Re: Toa'n tu+? De Rham Re: Gelfand-Kolmogorov, Najmark & Stone Re:Gelfand-Naimark

[Phung Ho Hai <phung@msri.org>]

On Sun, 17 Oct 1999, Aiviet Nguyen wrote:

> Hi,
> 
> >                                       ^^^^^^^^
> > Trong di.nh ly' Gelfand-Najmark kho^ng co' toa'n
> > tu+? DeRham na`o ca? vi`
> > chi? xe't o+? level topo. Ba'c nha^`m vo+'i geometry
> > (differential
> > geometry) co' le~ lu'c do' mo+'i ca^`n de^'n toa'n
> > tu+? DeRham-
> 
>   Chuye^.n ddo' ra^'t co' the^?. Tuy nhie^n Connes va`
> ca'c NCGeometrists hay refer to+'i Naimark-Gelfand
> trong mo^.t specific context o+? geometry level va`
> la`m mo^.t so^' ngu+o+`i bi. confused. Ddo' ch'nh la`
> starting point cu?a NCG. Ne^'u no'i ro~ ra`ng va`
> ta'ch ba.ch algebra ~ topo, DeRham ~ geometry thi` OK.
> Nhu+ng ho. hay no'i mo^.t ca^u xanh ro+`n:
> non-commutative algebra of operators replaces commutative geometry.
 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
what is it? Ba'c co' quote thi` cu~ng pha?i quote the^' na`o cho nguo+`i
do.c hie^?u no^?i chu+'. Ca^u na`y cha(?ng co' nghia~ gi`  ca?

Ba'c Ai' Vie^.t a., theo to^i ba'c ne^n nghie^n cu+'c ky~ ho+n ve^`
Non-commutative geometry truo+'c khi ba`n lua^.n. De^'n ba^y gio+` thi`
to^i hie^?u ra(`ng kie^'n thu+'c cu?a ba'c ve^` Non-comm geom ra^'t la`
"cuo+~i ngu+a. xem hoa".( Va^.y ma` ba'c vie^'t ma^'y ba`i ve^` Alain
Connes hay dao' de^?!)

Truo+'c tie^n to^i xin no'i ro~ ra(`ng co' 2 di.nh ly' lie^n quan to+'i
Gelfand (trong mo^t message truo+'c chi'nh to^i cu~ng lo^.n). Di.nh ly'
thu+ nha^'t (ma` to^i pha't bie^?u duo+'i da.ng "so+` ga'i") la` cu?a
Gelfand chu+' kho^ng pha?i chung vo+'i Najmark. Di.nh ly' Gelfand-Najmark
mo^ ta? ca'c C^*-da.i so^' di.n nghia~ mo^.t ca'ch tru+` tuo+.ng nhu+ la`
da.i so^' ca'c toa'n tu+? lie^n tu.c tre^n mo^.t kgian Hilbert. 
(Bac Sonnet cu~ng da~ nha^.n xe't die^`u na`y).

Di.nh ly' ma` ca'c N-Comm Geometers hay refer nhu+ la` starting point la`
di.nh ly' Gelfand chu+' kho^ng phai? Gelfand-Najmark. Argument cu?a ho.
nhu+sau:
Theo Gelfand thi` mo^.t kgian topo (compact, Hausdorff) duo.c xa'c di.nh
bo+?i mo^.t (commutative) C^*-da.i so^' ca'c ha`m tre^n no'. Ca'i na`y
duo.c go.i la` do^'i nga^~u Gelfand

Va^.y ne^'u consider mo^.t da.i so^' kho^ng giao hoa'n thi` ta thu duo.c
-qua do^'i nga^~u gelfand (gia? di.nh)-mo^.t non-comm. geometry theory.

Motivation cu?a ca'i nay` duo.c la^'y tu+` va^.t ly. Trong co+ co^?
die^.n, ca'c observation duo.c cho bo+?i ca'c ha`m-vi' du. ca'c phe'p so+`
mo^.t co^ ga'i cho ta ca'c ha`m. Ta^p ho+.p ca'c ha`m na`y la^.p tha`nh
mo^.t dai so giao hoa'n.
Trong co+ luong tu+?, ke^'t qua? ca'c phe'p thu+? duo.c mie^u ta? ba(`ng
ca'c toa'n tu+?-chu'ng kho^ng giao hoa'n.

Logic cu?a o^ng Connes trong quye^?n NCG deep ho+n va` kho^ng the^? gia?i
thi'ch mo^.t ca'ch do+n gia?n duo.c. Hie^?N nhie^n no' lie^n quan de^n
Geometry nhung kho^ng pha?i in the way ma` ba'c Ai' vie^.t hie^?u. For
short, ca'i da.i so^' ma` o^ng Connes considers "lo+'n ho+n"  ca'i dai.
so^' trong do^'i nga^~u o+? tre^n.

ha?i