Динамічна ентропія в операторних алгебрах та квантові системи Колмогорова: Автореф. дис… канд. фіз.-мат. наук / С.В. Нешвєєв, НАН України. Фіз.-техн

IAOe?IIAEUeIA AEAAeAI?ss IAOE OE?A?IE

O?CEEI-OAOI?*IEE ?INOEOOO

IECUeEEO OAIIA?AOO?

?I. A.?.A??E?IA

Ia i?aaao ?oeiieno

IAOA??A Na?a?e A?oae?eiae/

OAeE 517.98

AeEIAI?*IA AIO?II?ss A IIA?AOI?IEO

AEAAA?AO

OA EAAIOIA? NENOAIE EIEIIAI?IAA

01.01.01-iaoaiaoe/iee aiae?c

AAOI?AOA?AO

aeena?oaoe?? ia caeiaoooy iaoeiaiai nooiaiy

eaiaeeoaoa o?ceei-iaoaiaoe/ieo iaoe

Oa?e?a-1999

Aeena?oaoe??th ? ?oeiien.

?iaioa aeeiiaia a O?ceei-oaoi?/iiio ?inoeooo? iecueeeo oaiia?aoo? IAI
Oe?a?ie.

Iaoeiaee ea??aiee aeieoi? o?ceei-iaoaiaoe/ieo iaoe

Aieiaeaoeue Aaeaioei sseiae/,

O?ceei-oaoi?/iee ?inoeooo iecueeeo

oaiia?aoo? IAI Oe?a?ie,

i?ia?aeiee iaoeiaee ni?a?ia?oiee.

Io?oe?ei? iiiiaioe:

aeieoi? o?ceei-iaoaiaoe/ieo iaoe, i?ioani?

Naiieeaiei TH??e Noaoaiiae/,

?inoeooo iaoaiaoeee IAI Oe?a?ie;

eaiaeeaeao o?ceei-iaoaiaoe/ieo iaoe, aeioeaio

Ianniiia Ieeiea ?aaiiae/,

Oa?e?anueeee aea?aeaaiee oi?aa?neoao

n?eueneiai ainiiaea?noaa.

I?ia?aeia onoaiiaa:

Oa?e?anueeee aea?aeaaiee oi?aa?neoao, eaoaae?a iaoaiaoe/iiai
aiae?co

Caoeno a?aeaoaeaoueny “ 8 ” /a?aiy 1999?. i 11 aiaeei? ia
can?aeaii? niaoe?ae?ciaaii? a/aii? ?aaee Ae 64.175.01 O?ceei-oaoi?/iiai
?inoeoooo iecueeeo oaiia?aoo? IAI Oe?a?ie ca aae?anith: 310 164, i.
Oa?e?a, i?. Eai?ia, 47, e. 216.

C aeena?oaoe??th iiaeia iciaeiieoenue o iaoeia?e a?ae?ioaoe?
O?ceei-oaoi?/iiai ?inoeoooo iecueeeo oaiia?aoo? , i. Oa?e?a, i?. Eai?ia,
47.

Aaoi?aoa?ao ?ic?neaiee “28” ea?oiy 1999?.

A/aiee nae?aoa?

niaoe?ae?ciaaii? a/aii? ?aaee A.I. Eioey?ia

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

Aeooaeuei?noue oaie. Iiiyooy aio?ii?? aaoiii?o?cio i?inoi?o Eaaaaa aoei
aaaaeaia Eieiiai?iaei o 1958 ?ioe?, a iio?i oaeineiiaeaii N?ia?i o 1959
?ioe?. A o?e ae ?iaio? Eieiiai?ia aa?a iiiyooy eaac??aaoey?ii? nenoaie.
Iio?i oea iiiyooy aoei aeine?aeaeoaaia ?ioe?iei oa N?ia?i, a nai?
nenoaie ae?noaee iacae E-nenoai.

Eean E-nenoai aeyaeany aeineoue oe?ieei. Aei iueiai iaeaaeaoue,
iai?eeeaae, ia?ai?ooth/? ia?e?anuee? aaoiii?o?cie, a?aiaee/i?
aaoiii?o?cie eiiiaeoieo aaaeaaeo a?oi, ?aeaaeueiee aac, aaiaeace/i?
oa/?? ia iiaa?oiyo a?ae’?iii? e?eaeie. I?e oeueiio aio?ii?y a oeueiio
eean? ? neeueiei ?iaa??aioii: caaea/o iiaoaeiae iani?yaeaieo E-nenoai c
iaeiaeiaith aio?ii??th aoei ?ica’ycaii eeoa o 1973 ?ioe? I?inoaeiii.

C oi/ee ci?o iia?aoi?ieo aeaaa? a?aiaee/ia oai??y aea/a? aaoiii?o?cie
(aeeooc?eii?, c naia?aaaeueiei i?aaeni?yaeaiei) aaaeaai? W*-aeaaa?e. I?e
oeueiio o ia?oo /a?ao noaiiaeyoue ?ioa?an o? aaoiii?o?cie, aeey yeeo
?nio? ?iaa??aioiee ii?iaeueiee noai. Oiae? aio?ii?y Eieiiai?iaa-N?iay ?
/enei, yea ni?anoaaeth?oueny aaoiii?o?cio ( oa (-?iaa??aioiiio noaio (,
oa ? ?iaa??aioii ni?yaeaiino? ia?e ((,(). sseui aaoiii?o?ci a?aiaee/iee,
oi ?iaa??aioiee noai ?aeeiee, ? aio?ii?y ? ?iaa??aioii ni?yaeaiino?
aaoiii?o?cio.

Iai?ee?ioe? 60-eo — ii/aoeo 70-eo ?ie?a aoee c?iaeai? ni?iae ia?aianoe
iiiyooy aio?ii?? ia aeiaaeie aaoiii?o?ci?a aeia?eueii? W*-aeaaa?e.
Aaeaea icia/aiiy aoei aeaia nii/aoeo aeey ne?aeiaiai aeiaaeeo Eiiii oa
Noa?ia?ii o 1976 ?ioe?, a iio?i aeey caaaeueiiai aeiaaeeo Eiiii,
Ia?ioioa? oa O???iaii o 1987 ?ioe?. Cai?iaaaeaeoaaia ieie aaee/eia
iaceaa?oueny aeeiai?/iith aio?ii??th, aai CNT-aio?ii??th. sse ? a
eiiooaoeaiiio aeiaaeeo, aeey a?aiaee/iiai aaoiii?o?cio ?iaa??aioiee
ii?iaeueiee noai ?aeeiee (yeui ?nio?), ? aio?ii?y ? ?iaa??aioii
ni?yaeaiino? aaoiii?o?cio.

Ia nueiaiaei? CNT-aio?ii?y ia/eneaia aeey aaaaoueio iiaeaeae:
iaeiiooaoeaieo aa?ioe??anueeeo oa ia?e?anueeeo cnoa?a, cnoa?a aeaaa?
Oaiia?e?-E?aa, aaeeeiai eeano a?ia?ieo cnoa?a oa aiaietha?anueeeo
aaoiii?o?ci?a aeaaa? eaiiii/ieo eiiooaoe?eieo oa aioeeiiooaoe?eieo
ni?aa?aeiioaiue. Io?eiai? oaeiae ioe?iee aeey aio?ii?? aaoiio?o?ci?a
iaeiiooaoeaieo oi??a oa eaiii?/ieo cnoa?a aeaaa? Ieiaaio. Ia/eneaiith
aio?ii?? i?enay/ai? ?iaioe Eiia, Aieeoeaneo, Noa?ia?a, Aieiaeoey, *iaee
oa aaaaoueio ?ioeo.

*eneaii?noue nenoai c ia/eneaiith aio?ii??th aea? ciiao ia?aeoe aei
ye?niiai aiae?co aeeiai?ee iaeiiooaoeaieo nenoai. I?e oeueiio aaaeeeaith
caaea/ath ? ciaoiaeaeaiiy aiaeiaa E-aeanoeaino?. Oaeee aiaeia aoei
cai?iiiiiaaii Ia?ioioa? oa O???iaii. I?e?iaeiith caaea/ath ?
aeine?aeaeaiiy ??cieo iiaeaeae ia i?aaeiao iayaiino? o ieo
E-aeanoeaino?.

Ca’ycie c iaoeiaeie i?ia?aiaie, ieaiaie, oaiaie. Oaia ?iaioe
caoaa?aeaeaia ia can?aeaii? A/aii? ?aaee iaoaiaoe/iiai a?aeae?eaiiy
OO?IO IAI Oe?a?ie (i?ioieie ?4 a?ae 29 eenoiiaaea 1995 ?ieo).
Aeine?aeaeaiiy, aeeiiai? o ?iaio?, ? /anoeiith ieaiiai? oaie 1.4.10.22.6
iaoaiaoe/iiai a?aeae?eaiiy OO?IO IAI Oe?a?ie.

Iaoa ? caaea/? aeine?aeaeaiiy. Iniiaia iaoa aeine?aeaeaiiy – oea
ciaoiaeaeaiiy aeinoaoi?o oiia aeey aeeiiaiiy E-aeanoeaino? oa ia?aa??ea
E-aeanoeaino? aeey a?aeiieo iiaeaeae oai??? iia?aoi?ieo aeaaa? oa
eaaioiai? noaoenoe/ii? iaoai?ee.

Iaoeiaa iiaecia iaea?aeaieo ?acoeueoao?a. On? ?acoeueoaoe aeena?oaoe?? ?
iiaeie.

I?aeoe/ia cia/aiiy iaea?aeaieo ?acoeueoao?a. ?iaioa iineoue oai?aoe/iee
oa?aeoa?. ?? ?acoeueoaoe iiaeooue aooe aeei?enoai? i?e aea/aii?
aaoiii?o?ci?a iia?aoi?ieo aeaaa?, a oaeiae a eaaioia?e noaoenoe/i?e
iaoai?oe? i?e aea/aii? aeeiai?ee iane?i/aiiiaei??ieo eaaioiaeo nenoai.

Iniaenoee aianie caeiaoaa/a. Oaia aeine?aeaeaiiy oa iinoaiiaee caaea/
iaeaaeaoue iaoeiaiio ea??aieeo Aieiaeoeth Aaeaioeio sseiae/o.
?acoeueoaoe, ui iaea?aeai? coi?nii c iei ia?a?aoiaai? a aeena?oaoe?ei?e
?iaio?.

Ai?iaaoe?y ?acoeueoao?a aeena?oaoe??. ?acoeueoaoe ?iaioe aeiiia?aeaeeny
ia “I?aeia?iaeiiio neiiic?oi? c iaoaiaoe/ii? oa oai?aoe/ii? o?ceee”
(Ee?a, o?aaaiue 1997 ?ieo) [5], a oaeiae ia nai?ia?? OO?IO IAI Oe?a?ie c
a?aiaee/ii? oai??? oa iia?aoi?ieo aeaaa? (Oa?e?a, 1996 – 1998 ?iee,
ea??aiee nai?ia?o – A.ss.Aieiaeaoeue).

Ioae?eaoe??. Ca iaoa??aeaie aeena?oaoe?? iioae?eiaaii noaoo? [1-4].

Ia’?i oa no?oeoo?a aeena?oaoe??. Aeena?oaoe?y neeaaea?oueny c anooio,
oanoe ?icae?e?a, aeniiae?a oa nieneo aeei?enoaieo aeaea?ae, yeee
i?noeoue 53 iaeiaioaaiiy. Ianya aeena?oaoe?? – 116 noi??iie, ianya
nieneo aeei?enoaieo aeaea?ae – 5 noi??iie.

Aaoi? ue?i aeyeo? nai?io iaoeiaiio ea??aieeo Aieiaeoeth Aaeaioeio
sseiae/o ca iinoaiiaee caaea/ oa oaaao aei ?iaioe.

INIIAIEE CI?NO ?IAIOE

A ?icae?e? 1 iaaaaeothoueny iniiai? oaeoe c oai??? aeeiai?/ii? aio?ii??
oa aeiaiaeyoueny aeaye? ?? i?ino? iane?aeee.

, aea M — W*-aeaaa?a, ( — aaoiii?o?ci M, ( — ii?iaeueiee (-?iaa??aioiee
noai.

,

.

A ?icae?e? 2 aaiaeeoueny iiiyooy E-nenoaie, aeiaiaeyoueny aeaye?
caaaeuei? aeanoeaino? aeeiai?/ieo nenoai oa aeaoaeueii ?icaeyaea?oueny
aeaa eeane i?eeeaae?a — aa?ioe??anuee? nenoaie oa aaoiii?o?cie
iaeiiooaoeaieo oi??a.

aeey aoaeue-yei? ne?i/aiiiaei??ii? C*-i?aeaeaaa?e B a M.

, a ye?e ( ? oi/iei ne?aeii, ia? aooe neeueii aneiioioe/ii aaaeaaith.
?o aeiaaaeaiiy aeei?enoiao? eeoa oa, ui

.

O i?ae?icae?e? 2.2 ie aeiaiaeeii, ui oey ca?aei?noue aeey II1-oaeoi??a ?
ia o?eueee aeinoaoiueith, a e iaiao?aeiith oiiaith neeueii?
aneiioioe/ii? aaaeaaino?.

aoaeaii iaceaaoe aa?ioe??anueeith. Oeei nenoaiai i?enay/aiee
i?ae?icae?e 2.3.

Aio?ii?y aa?ioe??anueei? nenoaie aeey aoiia?ii? aeaaa?e N ia/eneaia
Eieiiai?iaei, Eiiii oa Noa?ia?ii. Aeey caaaeueiiai aeiaaeeo ia?ii
ianooiiee ?acoeueoao.

.

Aeey ian aaaeeeaei ?

iaaa?ioe??anueea.

A i?ae?icae?e? 2.4 ie aeiaiaeeii aeinoaoith oiiao aeey E-aeanoeaino?,
yea aeicaiey? io?eiaoe iaio o?ea?aeuei? i?eeeaaee, i?ae aa?ioe??anuee?
nenoaie.

— W*-aeeiai?/ia nenoaia. I?eionoeii, ui noai ( oi/iee oa ?nio? oaea
W*-i?aeaeaaa?a M0 a M, ui

;

;

neaai u?eueia a M.

? aio?ii?eiith E-nenoaiith.

neaai u?eueia a M” cai?noue (iii) a oai?ai? ia aeinoaoiuei aeey
E-aeanoeaino?: oaea nenoaia iiaea iaa?oue iaoe ioeueiao aio?ii?th.

Aeiaaaeaiiy oai?aie 2.4.1 nie?a?oueny ia ianooiio eaio.

,

.

a N ia?ii

,

.

Iai a?aeiiee eeoa iaeei eean iaeiiooaoeaieo nenoai, aeey yeiai
aeiaaaeaiiy E-aeanoeaino? ia nie?a?oueny ia oai?aio 2.4.1. Oea aeaye?
aaoiii?o?cie iaeiiooaoeaieo oi??a. ?i i?enay/aiee i?ae?icae?e 2.5.

.

).

? aio?ii?eiith E-nenoaiith.

Oeae ?acoeueoao aoa coi?ioeueiaaiee a iaei?e ?iaio? Ia?ioioa?, aea eiai
aeiaaaeaiiy aoei iaa??iei. A?eueo caaaeueiei ?acoeueoaoii ?

Oai?aia 2.5.2. Iaoae G — aeia?eueia aeene?aoia aaaeaaa a?oia, T —
aia??iaee/iee aaoiii?o?ci G, caa??aath/ee (. I?eionoeii, ui

.

? aio?ii?eiith E-nenoaiith.

sse aeiaaea Ia?ioioa?, a oiiaao oai?aie 2.5.1 ie ia?ii

.

Oiio ae?enii oai?aia 2.5.1 aeoiaeeoue c oai?aie 2.5.2.

oai?aia 2.5.2 aea? eeane/iee ?acoeueoao ?ioe?ia, yeee noaa?aeaeo?, ui
a?aiaee/iee aaoiii?o?ci eiiiaeoii? aaaeaai? a?oie ? E-aaoiii?o?ciii.

?icae?e 3 i?enay/aiee iiaoaeia? iaoiaeii A?ae? a?an?anueeeo noai?a ia
aeayeeo AF-aeaaa?ao oa aeine?aeaeaiith aio?ii?eieo aeanoeainoae oe?o
noai?a.

. Ie i?eionea?ii, ui a??i?

I?eiouaiiy 3.1.1.

eiiooothoue.

.

.

E??i iaeiii??ii? eaaioiai? ni?iiai? a?aoee ie ia?ii ianooii? i?eeeaaee.

.

Aaoiii?o?ci (2 c oeueiai i?eeeaaeo ? /anoeiiei aeiaaeeii eaiii?/iiai
cnoao ia aaoi? a?aeiinieo eiiooaio?a, ? oai??y i?aeoaeoi??a aea? iai
aaeeeo e?euee?noue i?eeeaae?a nenoai, caaeia?eueiyth/eo i?eiouaiiyi
3.1.1.

aeey aoaeue-yeiai iaii?iaeiueiai neiaa c si-eo.

, yeui ae?aiao? X a?eueoa, i?ae o?eniaaia r.

. Eieaeueiee noai A?ana caaea?oueny ??ai?noth

.

A i?ae?icae?e? 3.2 ie aeiaiaeeii ianooii? oai?aie.

Oai?aia 3.2.1.

?nio? ? ne?i/aiia.

. A?eueo oiai, ?niothoue C,q>0, oae? ui

.

.

Oai?aia 3.2.2. A?an?anueeee noai ? ?aaiii??ii aeniiiaioe?aeueii
eeanoa?iei, oiaoi ?niothoue C,q>0, oae? ui

.

,

,

,

,

.

. Oeae noai ? (-?iaa??aioiei, oiio iaeiicia/ii i?iaeiaaeo?oueny aei
(-?iaa??aioiiai noaio ia A. Oea ? ? ooeaiee noai A?ana.

. Oea c?iaeaii o i?ae?icae?e? 3.3. A oeueiio ae i?ae?icae?e? ie
iaea?aeo?ii aeaye? ioe?iee aeey aio?ii?? ?icaeyaeoaaieo nenoai.

?icae?e 4 i?enay/aiee iaeiiooaoeaiei ia?e?anueeei nenoaiai.

, aea

,

:

ne?i/aiia,

.

? aio?ii?eieie E-nenoaiaie, a ?o aio?ii?? aei??aiththoue aio?ii??
aeo?aeiiai aaoiii?o?cio Ia?eiaa.

.

? iioi/eiai ?iaa??aioiith aeey Fn.

Aeine?aeaeaiith oeeo noai?a i?enay/ai? i?ae?icae?ee 4.2 oa 4.3. Iniiaiei
?acoeueoaoii i?ae?icae?eo 4.2 ? ianooiia oai?aia.

Oai?aia 4.2.3. Iaoae ( — (-?iaa??aioiee noai ia A. Oiae? ianooii? oiiae
aea?aaeaioi?:

noai ( ? ia?e?anueeei;

, oi/iee noai ( ia N oa oiiaia niiae?aaiiy

,

,

;

.

, aea Z(N) — oeaio? aeaaa?e N.

? oaeoi?ii, a oeaio?ae?caoi? M( ? a?ia?o?i?oiei II1-oaeoi?ii.

Aaciina?aaeiuei c oaeoi?iino? aeoiaeeoue E-aeanoea?noue. Aio?ii?y
ia?e?anueei? nenoaie ia/eneaia Iaooeai. Oaeee aea ?acoeueoao aeoiaeeoue
aeey i?aenenoaie ia oeaio?ae?caoi??. Oaeei /eiii ni?aaaaeeeaa

? aio?ii?eieie E-nenoaiaie oa

.

?icae?e 5 i?enay/aiee aiaietha?anueeei aaoiii?o?ciai aeaaa?e eaiii?/ieo
aioeeiiooaoe?eieo ni?aa?aeiioaiue (CAR).

aioee?i?eia oa

.

, — eaac?a?eueiee noai (A, yeee ? (U-?iaa??aioiei, yeui A oa U
eiiooothoue.

), oi ie io?eia?ii aaeeeo e?euee?noue E-nenoai. A naia, ni?aaaaeeeaa

. Oiae? ianooii? oiiae aea?aaeaioi?:

? aio?ii?eiith E-nenoaiith;

iia?aoi? U ia? aaniethoii iaia?a?aiee niaeo?.

Aeiaaaeaiiy oai?aie iaaiaeeoueny o i?ae?icae?e? 5.2. Aiii nie?a?oueny ia
oaeoi?i?noue eaac?a?eueieo noai?a oa ia ?acoeueoao Aieeoeaneo-Noa?ia?a,
yeee noaa?aeaeo?, ui aio?ii?y aiaietha?anueeiai aaoiii?o?cio,
a?aeiia?aeath/iai oi?oa?iiio iia?aoi?o c neiaoey?iei niaeo?ii,
aei??aith? ioeth.

A i?ae?icae?e? 5.2 ie oaeiae ia/eneth?ii oei oaeoi?a Me. A?aecia/eii
eeoa, ui yeui iaia?a?aia /anoeia niaeo?a A iaii?iaeiy, oi Me ?
a?ia?o?i?oiei oaeoi?ii oeio III1, i?e oeueiio yeui A ia? /enoi
iaia?aaiee niaeo?, oi oeaio?ae?caoi? noaio (A o?ea?aeueiee.

Aio?ii?y aiaietha?anueeiai aaoiii?o?cio a?aeiinii eaac?a?eueiiai noaio
(A, a?aeiia?aeiiai aei iia?aoi?a A c /enoi oi/eiaei niaeo?ii, ia/eneaia
Aieeoeaneo oa Noa?ia?ii. I?e oeueiio oaeoe/ii aiie ?icaeyaeathoue aeaa
aeiaaeee:

niaeo? iia?aoi?a U no?iai neiaoey?iee;

niaeo? iia?aoi?a U aaniethoii iaia?a?aiee, a iia?aoi? A neaey?iee.

A i?ae?icae?e? 5.3 ie aea?ii iiaa aeiaaaeaiiy oi?ioee
Aieeoeaneo-Noa?ia?a o aeiaaeeo (ii). Aiii yaey? ?ioa?an c ianooiieo
i?e/ei: aiii ia ? aeiaaaeaiiyi a?ae noi?ioeaiiai ? iiaea aooe ia?aianaii
ia aiaietha?anuee? ae?? aeia?eueieo aaaeaaeo a?oi aac ne?ooo. Oaeei
/eiii, i?ae?icae?e 5.3 i?enay/aiee aeiaaaeaiith ianooiiiai ?acoeueoaoo.

ia?ii

.

Iniiaith iaoiai i?aeoiaeo ? ianooii? aeaa ?acoeueoaoe.

,

.

ia?ii

.

.

(yeui oeae ?ioaa?ae ne?i/aiiee). Caeeoa?oueny ia/eneeoe aio?ii?th aeey
aeayeeo aeiaaee?a.

?icae?e 6 i?enay/aiee aiaietha?anueeei aaoiii?o?ciai aeaaa?e eaiii?/ieo
eiiooaoe?eieo ni?aa?aeiioaiue (CCR). Ia a?aei?io a?ae CAR, CCR-aeaaa?a
ia ia? i?e?iaeieo ne?i/aiiiaei??ieo i?aeaeaaa?. Oea ?iaeoue neeaaei?oei
ia/eneaiiy aio?ii??. C ?ioiai aieo, aeey io?eiaiiy E-nenoaie a ian iaia?
iaiao?aeiino? ?icaeyaeaoe ia?io /anoeio aeaaa?e. Oea iieaaoo?
?icaeyaeaiiy ieoaiue, iia’ycaiieo c E-aeanoea?noth oa ni?yaeai?noth, ?
aea? ciiao iaea?aeaoe iia? i?eeeaaee.

, oaeeie ui

.

, a aeiaeaoiee iia?aoi? A — eaac?a?eueiee noai (A.

Ia?ii ianooiiee aiaeia oai?aie 5.2.1.

.Oiae? ianooii? oiiae aea?aaeaioi?:

? aio?ii?eiith E-nenoaiith;

iia?aoi? U ia? aaniethoii iaia?a?aiee niaeo?.

Aeiaaaeaiiy oai?aie i?noeoueny o i?ae?icae?e? 6.2, aea ie oaeiae
?icaeyaea?ii aeaye? i?eeeaaee.

I?eeeaae 6.2.4. Aeeiai?ea ?aeaaeueiiai aaco Aica ia? E-aeanoea?noue.

. Aeey eiaeii? iaa?ae’?iii? iaiaaeaii? aei??ii? ooieoe?? K aecia/eii
iia?aoi? AK,

,

? aio?ii?eiith E-nenoaiith. Oae? nenoaie aeathoue aeineoue aaeeeee
caian i?eeeaae?a, ? ie aea/a?ii ?o ia i?ioyc? onueiai ?icae?eo.

I?ae?icae?e 6.3 i?enay/aiee aio?ii??.

. Iaoae

?iceeaaeaiiy a i?yiee ?ioaa?ae. Oiae?

.

ia? /enoi oi/eiaee niaeo?, oi ia? i?noea ??ai?noue.

ia? /enoi oi/eiaee niaeo?, ? aiaeiaii oi?ioee Aieeoeaneo-Noa?ia?a aeey
CCR-aeiaaeea.

Iaeiei c iane?aee?a oai?aie ? oa, ui aio?ii?y aiaietha?anueeiai
aaoiii?o?cio, a?aeiia?aeiiai aei oi?oa?iiai iia?aoi?a c? no?iai
neiaoey?iei niaeo?ii, ioeueiaa a?aeiinii aoaeue-yeiai eaac?a?eueiiai
?iaa??aioiiai noaio. Oeae iane?aeie aeei?enoiao?oueny i?e aeiaaaeaii?
oai?aie 6.2.1.

sseui iia?aoi?e Az iathoue /enoi oi/eiaee niaeo? aeey iaeaea on?o z (ui
? iaiao?aeiith oiiaith ne?i/aiiino? ?ioaa?aeo a oai?ai? 6.3.1), a
iia?aoi? U ia? ce?/aiie?aoiee eaaaa?anueeee niaeo?, oi iaea?aeo?ii
nenoaie c i?eeeaaeo 6.2.5. Oaeei /eiii, ni?aaaaeeeaee

Iane?aeie 6.3.8. Aeey aoaeue-yei? iaiaaeaii? iaa?ae’?iii? aei??ii?
ooieoe?? K ia?ii

I?e oeueiio yeui K ce?/aiicia/ia, oi ia? i?noea ??ai?noue.

?acoeueoaoe iiia?aaei?o i?ae?icae?e?a aeicaieythoue iaea?aeaoe
iani?yaeai? E-nenoaie c iaeiaeiaith aio?ii??th. Oea c?iaeaii a
i?ae?icae?e? 6.4.

.

oa

.

, ? iiia?ii iani?yaeaieie iaaa?ioe??anueeeie aio?ii?eieie E-nenoaiaie c
iaeiaeiaith ne?i/aiiith aio?ii??th.

.

Ie aa/eii, ui iiaeoey?ia no?ooo?a iaeiiooaoeaii? aeaaa?e iiaea aooe
aoaeoeaiei ?iaa??aioii aeeiai?/ii? nenoaie. Oeae oac?n ?ethno?o?oueny
oaeiae aeiaaaeaiiyi ianooiiiai ?acoeueoaoo.

ni?yaeai? oiae? ? o?eueee oiae?, eiee

.

Oey oai?aia ciiao aea? ciiao iaea?aeaoe iani?yaeai? E-nenoaie c
iaeiaeiaith aio?ii??th.

Iane?aeie 6.4.3. Iaoae K — iaioeueiaa iaa?ae’?iia ooieoe?y, yea ia?
aiae?oe/ia i?iaeiaaeaiiy a aeayeee ie?e ae?enii? in?,

.

, ? iiia?ii iani?yaeaieie iaaa?ioe??anueeeie aio?ii?eieie E-nenoaiaie c
iaeiaeiaith ne?i/aiiith aio?ii??th.

AENIIAEE

Iaea?aeaia aeinoaoiy oiiaa aeey aeeiiaiiy E-aeanoeaino? a iaeiiooaoeai?e
aeeiai?/i?e nenoai?.

Aeiaaaeaii, ui E-aeanoea?noue iathoue iaeiiooaoeai? aa?ioe??anuee? oa
ia?e?anuee? cnoae, a oaeiae aeaye? aaoiii?o?cie iaeiiooaoeaieo oi??a.

Oai??th A?ae? eiino?othaaiiy oa ioe?iee oaeaeeino? ?iciaaeo
ei?aeyoe?eieo ooieoe?e a?an?anueeeo noai?a ia iaeiii??i?e eaaioia?e
ni?iia?e a?aooe? iioe?aii ia eean nenoai, yeee aeeth/a? eaiii?/i? cnoae
ia aeaaa?ao Ieiaaio oa aneiioioe/ii aaaeaa? a?ia?i? cnoae. Aeey
io?eiaieo nenoai aeiaaaeaia E-aeanoea?noue oa ioe?iaia aio?ii?y.

Aeiaaaeaii, ui aiaietha?anuee? aaoiii?o?cie CCR-aeaaa?e aai ia?ii?
/anoeie CAR-aeaaa?e c eaac?a?eueieie noaiaie ? E-nenoaiaie oiae? ?
o?eueee oiae?, eiee niaeo? oi?oa?iiai iia?aoi?a, ui aecia/a?
aaoiii?o?ci, ? aaniethoii iaia?a?aiei.

Aeaii iiaa aeiaaaeaiiy oi?ioee Aieeoeaneo-Noa?ia?a aeey aio?ii??
aiaietha?anueeiai aaoiii?o?cio CAR-aeaaa?e, a oaeiae iaea?aeaiee ??
aiaeia aeey CCR-aeaaa?e.

Ia a?ia?o?i?oiiio III1-oaeoi?? ceiino?oeiaaii eiio?iooi iani?yaeaieo
iaaa?ioe??anueeeo E-nenoai c iaeiaeiaith ne?i/aiiith aio?ii??th.

?acoeueoaoe ?iaioe iiaeooue ciaeoe canoinoaaiiy i?e aea/aii?
aaoiii?o?ci?a iia?aoi?ieo aeaaa?, a oaeiae a eaaioia?e noaoenoe/i?e
iaoai?oe? i?e aea/aii? iane?/aiiiaei??ieo eaaioiaeo nenoai.

IOAE?EAOe?? CAeIAOAA*A

CA OAIITH AeENA?OAOe??

[1] Golodets V.Ya., Neshveyev S.V. Non-isomorphic quantum K-systems of
type III1 // Aeiiia?ae? IAI Oe?a?ie. — 1998. — ?10. — N. 12-15.

[2] Iaoaaaa N.A. Eaaioiaua ia?eianeea E-nenoaiu // Iaoaiaoe/aneay
oeceea, aiaeec, aaiiao?ey. — 1998. — ?1/2. — N. 87-94.

[3] Golodets V.Ya., Neshveyev S.V. Non-Bernoullian quantum K-systems //
Communications in Mathematical Physics. — 1998. — V. 195. — P. 213-232.

[4] Golodets V.Ya., Neshveyev S.V. Gibbs states for AF algebras //
Journal of Mathematical Physics. — 1998. — V. 39. — P. 6329-6344.

[5] Golodets V.Ya., Neshveyev S.V. Dynamical entropy and quantum
K-systems // Oe?aeineee oece/aneee aeo?iae. — 1998. — ?6/7. — N.
688-696.

Iaoa??a N.A. Aeeiai?/ia aio?ii?y a iia?aoi?ieo aeaaa?ao oa eaaioia?
nenoaie Eieiiai?iaa. – ?oeiien.

Aeena?oaoe?y ia caeiaoooy iaoeiaiai nooiaiy eaiaeeaeaoa
o?ceei-iaoaiaoe/ieo iaoe ca niaoe?aeuei?noth 01.01.01 – iaoaiaoe/iee
aiae?c. – O?ceei-oaoi?/iee ?inoeooo iecueeeo oaiia?aoo? IAI Oe?a?ie,
Oa?e?a, 1999.

Iaea?aeaii aeinoaoith oiiao aeey aeeiiaiiy E-aeanoeaino? a
iaeiiooaoeai?e aeeiai?/i?e nenoai?. Oai??th A?ae? a?an?anueeeo noai?a ia
iaeiii??i?e eaaioia?e a?aooe? iioe?aii ia eean nenoai, yeee aeeth/a?
eaiii?/i? cnoae ia aeaaa?ao Ieiaaio oa aneiioioe/ii aaaeaa? a?ia?i?
cnoae. Aeiaaaeaii E-aeanoea?noue aeey iaeiiooaoeaieo aa?ioe??anueeeo oa
ia?e?anueeeo cnoa?a, cnoa?a ia AF-aeaaa?ao c a?an?anueeeie noaiaie,
aeayeeo aaoiii?o?ci?a iaeiiooaoeaieo oi??a. Iaea?aeaii iaiao?aeio oa
aeinoaoith oiiao aeeiiaiiy E-aeanoeaino? aeey aiaietha?anueeeo
aaoiii?o?ci?a CCR-aeaaa?e oa ia?ii? /anoeie CAR-aeaaa?e c eaac?a?eueieie
noaiaie. Cai?iiiiiaaii iiaa aeiaaaeaiiy oi?ioee Aieeoeaneo-Noa?ia?a aeey
aio?ii?? aiaietha?anueeiai aaoiii?o?cio CAR-aeaaa?e, a oaeiae aeiaaaeaii
?? aiaeia aeey CCR-aeaaa?e. Ia a?ia?o?i?oiiio III1-oaeoi?? ceiino?oeiaai
eiio?iooi iiia?ii iani?yaeaieo iaaa?ioe??anueeeo E-nenoai c iaeiaeiaith
ne?i/aiiith aio?ii??th.

Eeth/ia? neiaa: iia?aoi?i? aeaaa?e, aeeiai?/ia aio?ii?y, nenoaie
Eieiiai?iaa, iaeiiooaoeaiee oi?, noai A?ana, noai Ia?eiaa, oi?ioea
Aieeoeaneo-Noa?ia?a, a?ia?o?i?oiee oaeoi?.

Iaoaaaa N.A. Aeeiaie/aneay yio?iiey a iia?aoi?iuo aeaaa?ao e eaaioiaua
nenoaiu Eieiiai?iaa. – ?oeiienue.

Aeenna?oaoeey ia nieneaiea o/?iie noaiaie eaiaeeaeaoa
oeceei-iaoaiaoe/aneeo iaoe ii niaoeeaeueiinoe 01.01.01 – iaoaiaoe/aneee
aiaeec. – Oeceei-oaoie/aneee einoeooo ieceeo oaiia?aoo? IAI Oe?aeiu,
Oa?ueeia, 1999.

Iieo/aii ia?auaiea oai?aiu Aaiaooe-Ia?ioioa?, ooaa?aeaeathuaa, /oi
neeueiay aneiioioe/aneay aaaeaainoue aeeiaie/aneie nenoaiu ia
AFD-oaeoi?a aea/ao eeanoa?iia naienoai aeey yio?iieeiuo ooieoeeiiaeia.
Iieo/aii aeinoaoi/iia oneiaea aeey auiieiaiey E-naienoaa a
iaeiiiooaoeaiie aeeiaie/aneie nenoaia, yaeythuaany aiaeiaii
eeanne/aneiai ii?aaeaeaiey E-nenoaiu /a?ac E-?acaeaiey. Aeieacaii
E-naienoai aeey eeanna aaoiii?oeciia ne?o/aiiuo a?oiiiauo C*-aeaaa?,
aeeth/athuaai iaeioi?ua aaoiii?oeciu iaeiiiooaoeaiuo oi?ia, a oaeaea
i?iecaieueiua y?aiaee/aneea aaoiii?oeciu eiiiaeoiuo aaaeaauo a?oii, /oi
aeaao iiaia aeieacaoaeuenoai eeanne/aneiai ?acoeueoaoa ?ioeeia.
Aeieacaii, /oi aa?ioeeeaaneay aeeiaie/aneay nenoaia n eiia/iie yio?iieae
eiaao aeeaaiiaeecoaiue iiaeoey?iue iia?aoi?.

Oai?ey A?aee iino?iaiey e ioeaiee nei?inoe ?aniaaeaiey ei??aeyoeeiiiuo
ooieoeee aeaanianeeo ninoiyiee ia iaeiiia?iie eaaioiaie nieiiaie
?aoaoea, ioaa/athueo iioaioeeaeai n eiia/iie iaeanoueth acaeiiaeaenoaey,
ia?aianaia ia eeann nenoai, aeeth/athueo aneiioioe/anee aaaeaaua
aeia?iua naeaeae e naeaeae aeaaa? Oaiia?ee-Eeaa, a oaeaea eaiiie/aneea
naeaeae ia aeaaa?ao Ieiaaio. Eiaiii, aeieacaii, /oi eieaeueiua ninoiyiey
Aeaana iioi/a/ii noiaeyony e iaeioi?iio ninoiyieth, e yoi i?aaeaeueiia
ninoiyiea yaeyaony ?aaiiia?ii yeniiiaioeeaeueii eeanoa?iui. Aeey
iieo/aiiuo nenoai aeieacaii E-naienoai e ioeaiaia yio?iiey.

Eco/aiu iaeiiiooaoeaiua ia?eianeea nenoaiu ia iaeiiia?iie eaaioiaie
nieiiaie ?aoaoea. A /anoiinoe, aeieacaii, /oi ethaia ia?eianeia
ninoiyiea no?ieony ii iaeioi?iio oneiaiiio iaeeaeaieth ia eiia/iiia?iie
aeaaa?a, yaeythuaiony iaeiiiooaoeaiui aiaeiaii noioanoe/aneie iao?eoeu.
Aeaia oaeaea oa?aeoa?ecaoeey ia?eianeeo ninoiyiee /a?ac naienoaa
iiaeoey?iie a?oiiu, iaiauathuay oio oaeo, /oi aeaanianeay ia?a,
ioaa/athuay iioaioeeaeo acaeiiaeaenoaey aeeaeaeoaai ninaaea, yaeyaony
ia?eianeie. Iieacaii, /oi oioy ia anyeia ia?eianeia ninoiyiea
ii?aaeaeyaony ia?eianeie ia?ie, oeaio?aeecaoi? ethaie ia?eianeie nenoaiu
niaea?aeeo eeanne/aneoth ia?eianeoth iiaenenoaio ni naienoaaie,
aiaeiae/iuie naienoaai ea?oaiianeie iiaeaeaaa?u. Aeieacaii, /oi
oeaio?aeecaoi? ia?eianeiai ninoiyiey anaaaea yaeyaony aeia?oeieoiui
II1-oaeoi?ii. Aeieacaii, /oi iaeiiiooaoeaiua ia?eianeea nenoaiu
yaeythony E-nenoaiaie.

Iieiinoueth iienaiu E-nenoaiu a eeanna aiaiethaianeeo aaoiii?oeciia
aeaaa?u eaiiie/aneeo eiiiooaoeeiiiuo (CCR) e /aoiie /anoe aeaaa?u
eaiiie/aneeo aioeeiiiooaoeeiiiuo (CAR) niioiioaiee n eaacenaiaiaeiuie
ninoiyieyie. Eiaiii, aeieacaii, /oi oaeea nenoaiu yaeythony E-nenoaiaie
a oii e oieueei oii neo/aa, eiaaea oieoa?iue iia?aoi?, ii?aaeaeythuee
aiaiethaianeee aaoiii?oeci, eiaao aaniethoii iai?a?uaiue niaeo?.
I?aaeeiaeaii iiaia aeieacaoaeuenoai oi?ioeu Aieeoeaneo-Noa?ia?a aeey
yio?iiee aiaiethaianeiai aaoiii?oecia CAR-aeaaa?u ioiineoaeueii
eaacenaiaiaeiiai ninoiyiey, ioaa/athuaai iia?aoi?o n /enoi oi/a/iui
niaeo?ii, a oaeaea iieo/ai aa aiaeia aeey CCR-aeaaa?u. Iieo/aia oaeaea
ioeaiea naa?oo aeey yio?iiee a iauai neo/aa. A /anoiinoe, aeieacaii, /oi
aaoiii?oeci, ioaa/athuee iia?aoi?o ni no?iai neiaoey?iui niaeo?ii, eiaao
ioeaaoth yio?iieth ioiineoaeueii ethaiai eaacenaiaiaeiiai ninoiyiey. Ia
aeia?oeieoiii III1-oaeoi?a iino?iai eiioeiooi iiia?ii ianii?yaeaiiuo
E-nenoai n iaeeiaeiaie eiia/iie yio?iieae.

Eeth/aaua neiaa: iia?aoi?iua aeaaa?u, aeeiaie/aneay yio?iiey, nenoaiu
Eieiiai?iaa, iaeiiiooaoeaiue oi?, ninoiyiea Aeaana, ninoiyiea Ia?eiaa,
oi?ioea Aieeoeaneo-Noa?ia?a, aeia?oeieoiue oaeoi?.

Neshveyev S.V. Dynamical entropy in operator algebras and quantum
Kolmogorov systems. – Manuscript.

Thesis for a candidate’s degree by speciality 01.01.01 – mathematical
analysis. – Institute for Low Temperature Physics and Engineering,
Kharkov, 1999.

A sufficient condition for a non-commutative dynamical system to have
the K-property is obtained. The Araki theory of Gibbs states on a
one-dimensional quantum spin lattice is extended to a class of systems
containing canonical shifts on Ocneanu algebras and asymptotically
abelian binary shifts. The K-property for non-commutative Bernoulli and
Markov shifts, shifts of AF-algebras with Gibbs states and some
automorphisms of non-commutative tori is proved. A necessary and
sufficient condition for the K-property of a Bogoliubov automorphism of
the CCR-algebra and the even part of the CCR-algebra with a quasi-free
state is obtained. A new proof of Stormer-Voiculescu’s formula for the
entropy of a Bogoliubov automorphism of the CAR-algebra is suggested and
its analogue for the CCR-algebra is proved. A continuum of pairwise
non-conjugate non-Bernoullian K-systems with the same finite entropy on
the hyperfinite III1-factor is constructed.

Key words: operator algebras, dynamical entropy, Kolmogorov systems,
non-commutative torus, Gibbs state, Markov state, the Stormer-Voiculescu
formula, hyperfinite factor.

A?aeiia?aeaeueiee ca aeione ?.ss. Eoaeaoeue

I?aeienaii aei ae?oeo 3.03.99. Oi?iao iaia?o ? /anoea

a?eooa 60×90/16. O?c.ae?.a?. 1, iae.aeae.a?. 1. Caiiaeaiiy ? 12-99.

Oe?aae 100 i?ei.

?ioai??io OO?IO IAI Oe?a?ie, 310164, Oa?e?a, i?. Eai?ia 47

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