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Re: Gelfand, NCG
Hi Hai,
Dda^u co' bu+.c mi`nh gi` dda^u. Sai thi` pha?i no'i
tho^i.
--- Phung Ho Hai <phung@msri.org> wrote:
> Any way, vie^.c anh ap dung NCG va`o va^.t ly'
kho^ng co' nghi~a anh
> hie^?u NCG mo^.t ca'ch toa'n ho.c. Thu+.c ra die^`u
nay cha(?ng co' gi`
> la.. Ca'c nha va^.t ly' invented tensor product song
ra^'t nhie^`u nguo+.i
> la`m va^.t ly' ly' thuye^'t kho^ng hie^?u duo.c
kha'i nie^.n tensor
> product trong toa'n ho.c.
Ai va^.y :-))
To^i nghi~ pha^`n "geometry" thi` to^i hie^?u NCG
ve^` ma(.t toa'n ho.c co' le~ kho^ng thua ai ( dda(.c
bie^.t pha^`n noncom Riemamnian geometry thi` ke^? ca?
Connes).
To^i kho^ng quan ta^m la('m to+'i pha^`n measure
theory hay topological spaces trong cuo^'n sa'ch cu?a
Connes. Physicists chi? quan ta^m to+'i ti'nh toa'n
ne^n ne^'u kho^ng differetial calculus thi` kho^ng
co' y' nghi~a thu+.c tie^~n.
Ba?n tha^n NCG cu?a Connes cha^'m du+'t o+? metric
spaces ma` kho^ng ddi tie^'p ve^` Quantum
Groups....ddu? tha^'y boundary cu?a cuo^'n sa'ch
du+`ng la.i o+? "geometry".
> Toi no'i anh "cuoi ngua xem hoa" vi` anh nha^`m
la^~n giu+a hai di.nh ly'
> cua? Gelfand va` Gelfand-Najmark khi noi ve^` NCG.
Tui dda~ no'i ngay tu+` dda^`u la` tui chu+a bao
gio+` ddo.c Gelfand-Naimark chi? ddo.c quote la.i
tho^i : va` co' the^? ho. quote la.i trong mo^.t
context he.p cho differential geometry (tui cu~ng dda~
no'i).
Hi`nh nhu+ pha?i co' co^ng tri`nh chung vo+'i Naimark
thi` mo+'i co' chuye^.n bie^?u die^~n dda.i so^' giao
hoa'n ba(`ng phe'p ti'nh nha^n vo+'i mo^.t so^' phu+'c
tre^n module be^n pha?i.
Connes qua? tha^.t chi? refer to+'i Gelfand
(theory chu+' kho^ng pha?i theorem) trong context
topo, nhu+ng add the^m mo^.t ca^u: ca^`n Fredholm
module dde^? construct geometry.
Anyway, to^i chi? start vo+'i Connes. O^ng na`y
trong NCG co' tri`nh ba`y measure theory va`
topo...nhu+ng hi`nh nhu+ co^'ng hie^'n cu?a o^ng chi?
tu+` pha^`n cyclic cohomology tro+? ddi.
> A` anh co`n ba('t be? hai ca'i thua^.t ngu+~ topo
va` geometry. Topo la`
> gi` ne^'u nhu+ kho^ng pha?i la` hi`nh ho.c. Nghie^n
cu+'u hi`nh ho.c trong
> toa'n kho^ng nha^'t thie^'t la` cu+' pha?i co' mo^.t
toa'n tu+? De Rham.
> Thua^.t ngu+~ geometry anh hie^?u trong toa'n go.i
la` mo^.t differential
> structure.
Ho^m tru+o+'c hi`nh nhu+ co' ai va.ch cho to^i
tha^'y diiference ve^` level topo va` level geometry
no' kha'c nhau ra sao :-))
> Ca'i triple ma` anh no'i de^'n co' the^? hie^?u la`
"mo^.t kg topo (non-comm), mo^.t diff. structure va`
mo^.t phe'p ti'ch pha^n"
> Ca'i da.i so^' A trong ca'i triple a^'y no' pha^n
ba^.c va` tha`nh pha^`n
> ba^.c 0 xa'c di.nh kg. topo (non-comm)-bo+?i do^'i
> nga^~u Gelfand.
Ddo^'i nga^~u Gelfand sao kho^ng tha^'y Connes
du`ng bao gio+`.
No'i cho ddu'ng ra dda.i so^' A na`y kho^ng
pha^n ba^.c
gi` ca?. No' chi'nh la` tha`nh pha^`n ba^.c 0 cu?a
mo^.t dda.i so^' bao
(qua? ti`nh co' pha^n ba^.c) thu+o+`ng ky' hie^.u la`
\Omega (A).
Ma(.t kha'c y' nghi~a cu?a kho^ng gian HIlbert
trong triple kia kho^ng pha?i la` dde^? mang la.i
phe'p ti'nh ti'ch pha^n. No' chi? la` module dde^?
ca'c toa'n tu+? ta'c du.ng le^n ma` tho^i.
Phe'p ti'nh tu+o+ng tu+. nhu+ ti'ch pha^n tre^n dda
ta.p chu+a co' ddi.nh nghi~a na`o kha? di~. Thi' du.
dde^? ddi.nh nghi~a mo^.t ca'i gi` tu+o+ng tu+. nhu+
ti'ch pha^n chuye^?n ddo^.ng hie^.n nay ngu+o+`i ta
co' the^? du`ng Dixmier trace, Wodzicki
residue,...Ga^`n dda^y Connes&Chamsedine
( sau khi dda~ thu+? he^'t nhu+~ng ddi.nh nghi~a kia)
dde^` nghi. ca'i go.i la` Spectral Action. Theo to^i
va^~n chu+a tho?a dda'ng vi` no' gio^'ng nhu+ la`
ti'nh ca'c characteristic class ( du`ng heat
kernel...). Trong va^.t ly' ca'c dda.i lu+o+.ng na`y
thu+o+`ng go.i la` anomaly chu+' kho^ng pha?i Action
integral. Ma` qua? tha^.t ma^'y o^ng na`y ti'nh ra
R^2....va` ca'c action ra^'t qua'i chie^u cho gravity
chu+' kho^ng pha?i Einstein-Hilbert action ddo+n
thua^`n.
Cheers
Aiviet
=====
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