so many fruitful interactions with other fields of mathematics.

For the first edition of this book, please click [here](https://ems.press/books/etm/37).

07aMathematics2bicssc07aResearch exposition (monographs, survey articles) pertaining to topological groups2msc07aStructure of general topological groups2msc07aGeneral properties and structure of locally compact groups2msc07aGeneral properties and structure of other Lie groups2msc07aInfinite-dimensional Lie groups and their Lie algebras: general properties2msc1 aMorris4aut40uhttps://ems.press/doi/10.4171/etm/36423cover imageuhttps://content.ems.press/assets/public/images/books/cover-270.png03820nam 22008175a 450000100160000000300080001600500110002400600190003500700150005400800410006902000180011002400250012804000120015307200150016508400150018008400150019508400150021008400150022508400150024008400150025508400150027008400150028508400150030008400150031508400150033008400150034508400150036008400150037508400150039008400150040508400150042008400150043508400150045008400150046508400150048010000220049524500550051726000200057230000350059233600260062733700260065333800360067934700240071549000750073950600560081452006240087065000240149465000680151865000950158665000890168165000610177065000450183165001370187665001220201365000750213565000370221065000540224765000730230165000720237465000440244665000410249065000440253165000350257565000210261065000870263165000400271865000610275865000570281985600420287685600840291810.4171/irma/34DE-40842023-08-08a fot 00| 0|cr nn mmmmamaa230808e20230808gw fot 00| 0|eng d a97839854752477 a10.4171/irma/342doi aDE-4084 7aPB2bicssc a00B152msc a01-022msc a01-062msc a01A702msc a14B052msc a34M352msc a57K202msc a57R452msc a32S052msc a32S552msc a57K122msc a57R172msc a57K452msc a57K302msc a57K452msc a00A302msc a57K102msc a57K162msc a58K302msc a51M152msc a53C702msc1 aPapadopoulos4edt10aEssays in Geometry ;bDedicated to Norbert ACampo.3 aEMS Pressc2023 a1 online resource (1028 pages) atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda1 aIRMA Lectures in Mathematics and Theoretical Physics (irma)x2523-51411 aRestricted to subscribers.uhttps://ems.press/books aThis volume consists in a collection of essays dedicated to Norbert A'Campo on the occasion of his 80th birthday. The subject is geometry in the broadest sense. The topics include hyperbolic and super hyperbolic geometry, 3-manifolds, metric geometry, mapping class groups, linear groups, Riemann surfaces, Teichmčuller spaces, high-dimensional complex geometry, differential topology, symplectic geometry, singularity theory, number theory, algebraic geometry, dynamics, mathematical physics and philosophy of mathematics. The book gives a fairly comprehensive overview of the wealth of current research in geometry. 07aMathematics2bicssc07aCollections of articles of miscellaneous specific interest2msc07aResearch exposition (monographs, survey articles) pertaining to history and biography2msc07aProceedings, conferences, collections, etc. pertaining to history and biography2msc07aBiographies, obituaries, personalia, bibliographies2msc07aSingularities in algebraic geometry2msc07aSingularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms2msc07a2-dimensional topology (including mapping class groups of surfaces, Teichmčuller theory, curve complexes, etc.)2msc07aSingularities of differentiable mappings in differential topology2msc07aLocal complex singularities2msc07aMilnor fibration; relations with knot theory2msc07aGeneralized knots (virtual knots, welded knots, quandles, etc.)2msc07aSymplectic and contact topology in high or arbitrary dimension2msc07aHigher-dimensional knots and links2msc07aGeneral topology of 3-manifolds2msc07aHigher-dimensional knots and links2msc07aPhilosophy of mathematics2msc07aKnot theory2msc07aFinite-type and quantum invariants, topological quantum field theories (TQFT)2msc07aGlobal theory of singularities2msc07aGeometric constructions in real or complex geometry2msc07aDirect methods (\(G\)-spaces of Busemann, etc.)2msc40uhttps://ems.press/doi/10.4171/irma/34423cover imageuhttps://content.ems.press/assets/public/images/books/cover-265.png03371nam 22004935a 450000100150000000300080001500500110002300600190003400700150005300800410006802000180010902400240012704000120015107200150016308400150017808400150019308400150020808400150022308400150023808400150025308400150026810000190028324500470030226000200034930000340036933600260040333700260042933800360045534700240049149000700051550600410058552016800062665000240230665000210233065000730235165000630242465000490248765000880253665000480262465000640267270000160273685600410275285600840279310.4171/mems/7DE-40842023-10-07a fot 00| 0|cr nn mmmmamaa231007e20231007gw fot 00| 0|eng d a97839854755517 a10.4171/mems/72doi aDE-4084 7aPB2bicssc a76T202msc a35R602msc a76M502msc a35Q352msc a76D032msc a76D072msc a60G552msc1 aDuerinckx4aut10aOn Einstein's Effective Viscosity Formula.3 aEMS Pressc2023 a1 online resource (196 pages) atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda1 aMemoirs of the European Mathematical Society (mems)v7x2747-90990 aOpen Accessuhttps://ems.press/books aIn his PhD thesis, Einstein derived an explicit first-order expansion for the effective viscosity of a Stokes fluid with a suspension of small rigid particles at low density. His formal derivation relied on two implicit assumptions: (i) there is a scale separation between the size of the particles and the observation scale; and (ii) at first order, dilute particles do not interact with one another. In mathematical terms, the first assumption amounts to the validity of a homogenization result defining the effective viscosity tensor, which is now well understood. Next, the second assumption allowed Einstein to approximate this effective viscosity at low density by considering particles as being isolated. The rigorous justification is, in fact, quite subtle as the effective viscosity is a nonlinear nonlocal function of the ensemble of particles and as hydrodynamic interactions have borderline integrability.

In the present memoir, we establish Einstein's effective viscosity formula in the most general setting. In addition, we pursue the low-density expansion to arbitrary order in form of a cluster expansion, where the summation of hydrodynamic interactions crucially requires suitable renormalizations. In particular, we justify a celebrated result by Batchelor and Green on the second-order correction and we explicitly describe all higher-order renormalizations for the first time. In some specific settings, we further address the summability of the whole cluster expansion. Our approach relies on a combination of combinatorial arguments, variational analysis, elliptic regularity, probability theory, and diagrammatic integration methods.

07aMathematics2bicssc07aSuspensions2msc07aPDEs with randomness, stochastic partial differential equations2msc07aHomogenization applied to problems in fluid mechanics2msc07aPDEs in connection with fluid mechanics2msc07aExistence, uniqueness, and regularity theory for incompressible viscous fluids2msc07aStokes and related (Oseen, etc.) flows2msc07aPoint processes (e.g., Poisson, Cox, Hawkes processes)2msc1 aGloria4aut40uhttps://ems.press/doi/10.4171/mems/7423cover imageuhttps://content.ems.press/assets/public/images/books/cover-269.png02555nam 22004335a 450000100150000000300080001500500110002300600190003400700150005300800410006802000180010902400240012704000120015107200150016308400150017808400150019308400150020808400150022310000140023824501460025226000200039830000340041833600260045233700260047833800360050434700240054049000560056450600560062052009090067665000240158565000980160965000830170765000870179065000860187770000160196370000170197985600410199685600840203710.4171/ecr/19DE-40842023-11-30a fot 10| 0|cr nn mmmmamaa231130e20231130gw fot 10| 0|eng d a97839854755447 a10.4171/ecr/192doi aDE-4084 7aPB2bicssc a16-062msc a18-062msc a13-062msc a14-062msc1 aBuan4edt10aRepresentations of Algebras and Related Structures ;bInternational Conference on Representations of Algebras, ICRA 2020, 9-25 November 2020.3 aEMS Pressc2023 a1 online resource (428 pages) atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda1 aEMS Series of Congress Reports (ecr)v19x2523-51681 aRestricted to subscribers.uhttps://ems.press/books aThis volume presents a collection of articles devoted to representations of algebras and related topics. Distinguished experts in this field presented their work at the International Conference on Representations of Algebras in 2020. The book reflects recent trends in the representation theory of algebras and its interactions with other central branches of mathematics, including combinatorics, commutative algebra, algebraic geometry, topology, data analysis, Lie algebras, quantum groups, homological algebra, and theoretical physics. There are thirteen independent articles, written by leading experts in the field. Most are expository survey papers, but some are also original research contributions. This collection is addressed to researchers and graduate students in algebra as well as to a broader mathematical audience. It contains open problems and new perspectives for research in the field.07aMathematics2bicssc07aProceedings, conferences, collections, etc. pertaining to associative rings and algebras2msc07aProceedings, conferences, collections, etc. pertaining to category theory2msc07aProceedings, conferences, collections, etc. pertaining to commutative algebra2msc07aProceedings, conferences, collections, etc. pertaining to algebraic geometry2msc1 aKrause4edt1 aSolberg4edt40uhttps://ems.press/doi/10.4171/ecr/19423cover imageuhttps://content.ems.press/assets/public/images/books/cover-271.png01978nam 22003375a 450000100160000000300080001600500110002400600190003500700150005400800410006902000180011002400250012804000120015307200150016508400150018010000170019524500640021226000200027630000350029633600260033133700260035733800360038334700240041950600410044352009180048465000240140265000710142670000170149785600420151485600840155610.4171/icm2022DE-40842023-12-15a fot 10| 0|cr nn mmmmamaa231215e20231215gw fot 10| 0|eng d a97839854755827 a10.4171/icm20222doi aDE-4084 7aPB2bicssc a00B252msc1 aBeliaev4edt10aInternational Congress of Mathematicians ;b2022 July 6-14.3 aEMS Pressc2023 a1 online resource (5940 pages) atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda0 aOpen Accessuhttps://ems.press/books aFollowing the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library. The contents of the ICM 2022 Proceedings are available online with open access.07aMathematics2bicssc07aProceedings of conferences of miscellaneous specific interest2msc1 aSmirnov4edt40uhttps://ems.press/doi/10.4171/icm2022423cover imageuhttps://content.ems.press/assets/public/images/books/cover-272.png02009nam 22003375a 450000100180000000300080001800500110002600600190003700700150005600800410007102000180011202400270013004000120015707200150016908400150018410000170019924500900021626000200030630000340032633600260036033700260038633800360041234700240044850600410047252009180051365000240143165000710145570000170152685600440154385600840158710.4171/icm2022-1DE-40842023-12-15a fot 10| 0|cr nn mmmmamaa231215e20231215gw fot 10| 0|eng d a97839854755997 a10.4171/icm2022-12doi aDE-4084 7aPB2bicssc a00B252msc1 aBeliaev4edt10aInternational Congress of Mathematicians ;b2022 July 6-14.bVolume I. Prize Lectures3 aEMS Pressc2023 a1 online resource (596 pages) atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda0 aOpen Accessuhttps://ems.press/books aFollowing the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library. The contents of the ICM 2022 Proceedings are available online with open access.07aMathematics2bicssc07aProceedings of conferences of miscellaneous specific interest2msc1 aSmirnov4edt40uhttps://ems.press/doi/10.4171/icm2022-1423cover imageuhttps://content.ems.press/assets/public/images/books/cover-273.png02012nam 22003375a 450000100180000000300080001800500110002600600190003700700150005600800410007102000180011202400270013004000120015707200150016908400150018410000170019924500930021626000200030930000340032933600260036333700260038933800360041534700240045150600410047552009180051665000240143465000710145870000170152985600440154685600840159010.4171/icm2022-2DE-40842023-12-15a fot 10| 0|cr nn mmmmamaa231215e20231215gw fot 10| 0|eng d a97839854756057 a10.4171/icm2022-22doi aDE-4084 7aPB2bicssc a00B252msc1 aBeliaev4edt10aInternational Congress of Mathematicians ;b2022 July 6-14.bVolume II. Plenary Lectures3 aEMS Pressc2023 a1 online resource (880 pages) atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda0 aOpen Accessuhttps://ems.press/books aFollowing the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library. The contents of the ICM 2022 Proceedings are available online with open access.07aMathematics2bicssc07aProceedings of conferences of miscellaneous specific interest2msc1 aSmirnov4edt40uhttps://ems.press/doi/10.4171/icm2022-2423cover imageuhttps://content.ems.press/assets/public/images/books/cover-274.png02009nam 22003375a 450000100180000000300080001800500110002600600190003700700150005600800410007102000180011202400270013004000120015707200150016908400150018410000170019924500900021626000200030630000340032633600260036033700260038633800360041234700240044850600410047252009180051365000240143165000710145570000170152685600440154385600840158710.4171/icm2022-3DE-40842023-12-15a fot 10| 0|cr nn mmmmamaa231215e20231215gw fot 10| 0|eng d a97839854756127 a10.4171/icm2022-32doi aDE-4084 7aPB2bicssc a00B252msc1 aBeliaev4edt10aInternational Congress of Mathematicians ;b2022 July 6-14.bVolume III. Sections 1-43 aEMS Pressc2023 a1 online resource (952 pages) atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda0 aOpen Accessuhttps://ems.press/books aFollowing the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library. The contents of the ICM 2022 Proceedings are available online with open access.07aMathematics2bicssc07aProceedings of conferences of miscellaneous specific interest2msc1 aSmirnov4edt40uhttps://ems.press/doi/10.4171/icm2022-3423cover imageuhttps://content.ems.press/assets/public/images/books/cover-275.png02009nam 22003375a 450000100180000000300080001800500110002600600190003700700150005600800410007102000180011202400270013004000120015707200150016908400150018410000170019924500890021626000200030530000350032533600260036033700260038633800360041234700240044850600410047252009180051365000240143165000710145570000170152685600440154385600840158710.4171/icm2022-4DE-40842023-12-15a fot 10| 0|cr nn mmmmamaa231215e20231215gw fot 10| 0|eng d a97839854756297 a10.4171/icm2022-42doi aDE-4084 7aPB2bicssc a00B252msc1 aBeliaev4edt10aInternational Congress of Mathematicians ;b2022 July 6-14.bVolume IV. Sections 5-83 aEMS Pressc2023 a1 online resource (1016 pages) atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda0 aOpen Accessuhttps://ems.press/books aFollowing the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library. The contents of the ICM 2022 Proceedings are available online with open access.07aMathematics2bicssc07aProceedings of conferences of miscellaneous specific interest2msc1 aSmirnov4edt40uhttps://ems.press/doi/10.4171/icm2022-4423cover imageuhttps://content.ems.press/assets/public/images/books/cover-276.png02009nam 22003375a 450000100180000000300080001800500110002600600190003700700150005600800410007102000180011202400270013004000120015707200150016908400150018410000170019924500900021626000200030630000340032633600260036033700260038633800360041234700240044850600410047252009180051365000240143165000710145570000170152685600440154385600840158710.4171/icm2022-5DE-40842023-12-15a fot 10| 0|cr nn mmmmamaa231215e20231215gw fot 10| 0|eng d a97839854756367 a10.4171/icm2022-52doi aDE-4084 7aPB2bicssc a00B252msc1 aBeliaev4edt10aInternational Congress of Mathematicians ;b2022 July 6-14.bVolume V. Sections 9-113 aEMS Pressc2023 a1 online resource (812 pages) atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda0 aOpen Accessuhttps://ems.press/books aFollowing the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library. The contents of the ICM 2022 Proceedings are available online with open access.07aMathematics2bicssc07aProceedings of conferences of miscellaneous specific interest2msc1 aSmirnov4edt40uhttps://ems.press/doi/10.4171/icm2022-5423cover imageuhttps://content.ems.press/assets/public/images/books/cover-277.png02011nam 22003375a 450000100180000000300080001800500110002600600190003700700150005600800410007102000180011202400270013004000120015707200150016908400150018410000170019924500920021626000200030830000340032833600260036233700260038833800360041434700240045050600410047452009180051565000240143365000710145770000170152885600440154585600840158910.4171/icm2022-6DE-40842023-12-15a fot 10| 0|cr nn mmmmamaa231215e20231215gw fot 10| 0|eng d a97839854756437 a10.4171/icm2022-62doi aDE-4084 7aPB2bicssc a00B252msc1 aBeliaev4edt10aInternational Congress of Mathematicians ;b2022 July 6-14.bVolume VI. Sections 12-143 aEMS Pressc2023 a1 online resource (880 pages) atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda0 aOpen Accessuhttps://ems.press/books aFollowing the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library. The contents of the ICM 2022 Proceedings are available online with open access.07aMathematics2bicssc07aProceedings of conferences of miscellaneous specific interest2msc1 aSmirnov4edt40uhttps://ems.press/doi/10.4171/icm2022-6423cover imageuhttps://content.ems.press/assets/public/images/books/cover-278.png02011nam 22003375a 450000100180000000300080001800500110002600600190003700700150005600800410007102000180011202400270013004000120015707200150016908400150018410000170019924500920021626000200030830000340032833600260036233700260038833800360041434700240045050600410047452009180051565000240143365000710145770000170152885600440154585600840158910.4171/icm2022-7DE-40842023-12-15a fot 10| 0|cr nn mmmmamaa231215e20231215gw fot 10| 0|eng d a97839854756507 a10.4171/icm2022-72doi aDE-4084 7aPB2bicssc a00B252msc1 aBeliaev4edt10aInternational Congress of Mathematicians ;b2022 July 6-14.bVolume VII. Sections 15-203 aEMS Pressc2023 a1 online resource (804 pages) atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda0 aOpen Accessuhttps://ems.press/books aFollowing the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library. The contents of the ICM 2022 Proceedings are available online with open access.07aMathematics2bicssc07aProceedings of conferences of miscellaneous specific interest2msc1 aSmirnov4edt40uhttps://ems.press/doi/10.4171/icm2022-7423cover imageuhttps://content.ems.press/assets/public/images/books/cover-279.png03172nam 22004455a 450000100150000000300080001500500110002300600190003400700150005300800410006802000180010902400240012704000120015107200150016308400150017808400150019308400150020808400150022308400150023810000170025324500770027026000200034730000340036733600260040133700260042733800360045334700240048949000700051350600410058352017210062465000240234565000410236965000320241065000440244265000700248665000330255670000120258985600410260185600840264210.4171/mems/8DE-40842024-02-02a fot 00| 0|cr nn mmmmamaa240202e20240202gw fot 00| 0|eng d a97839854756677 a10.4171/mems/82doi aDE-4084 7aPB2bicssc a19K352msc a19K332msc a46L052msc a46L802msc a46L852msc1 aWillett4aut10aThe Universal Coefficient Theorem for *-Algebras with Finite Complexity.3 aEMS Pressc2024 a1 online resource (108 pages) atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda1 aMemoirs of the European Mathematical Society (mems)v8x2747-90990 aOpen Accessuhttps://ems.press/books aA *-algebra satisfies the Universal Coefficient Theorem (UCT) of Rosenberg and Schochet if it is equivalent in Kasparov's KK-theory to a commutative *-algebra. This paper is motivated by the problem of establishing the range of validity of the UCT, and in particular, whether the UCT holds for all nuclear *-algebras. We introduce the idea of a *-algebra that "decomposes" over a class of *-algebras. Roughly, this means that locally there are approximately central elements that approximately cut the *-algebra into two *-subalgebras from that have well-behaved intersection. We show that if a *-algebra decomposes over the class of nuclear, UCT *-algebras, then it satisfies the UCT. The argument is based on a Mayer-Vietoris principle in the framework of controlled KK-theory; the latter was introduced by the authors in an earlier work. Nuclearity is used via Kasparov's Hilbert module version of Voiculescu's theorem, and Haagerup's theorem that nuclear *-algebras are amenable. We say that a *-algebra has finite complexity if it is in the smallest class of *-algebras containing the finite-dimensional *-algebras, and closed under decomposability; our main result implies that all *-algebras in this class satisfy the UCT. The class of *-algebras with finite complexity is large, and comes with an ordinal-number invariant measuring the complexity level. We conjecture that a *-algebra of finite nuclear dimension and real rank zero has finite complexity; this (and several other related conjectures) would imply the UCT for all separable nuclear *-algebras. We also give new local formulations of the UCT, and some other necessary and sufficient conditions for the UCT to hold for all nuclear *-algebras.07aMathematics2bicssc07aKasparov theory (\(KK\)-theory)2msc07aExt and \(K\)-homology2msc07aGeneral theory of \(C^*\)-algebras2msc07a\(K\)-theory and operator algebras (including cyclic theory)2msc07aNoncommutative topology2msc1 aYu4aut40uhttps://ems.press/doi/10.4171/mems/8423cover imageuhttps://content.ems.press/assets/public/images/books/cover-280.png03335nam 22005655a 450000100150000000300080001500500110002300600190003400700150005300800410006802000180010902400240012704000120015107200150016308400150017808400150019308400150020808400150022308400150023808400150025308400150026808400150028308400150029808400150031310000180032824500710034626000200041730000340043733600260047133700260049733800360052334700240055949000700058350600410065352012510069465000240194565000760196965000800204565000430212565000500216865000680221865000880228665000620237465000360243665000840247265000740255670000140263085600410264485600840268510.4171/mems/9DE-40842024-02-02a fot 00| 0|cr nn mmmmamaa240202e20240202gw fot 00| 0|eng d a97839854756747 a10.4171/mems/92doi aDE-4084 7aPB2bicssc a22E662msc a17B152msc a17B562msc a17B652msc a17B662msc a17B672msc a17B812msc a22E602msc a22E652msc a22E672msc1 aJanssens4aut10aPositive Energy Representations of Gauge Groups I ;bLocalization.3 aEMS Pressc2024 a1 online resource (156 pages) atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda1 aMemoirs of the European Mathematical Society (mems)v9x2747-90990 aOpen Accessuhttps://ems.press/books aThis is the first in a series of papers on projective positive energy representations of gauge groups. Let be a principal fiber bundle, and let