Το Θε‚ρηmα Μοναδικìτηταc των Stone και von...

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Κ Ε Το Θερηα Μοναδικτητα των Stone και von Neumann ιπλωατικ Εργασα Ειδκευση στα Θεωρητικ Μαθηατικ Πανεπιστιο Αθηνν Τα Μαθηατικν Αθνα 2011

Transcript of Το Θε‚ρηmα Μοναδικìτηταc των Stone και von...

  • Stone von Neumann

    2011

  • ii

    L2(R,m) unitary {S }R {Tt}tR, (S f )(x) = eix f (x) (Tt f )(x) = f (x t). Weyl:

    S Tt = eitTtS , , t R,

    Heisenberg -. Stone von Neumann , - SOT- unitary {U}, {Vt}t Hilbert H , Weyl , unitary R : L2(R,m) H , U = RS R Vt = RTtR, , t R.

    , - . von Neumann., unitary , S (n) , n N {+}. von Neumann, - Stone, SOT- unitary . , unitary , - T (n)t , S

    (n) .

    n = 1. .

  • iii

    Abstract

    On the Hilbert space L2(R,m), we define the one parameter unitary groups {S }R and{Tt}tR, where (S f )(x) = eix f (x) and (Tt f )(x) = f (x t). These two groups satisfy theWeyl relations :

    S Tt = eitTtS , , t R,

    which is a form of Heisenbergs Uncertainty Principle of Quantum Mechanics. The theo-rem of Stone-von Neumann states that for every two jointly irreducible SOT-continuousone parameter unitary groups {U}, {Vt}t on a separable Hilbert space H , which satisfythe Weyl relations, there is a unitary operator R : L2(R,m) H , mapping U to S and Vt to Tt.

    In this study we give a proof of the above theorem, using the theory of abelian vonNeumann algebras.In particular, we first prove that after a unitary equivalence, we may assume that U isS (n) , for some n N{+}. The crucial tools here are the multiplicity theory of abelianvon Neumann algebras and Stones theorem, which characterizes the SOT-continuousone parameter unitary groups.The second step is to prove that there is a unitary operator which carries Vt to T (n)t andleaves S (n) fixed. The fact that {U}, {Vt}t act jointly irreducibly on H then forces n = 1.Our main arguments in this step are measure-theoretic.

  • iv

    , Hilbert. , , : . - , Q P, :

    D(Q) = { f L2(R,m) : id f L2(R,m)} (Q f )(x) = (id f )(x) = x f (x)

    D(P) = { f L2(R,m) : f AC(R), f L2(R,m)} 1 (P f )(x) = i f (x)

    D(PQ) D(QP) L2(R,m), Schwarz, f :

    (PQ QP) f = i f

    (canonical commutation relation-CCR).

    P,Q, A, B, Hilbert , unitary U : H L2(R,m), UAU1 = P UBU1 = Q. A, B CCR . , :

    M f : L2[0, 1] L2[0, 1] : g 7 f g, f (x) = x,x [0, 1]

    P1 : D(P1) L2[0, 1] : g 7 ig, D(P1) = { AC[0, 1] : (0) = (1)}

    1AC(R) = { f : R 7 C : f }

  • v

    D(P1M f )D(M f P1), M f , P1 CCR, P1 {2k : k Z}, , , R.

    Stone([11]), 1-1 SOT- unitary . , ([10]) Q, P eiQ = S , (S f )(x) = eix f (x) eitP = Tt, (Tt f )(x) = f (x t), :

    S Tt = eitTtS

    , SOT- unitary {U}, {Vt}t, Weyl :

    UVt = eitVtU , , t R. (1)

    A, B , , A B, CCR.

    SOT- - unitary , Weyl . Johnvon Neumann Marshall Stone 1931 : unitarily -, {U}, {Vt}t , unitary W : H L2(R,m), WUW1 = S WVtW1 = Tt.

    Stone - von Neu-mann , vonNeumann . Lectures on InvariantSubspaces H. Helson. , Banach Fourier .

    . . -

  • vi

    . . . - , , , , .

  • 1 1

    1.1 B(H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    2 7

    2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 . . . . . . . . . . . . . . . . . . . . 16

    3 von Neumann 21

    3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 von Neumann . . . . . . . . . . . . . . . . . . . . . 223.3 . . . . . . . . . . . . . . . . . . . . . 253.4 von Neumann . . . . . . . . . . . . . . . . . . . . . . . 27

    4 37

    4.1 . . . . . . . . . . . . . . . . . . . . 37

    5 Weyl 47

    5.1 Stone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.2 Stone-Von Neumann . . . . . . . . . . . . . . 52

    vii

  • 1

    1.1 B(H)

    1.1. H Hilbert . x H , px : B(H) R : T 7 T x

    B(H). , {px : x H} - B(H), , (Strong Operator Topology, SOT). SOT- T0 B(H), :

    V(T0 : x1, . . . , xn; ) = {T B(H) : (T T0)x j < , ( j = 1, . . . ,m)}, x1, . . . , xn H > 0. (Ti)iI T B(H) SOT- Tix T x 0, x H .

    Hilbert H . H , :

    P Q P(H) Q(H), P,Q B(H).

    1.2. {Ni : i I} Hilbert H Ni , - Ni, Ni , Ni.

    Ni =iI

    Ni, Ni

    1

  • 2 1.

    iI

    Ni. - {Pi = pro j(Ni), i I} supremum Pi = pro j(Ni), in f imum Pi = pro j(Ni).

    1.1. (E)A Hilbert H E =

    A

    E. E =SOT-lim

    E

    . (E(H)) , A

    E(H)

    H , E(H) = A

    E(H). x H > 0. Ex E(H), y E(H), A, Ex y < . b , E Eb E, y E(H) Eb(H) E(H). :

    Ex Ebx = E(Ex y) Eb(Ex y) E Eb Ex y < .

    1.1. (E)A Hilbert H E =

    A

    E. A

    E = I A

    E E =SOT-lim

    E

    1.2. (E)A , Hilbert H E =

    AE. x H Ex =

    Ex.

    . , . .

    F = {F : F A}. F F GF =

    F

    E =F

    E. ({GF : F F },) ,

    FF

    GF =F

    E = E.

    : Ex =

    FFGF x =

    Ex, .

    1.3. HilbertH . (weak opera-tor topology,WOT) :

    x,y : B(H) C : T 7 T x, y. x,y(T ) = 0, x, y H , T = 0, {x,y : x, y H} B(H). WOT- .

  • 1.1. B(H) 3

    :V(T0 : x1, . . . , xn, y1, . . . , yn; ) = {T B(H) : |(T T0)x j, y j| < , ( j = 1, . . . , n)},

    x1, . . . , xn, y1, . . . , yn H > 0, T0 B(H), WOT-. , (Ti)iI Tix, y T x, y,x, y H .

    1.1. : B(H) C WOT- SOT-.

    . (i) SOT- WOT, WOT-, SOT-.

    (ii) SOT-. SOT- 0,

    W = {T B(H) :n

    k=1T xk2 < 2},

    |(T )| < 1, T W . yk = 2 xk p(T ) =( n

    k=1Tyk2

    )1/2,

    W = {T B(H) : p(T ) < 2}.

    : |(T )| ( nk=1Tyk2

    )1/2, T B(H).

    p(T ) = 0, mT W, |(mT )| < 1 m N, |(T )| = 0 . p(T ) , 0, Tp(T ) W, |( Tp(T ) )| < 1, |(T )| < p(T ), .

    K = {(Ty1,Ty2, . . . ,Tyn) : T B(H)} Hn. (Ty1,Ty2, . . . ,Tyn) = 0, , |(T )| = 0.

    : K C : (Ty1,Ty2, . . . ,Tyn) 7 (T )

    , , Hn. Hn, Riesz ([10]),

  • 4 1.

    (u1, u2, . . . , un) Hn, :

    (Ty1,Ty2, . . . ,Tyn) =(Ty1,Ty2, . . . ,Tyn), (u1, u2, . . . , un)Hn =n

    k=1

    Tyk, ukH =n

    k=1

    yk ,uk(T )

    (Ty1,Ty2, . . . ,Tyn) K,

    (T ) =n

    k=1

    yk ,uk(T )

    T B(H). WOT- WOT- yk ,uk , k = 1, 2, . . . , n.

    1.2. S B(H) SOT- WOT-.

    . WOT- SOT- B(H), S

    S OT S

    WOT.

    , T < SS OT T < S

    WOT.

    T < SS OT

    . Hahn - Banach ([14]), SOT- , R, Re(T ) > Re(S ) < , S S. WOT-, T < S

    WOT

    1.3. (B(H))1 B(H) WOT-.

    . x, y H Dx,y = {z C : |z| x y}.

    x,yH

    Dx,y ,

    : (B(H))1 [(B(H))1]

    x,yHDx,y : T 7 {T x, y : x, y H}

    , . Tycho-noff

    x,yH

    Dx,y ,

    [(B(H))1]

    x,yHDx,y.

    b {b(x, y) : x, y H} [(B(H))1]. x1, x2, y1, y2 H , C > 0

  • 1.1. B(H) 5

    T (B(H))1, :

    | b(x j, yk) T x j, yk| <

    |b(x j, yk) T x j, yk| <

    |b(x1 + x2, yk) T (x1 + x2, yk| <

    |b(x j, y1 + y2) T x j, y1 + y2| <

    j, k {1, 2}. , > 0, :

    |b(x1 + x2, y1) b(x1, y1) b(x2, y1)| < 3

    |b(x1, y1 + y2) b(x1, y1) b(x1, y2)| < 3

    , |b(x, y)| x y b(x, y) Dx,y, (x, y) 7 b(x, y) sesquilinear . T0 (B(H))1, b(x, y) = T0x, y, x, y H . b [(B(H))1], [(B(H))1]

    x,yH

    Dx,y.

  • 6 1.

  • 2

    2.1

    2.1. R {Et : t R} HilbertH , :

    s t Es Et

    SOT- limt

    Et = 0

    SOT- limt+

    Et = I

    t 7 Et SOT- , SOT limtt0

    Et = Et0 .

    {Et : t R} (a, b), Ea = Eb.

    2.1. {Et : t R} , Hilbert H , F R.

    f =n

    k=1

    k(k ,k+1]F F, :R

    f (t)dEto=

    nk=1

    k(Ek+1 Ek)=

    nk=1

    kE((k, k+1]) (2.1)

    7

  • 8 2.

    St(F) F, :

    : (St(F), ) (B(H), ) : f 7R

    f dE (2.2)

    - - . St(F) supremum, .

    . (i) -.

    f , g (St(F)), C.

    , f =n

    k=1

    k(k ,k+1]|F , g =n

    k=1

    k(k ,k+1]|F

    () ( f + g) =R

    ( f + g)dE =n

    k=1

    (k + k)E((k, k+1]) =

    nk=1

    kE((k, k+1]) +n

    k=1

    k E((k, k+1]) =R

    f dE +R

    gdE = ( f ) + (g)

    () ( f g) =R

    ( f g)dE =n

    k=1

    kkE((k, k+1]) =

    =

    nk=1

    kE((k, k+1])n

    i=1

    iE((i, i+1]) =R

    f dER

    gdE = ( f )(g)

    , (a, b] (c, d] = , E((a, b])E((c, d]) = 0.

    () ( f ) =R f dE =

    nk=1

    kE((k, k+1]) = n

    k=1

    kE((k, k+1]) =

    =

    R

    f dE = ( f )

    () ( f ) =R

    f dE =n

    k=1

    kE((k, k+1]) =n

    k=1

    [kE((k, k+1])] =

    =

    nk=1

    kE((k, k+1])

    = [R

    f dE]

    = [( f )]

    (ii) . h H , :

    ( f )h22 =

    nk=1

    kE(k, k+1])h

    2

    2

    ..1

    =

    nk=1

    |k|2E((k, k+1])h22

  • 2.1. 9

    f 2n

    k=1

    E((k, k+1])h)22.= f 2

    n

    k=1

    E(k, k+1])h

    2

    2

    6 f 2h22.

    ( f ) f . |k0 | = max {|k| : k = 1, 2, ..., n} = f . E() F, x ran(E((k0 , k0])),

    ( f )x22 =

    nk=1

    kE(k, k+1])x

    2

    2

    = |k0 |22x22, ( f ) |k0 | = f .

    , ( f ) = f . 0

    - F,

    C0(F) =

    C(F) F ,{

    f : F C : lim|x|

    f (x) = 0}

    2.2. - A Hilbert H - : A B(H).

    2.1. F R. C0(F) .

    . . f C0(F) f 0, g =

    f C0(F) , f = gg. ,

    : 0( f ) = 0(gg) = 0(g)[0(g)

    ]= |0(g)|2 0.

    f C0(F) f 1. 1 f f = 1 | f |2 , (1 | f |2) 0, (I ( f f ))x, x 0, x H . ( f )x2 = ( f )x, ( f )x = ( f f )x, x x2, ( f ) 1.

    F R. :W+(F) =

    {f : F R+ | f , fn C0(F) : fn

    f}

    W(F) W+(F ).1

  • 10 2.

    , W(F) , . f , g W+(F), f g W+(F). , f g . W+(F), fn, gn C0(F), n N, fn

    f gn

    g. fngn C0(F), n N fngn

    f g, f g W+(F).

    2.2. - 0 : C0(F) B(H) - :W(F) B(H) :

    fn, f W+(F), n N, fn

    f ( f ) = SOT- limn

    ( fn) (2.3)

    , .

    . (1) 0 W+(F) f W+(F) fn C0(F), fn

    f . : fn f 0( fn) = fn f {0( fn)} . Banach - Steinhaus ([14]) SOT-lim

    n0( fn)

    . +( f ) = SOT- lim

    n0( fn), f W+(F)

    : + gm C0(F),m N, gm

    f . n N :

    fn gm : F R+ : x 7 min{ fn(x), gm(x)}. fn gm

    m fn f = fn F., > 0 n N, K F fn(x) ,x < K. 0 ( fn gm)(x)

    fn(x) , x < K,m N, |( fn gm)(x) fn(x)| < , x < K,m N fn gm

    fn , Dini lim

    mfn gm = fn .

    , n N limm

    fn gm = fn F, lim

    m0( fn gm) = 0( fn).

    , , Banach - Steinhaus

  • 2.1. 11

    SOT-limm

    0( fn gm), n N :SOT-lim

    m0(gm) SOT- lim

    m0( fn gm) = 0( fn)

    SOT- limm

    0(gm) SOT- limn

    0( fn)

    fn, gm , SOT-lim

    m0(gm) =SOT-lim

    n0( fn)

    .

    f , g W+(F), fn, gn C0(F) fn

    f , gn

    g. fn + gn

    f + g, :+( f +g) = SOT- lim

    n0( fn+gn) = SOT- lim

    n0( fn)+SOT- lim

    n0(gn) = +( f )++(g),

    +( f + g) = +( f ) + +(g). , +(cg) = c+(g), c 0., 2 SOT- :+( f )+(g) = SOT- lim

    n0( fn)SOT- lim

    n0(gn) = SOT- lim

    n[0( fn)0(gn)] =

    = SOT- limn

    0( fngn) = +( f g), +( f g) = +( f )+(g).

    (2) W(F) f : F R W(F). f1, f2 W+(F), f = f1 f2. ( f ) = +( f1) +( f2) , f W(F), : f = Re f + iIm f , Re f , Im f : F R Re f , Im f W+(F). ( f ) = (Re f ) + i(Im f )

    (i) . f1, f2, g1, g2 W+(F), f1 f2 = g1 g2. f1 f2 = g1 g2 f1 + g2 = g1 + f2 +( f1 + g2) = +(g1 + f2) +( f1) + +(g2) = +(g1) + +( f2) +( f1) +( f2) = +(g1) +(g2) ( f1 f2) = (g1 g2).

    (ii) - ..

    2 SOT- , . An, Bn

    SOT 0, An < M, M > 0. x H AnBnx An Bnx MBnx 0.

  • 12 2.

    (iii) . f 0 ( f ) 0. f 0, f W+(F) fn C0(F) 0 fn

    f ., 0 0( fn)

    SOT ( f ).

    (iv) f W(F), 0 f f f 21, 1 : F R : x 7 1. (iii):

    0 ( f )2 = ( f )( f ) ( f f ) f 2(1) = f 2 I = f 2

    (v) fn, f W+(F) fn

    f , ( f ) = SOT- limn

    ( fn).

    n N ( fn,m) C0(F) :fn,m

    fn ( fn) = SOT- limm

    ( fn,m)

    gn = max{f1,n, f2,n, ..., fn,n

    } C0(F) : gn fn gn

    f , n N, n N fn,m

    fn fn,m fn, m N, fn

    f fi fn f , i n. : fi,m fi fn, i n,m N. max

    1in

    {fi,n

    }= gn fn.

    gn gn+1, x F, supn{gn(x)} = f (x).

    x. fn

    f , n1 N | fn1(x) f (x)| < 2 , fn1,m

    fn1 , m1 N | fn1,m1(x) f n1(x)| < 2 ,| fn1,m1(x) f (x)| | fn1,m1(x) f n1(x)| + | fn1(x) f (x)| < 2 +

    2 =

    :

    B n1 m1, gm1 = max1im1{fi,m1

    } fn1,m1 , f (x) gm1(x) f (x)B m1 < n1, gn1 = max1in1

    {fi,n1

    } fn1,n1 fn1,m1 , f (x) gn1(x) f (x)

    n0 = max{n1,m1}, gn0(x) ( f (x) , f (x)]. gn gn(x) ( f (x) , f (x)], n n0. gn

    f , ( f ) = SOT- limn

    0(gn).

  • 2.1. 13

    0(gn) ( fn) ( f ),n N. , , ( f ) = SOT- lim

    n( fn).

    (vi) .

    2.1. 2.1 2.2 {Et : t R} F, -

    : (St(F)), ) (B(H), ) : f 7R

    f dE =n

    k=1kE((k, k+1])

    -0 : C0(F) B(H) : f 7

    R

    f dE

    - :W(F) B(H)

    (2.3). f W(F)

    R

    f dE = ( f ).

    2.3. F R. - C0(F) Hilbert H {Et : t R}, F,

    R

    f dE = ( f ), f C0(F) (2.4)

    . , - - :W(F) B(H), (2.3). e = F e = F = 1W(F) (e) = I. t R et = (,t]F W(F). (et)

    = Et

    :

    1. E2t = [(et)]2 = (e2t ) = (et) = Et

    2. Et = [(et)] = (et ) = (et) = Et

    3. t s, et es 0 (et es) 0 (et) (es) Et Es

    4. t s, eet

    ees (eet)SOT (ees) IEt

    SOT IEs EtSOT Es

  • 14 2.

    5. t +, et

    e (et)SOT (e) Et

    SOT I

    6. t , et

    0 (et)SOT (0) Et

    SOT 0

    (1) - (6) {Et : t R} Hilbert H . , , R < , :

    E E = 0 (, ) F =

    {Et : t R} F., 2.1 -

    :W(F) B(H) : f 7

    Ff (t)dE

    (2.3). : ( f ) = ( f ), f W(F) f = e e. ( f ) = I E. (Bn), Bn = (, + n] F, :

    (Bn) =R

    (,+n]FdE = E((, + n]) = E+n ESOT I E

    , Bn

    f , (Bn)SOT ( f ).

    ( f ) = I E = ( f ) .

    2.1. x H , Fx : R R 7 Etx, x.

    , , limt

    Fx(t) = 0. Lebesgue - Stieltjes x : x((, t]) = Fx(t). f W(F), R

    f dEx, x =R

    f dx.

    f =

    nk=1k(k ,k+1]F F. :

  • 2.1. 15

    R

    f dx =n

    k=1

    k

    R

    (k ,k+1]dx =n

    k=1

    k((k, k+1]) =

    nk=1

    k{(, k+1] (, k]} =n

    k=1

    k[Ek+1 x, x Ek x, x] =

    n

    k=1

    kE((k, k+1])x, x = R

    f dEx, x

    , f W+(F). fn, fn

    f . 2.2 ( fn)

    SOT ( f ) =R

    f dE. , x H , ( fn)x, x ( f )x, x. ( fn)x, x =

    R

    fndx, , ( fn)x, x

    R

    f dx .

    2.2 ( ). A (H). {Et : t R} (), :

    f (A) =(A)

    f (t)dEt, f C((A)) (2.5)

    , A =(A)

    tdEt.

    . A (H) . () R. ,

    : P((A)) (H) : p 7 p(A) -, - (A), -:

    0 : C((A)) (H) : f 7 f (A) (A),

    f (A) =R

    f dE, f C((A)) (2.6)

  • 16 2.

    2.2

    2.3. {E() : B(R)} Hilbert H , - , :

    1. E() = E()

    2. E(1 2) = E(1)E(2)

    3. E() = 0 E(R) = I

    4. x, y H , x,y : 7 E()x, y B(R)

    2.2. (i) (1), (2) E() H . (x, y) 7 xy() sesquilinear,

    4xy() =3

    k=0

    ikx+iky,x+iky(), (4)

    :

    4. x H , x= x,x : 7 E()x, x

    B(R).

    , x Borel (R, | |), .

    (ii) , .

    x H . 1,2 B(R) , :

    {E(1) + E(2)} x, x = E(1)x, x + E(2)x, x = x(1) + x(1) =

    = x((1 2)) = E(1 2)x, x, E(1) + E(2) = E(1 2)

    {n : n N} B(R), E(n) , 0, n N

    n+1k

    E(k) nk

    E(k) = E(n+1) = 1,

    Cauchy, n

    E(n) .(iii)

    n

    E(n) SOT-.

  • 2.2. 17

    =

    n=1n. Vn = { : n}.

    (ii) E() = E(Vn) + E( \ Vn) =n

    k=1E(k) + E( \ Vn).

    limn+

    E( \ Vn)x = 0. :E( \ Vn)x2 = E( \ Vn)x, E( \ Vn)x = E( \ Vn)x, x = x( \ Vn) 0,

    x - .

    2.4. Hilbert H E((, t]) = Et, t R, -.

    . {Et : t R} H . 3 P(R), E() =

    {n

    E((an, bn]) :

    n=1(an, bn]

    }: {E() : B(R)} H .

    1. 1.2 E() E() = E()

    2. tn t, 0 E() E(t, tn]SOT 0, E() = 0.

    I E(R) E((, n]), n N, E((, n]) = EnS OT I,

    n +. E(R) = I.

    3. x H ,

    x : P(R) R+ : 7 E()x, x (2.7)

    x ., x() = E()x, x = 0, 1 2 E(1) E(2) E(1)x, x E(2)x, x x(1) x(2) {n : n N} Borel R. 1.1 n N {(an,i, bn,i] : i N}, n

    (iE(an,i, bn,i]) x E(n)x 2n .

    {(an,i, bn,i] : n N, i N

    } ,

    n=1

    n n,i

    (an,i, bn,i]. , (n,iE(an,i, bn,i]) x

    n=1E(n)x

    .3 P(R) R, R

  • 18 2.

    , {(an,i, bn,i] : n N, i N

    } ,

    Jm.

    Jm \m1k=1

    Jk ,

    -. , R, {In}n,

    n=1

    n

    n=1In n,i E(an,i, bn,i]x

    n=1E(n)x

    = n=1

    E(In)x

    n=1E(n)x .

    x H :

    E(

    n=1n)x, x

    n=1

    Inx, x

    n=1E(n)x, x + x(

    n=1

    n)

    n=1x(n).

    x . - , x - Borel R , x.

    4. E(1)E(2) = E(1

    2),1,2 B(R). E((a, b])E((c, d]) = E((a, b]

    (c, d]).

    a c :

    E((a, b])E((c, d]) = (Eb Ea)(Ed Ec) = EbEd EbEc EaEd + EaEc =

    =Emin{b,d} Emin{b,c} Ea + Ea = Emin{b,d} Emin{b,c} = E((a, b] (c, d])

    D = {D B(R) : E(D)E((a, b]) = E(D (a, b]), a, b R} Dynkin.

    R D, E(R)E((a, b]) = E() = E(R(a, b]) A, B D, A B, B \ A D. ,

    E(B \ A)E((a, b]) = E(B)E((a, b]) E(A)E((a, b]) == E(B

    (a, b]) E(A (a, b]) = E((B \ A) (a, b])

    {Bn} D, B =

    n=1Bn D.

    , x H , :

  • 2.2. 19

    E(Bn)x, x = x(Bn) x(B) = E(B)x, x E(Bn) = E(B). :

    E(B)E((a, b])x = limn+

    E(Bn)E((a, b])x = limn+

    E(Bn (a, b])x = E(B (a, b])x,

    E(B)E((a, b]) = E(B

    (a, b])

    D Dynkin - Q = {(a, b], a, b R}, Dynkin (Q). , Q , (Q) - Q, B(R).

    D = { B(R) : E(D)E() = E(D ), D B(R)} Dynkin, B(R)., E(1)E(2) = E(1

    2),1,2 B(R),

    .

    5. E1(), E2() H ,

    Et = E1((, t]) = E2((, t]), t R. x H , 1x, 2x , Lebesgue - Stieltjes Fx : R R : t 7 Etx, x. Borel R, 1x() =

    2x() E1()x, x = E2()x, x, x H .

    E1() = E2(), B(R). , E1(), E2() .

    {E() : B(R)} . t R Et = E((, t]). {Et : t R} H .

    s t, Es = Et + E((t, s]) Et.

    x H . t R {tn} R lim

    n+tn = t. Etn x Etx2 = (Etn Et)x, x = x((t, tn]) 0,

  • 20 2.

    t 7 x((, t]) . t 7 Et SOT- .

    SOT- limt

    Et = 0 SOT- limt+

    Et = I

    2.3. {Et}tR F W(F). 2.2, () = ()2 = (), () . E() =

    {n

    E((an, bn]) :

    n=1(an, bn]

    } Borel , () = E().

    , x H , : x(B) = E(B)x, x, 2x, t 7 Etx, x. , 2.1 :

    E()x, x = x() = x() =

    dx =

    dEx, x = ()x, x

    x H . , .

  • 3

    von Neumann

    3.1

    3.1. Hilbert H S B(H). S :

    S = {T B(H) : TS = S T,S S} (3.1)

    S SOT-. , (Ai)iI Ai S, lim

    iAix Ax = 0,x H , S S :

    S (Ax) = S (limi

    Aix) = limi

    S Aix = limi

    Ai(S x) = A(S x). S norm- .

    3.2. Hilbert H . von NeumannM H B(H), M =M.

    von Neumann C- , SOT-. S B(H), (SS) von Neumann, S, von Neumann S.

    3.3. A B(H). E H - A(E) E. , E - -. , -, AP = PAP P = pro j(E).

    21

  • 22 3. VON NEUMANN

    , - A-. , AP = PA, P {A}.

    S B(H), S- S (E) E,S S. , S S-, S, S S.

    3.2 von Neumann

    3.1. A von Neumann Hilbert H ,E A K = E(H).

    AE = {EA|K : A A} A|K = {A|K : A A} von Neumann , K.

    . AE, A|K - B(K) , A A, A A, x K :

    (EAE)(EAE)x EA=AE= EAAEx = EAAEx = (EAE)(EAE)x EA|KA|K = A|KEA|K .

    AE (A|K) A|K (AE).

    .

    (i) T (A|K). S =[ T 0

    0 0] B(H). S A,

    X A, :

    ES X =

    T 00 0X|K 00 X|K

    =T X|K 00 0

    =X|KT 00 0

    =X|K 00 X|K

    T 00 0

    = XS A = A, S A, T = ES |K AE

    (ii) (AE) A|K . (AE) von Neumann, unitary 1.

    1 A = A1 + iA2, A1 = A+A

    2 , A2 = iAA

    2 . A 1. I A2 , U = A + i

    1 A2. U unitary, A = U+U2 . ,

    von Neumann {A} norm-, U .

  • 3.2. VON NEUMANN 23

    unitary T (AE), T A|K . S A, S |K = T .

    H0 ={

    ni=1

    Aixi : Ai A, xi K, n N},

    H1 = H0 A- . S :

    x =n

    i=1Aixi H0, S

    (n

    i=1Aixi

    )=

    ni=1

    AiT xi,

    S (H1 ) = 0. ,

    ni=1

    Aixi =m

    i=1Biyi, Ai, Bi A, xi, yi K,

    ni=1

    AiT xi =m

    i=1BiT yi

    ni=1

    Aixi = 0,

    n

    i=1AiT xi = 0. :

    ni=1

    AiT xi2 = n

    i=1

    AiT xi,n

    j=1

    A jT x j

    =

    i, j

    AiT xi, A jT x jxi=Exi

    =

    T E=ET =

    i, j

    AiET xi, A jET x j =

    i, j

    (EAjAiE)T xi,T x jT (AE)

    =

    i, j

    T (EAjAiE)xi,T x jT unitary

    =

    i, j

    (EAjAiE)xi, x j = n

    i=1

    Aixi2

    n

    i=1Aixi = 0

    ni=1

    Aixi = 0 n

    i=1AiT xi

    = 0 ni=1

    AiT xi = 0,

    S .

    ni=1

    AiT xi = n

    i=1Aixi

    S |H0 , H1 S (H1 ) = 0, S

    , . , x K S x = T x, S |K = T . S A. A A. S (H1 ) = 0 H1 A-, x H0

  • 24 3. VON NEUMANN

    S Ax = AS x. x =n

    i=1Aixi H0.

    S Ax = S A

    ni=1

    Aixi

    = S ni=1

    AAixi

    = ni=1

    AAiT xi

    =A

    ni=1

    AiT xi

    = AS ni=1

    Aixi

    = AS x .

    3.2. {Ha} Hilbert. {Aa} von Neumann Aa Ha, A = aAa = {aAa : Aa A, supa{Aa} < +} von Neu-mann, Hilbert H = aHa = {x = (xa)a :

    axa < +}.

    . A - B(H). S < A, S < A. S = aS a B(H), S < A. S a0 < Aa0 , Aa0 von Neumann, S a0 < Aa0 . Ta0 Aa0 , S a0Ta0 , Ta0S a0 .

    T ={

    Ta0 , a = a00 ,

    , T A, TS , S T ,

    S < A.

    3.3. Hilbert H , n N A von Neumann , H . A A, A(n) = (Ai j)nn,2 Hilbert Hn =

    ni=1H . A(n) =

    {A(n) : A A

    }

    von Neumann, (A(n)) = (A)(n).

    . A(n) - B(Hn). , T B(H) S = (S i j) B(Hn), :

    T (n)S = (TS i j) S T (n) = (S i jT )

    2 Kronecker i j ={

    1 , i= j0 ,

  • 3.3. 25

    T (n) S T S i j. (A(n)) = Mn(A) (A)(n) Mn(A). . S = (S i j) Mn(A).

    Ei j ={

    IH , i j0 , A A, S AEi j = AEi jS ,

    S kiA jl = AkiS jl k, l, S i j = 0 i , j, S iiA = AS j j, i, j, S (A)(n).

    3.3

    3.1. Hilbert H A - B(H). A SOT- A

    . 1. T A, x H . Ax, A- H . P = pro j(Ax), P A, PT x = T Px. I A, x Ax. T x Ax, (An)nN A, T x = lim

    nAnx.

    2. k : B(H) B(H k) : A 7 A(n) - 3.3 k(T ) (k(A)), k N. - (1), x = (x1, . . . , xk) H k, (Am)mN A, k(T )x = lim

    kk(Am), T x j = lim

    mAm(x j), j = 1, . . . , k.

    3. : T AS OT

    W = {A B(H) : Axi T xi < , i = 1, . . . , k0} SOT- . (2), (Am)mN A, k0(T ) = limk

    k0(Am) T x j = limm Am(x j), j = 1, . . . , k0. N N AN(x j) T x j < , j = 1, . . . , k0, An W, W A , .

    3.1 ( ). A B(H) , A = AS OT = AWOT . , :

    () A von Neumann

  • 26 3. VON NEUMANN

    () A SOT-

    () A WOT-

    . 3.1 1.2.

    3.4. von Neumann norm .

    . A von Neumann. A M, A = A1 + iA2, A1, A2 M . M norm . . A [{E() : B((A))]. E() M, B((A)). T M. , f (A), f C((A))., x, y B(H) :

    f (A)T x, y = T f (A)x, y f (A)T x, y = f (A)x,T y

    fT x,y =

    fx,T y, f C((A)).

    T x,y, x,T y , T x,y =

    x,T y E()T x, y = E()x,T y

    E()T x, y = T E()x, y, (A),x, y B(H).

    E()T = T E(), B(H).

    3.1. A B(H) {Et : t R} H , A =

    R

    tdEt. - {A} = {Et : t R}.

    3.5. M von Neumann, HilbertH , E M , M von Neumann, E =M.

  • 3.4. VON NEUMANN 27

    . 1.3 B(H) WOT-, {An : n N} WOT- - M 3. , An norm M. , m N En,m = {En,m1 , . . . , E

    n,mkn,m} M

    1, . . . , k An kn,mi=1iEn,mi < 1m An < En,m >

    .

    E = nEn, En =

    mEn,m.

    E M M von Neumann E M. . T E T En, n N, T An = AnT, n N. {An : n N}

    WOT= (B(H))1 M, M, -

    M, E M, M =M E.

    3.4 von Neumann

    3.6. von Neumann M, - Hilbert H . A M, {A} = M. , 0 A I

    . 3.5, E = {En : n N} M, E = M. B =

    n

    14n (2En I) =

    n

    14n S n,

    S n = 2En I. B M : 13 I n

    14n (I) B

    n

    14n I

    13 I.

    T B(H) B, En. B E1, ...En1, :

    4n(B n1k=1

    14k

    S k) = 4n(+k=n

    14k

    S k) =+k=n

    4n

    4kS k = S n +

    k=1

    14k

    S n+k = S n + B1,

    3 H . = {, : , } , , = {n : n N}.

    d : ((B(H))1, (B(H))1) R+ : (T, S ) 7

    n=1

    12n|n(TS )|n

    . WOT- (B(H))1 , , .

  • 28 3. VON NEUMANN

    B1 =

    k=1

    14k S n+k

    T

    12

    [3(S n + B1) (S n + B1)3] =32

    S n +32

    B1 12

    S 3n 32

    S 2nB1 32

    S n B21 12

    B31S 2n=I=

    = (32 1

    2)S n + (

    32 3

    2)B1

    32

    S nB21 12

    B31 = S n + B2,

    B2 = 32S nB21 12 B

    31

    B2 :

    B2 = 3

    2S nB21

    12

    B31 3

    2S n B21

    12B31

    32B12

    12B13

    59B1,

    S n = 1 B1 13 . B2 B1, T - 12 [3(S n + B2) (S n + B2)3] = S n + B3, B3 =

    32S nB

    22 12 B32,

    B3 59B2 (59 )

    2B1. Bk Bk ( 59 )k1B1, T S n + Bk, k N. T lim

    k+(S n + Bk) = S n, En.

    , T E1 = ET1 TS 1 = S 1T .

    4B = 4+n=1

    14n

    S n =+n=1

    14n1

    S n = S 1 ++n=1

    14n

    S n+1 = S 1 + A1,

    A1 =+n=1

    14n

    S n+1.

    Ak Ak ( 59 )k1A1, T S 1 + Ak, k N., T E1. A = 32 (B +

    13 I),

    0 A I, {A} = {B}. , A .

    3.1. M von Neumann, Hilbert H A B(H) {A} = M. A =

    RdE, 3.4

  • 3.4. VON NEUMANN 29

    M = {E() : B(R}. P = E(0) , x P(H) :

    A|P(H)x = AE(0)x = E(0)AE(0)x = E(0)(R

    dE

    )E(0)x =

    R

    d(E(0)EE(0))x =R

    d(F)x, F = E|P(H)

    {F = R P(H). M|P(H) = {E|P(H) : B(R)}

    = {A|P(H)}.

    3.4. von Neumann Hilbert H (maximal abelian selfadjoint algebra) masa - H . Zorn vonNeumann Hilbert H masa. , von NeumannM M M, masa M =M.

    3.7. (X, ) - , M masa.

    . M , T (M), g L(X, ), T = Mg.

    (i) , (X) < , 1 L2(X, ). : T (M) T = Mg, g = T1. f L(X, ), :

    T f = T ( f1) = T M f1 = M f T1 = M f g = f g = g f .

    L(X, ) L2(X, ), T Mg L2(X, ). g L(X, ). > 0, U = {t X : |g(t)| > } . : T2U22 TU2 =

    |g|2d

    |U |2d = U22.

    T , T g.

  • 30 3. VON NEUMANN

    (ii) - , X, Xn . (i), T |Xn = Mgn , gn = Tn, Mgn T . g X, g|Xn = gn|Xn , n N g L(X, ), g supn gn T. Mg T L2(Xn, ), , L2(X, ).

    3.5. S B(H). H S, S , 0,S S \ {0}. S, S H .

    3.2. (i) S B(H) H S, S

    (ii) M von Neumann H M, S

    . (i) T S T = 0. S S, T (S ) = S (T) = 0. S = H , T = 0

    (ii) P = pro j(M). M M-, P (M) =M. , I M, P = (I P) = 0, I = P, M. M H .

    3.3. von Neumann M Hil-bert H . M masa .

    . von Neumann M , N masa M N . , N, N -. Zorn = {n : n N},

  • 3.4. VON NEUMANN 31

    . Pn = pro j(Nn). Nn N-, Pn N = N . : =

    n

    12nn N .

    , H - N, T, S N T, S n = ,T S (2nPn) = 0, T S Pn N ., NNn,n N, , - .

    3.6. H ,K Hilbert, S B(H), T B(K) unitarily , unitary U : H K ,

    adU : B(H) B(K) : A 7 UAU

    S T .

    3.4. von Neumann unitarily unitarily .

    . () M B(H), N B(K) von Neumann Uunitary adU : B(H) B(K) : A 7 UAU M N . T M, T M = MT,M M. (UTU)(UMU) = UT MU = UMTU = UMUUTU,UMU N . adU(M) = N , UTU N ., S N US U M, adU M N . M,N unitarily .

    () (M) = M = M, , () .

    3.2. von Neumann unitarily masa, masa.

  • 32 3. VON NEUMANN

    3.2. adU -. , C- unitarily , -. . , M masa, - - M(2). M(2) M2(M), M(2) masa . M,M(2) unitarily .

    3.3. E() . H von Neumann M = {E()}, , () = E(), B(R), E(), . , B(H), () = 0. :

    0 = () = E(), = E()2 E() = 0 E() = 0,

    , , M, , L(X, ) = L(X, ), X R. :

    f 0 L(X, ) ({x X : f (x) , 0}) = 0

    ({x X : f (x) , 0}) = 0 f 0 L(X, )

    3.8. {Et : t R} F Hilbert H Borel , {Et : t R}. 2.1 -

    :W(F) B(H) : f 7R

    f dE

    (F,B(F), ) , - f : F C g W(F), g = f -...M = {Et : t R}, :

    : L(F, )M : f 7 (g) :

    (i) -

    (ii) H M. unitary :

    U :M 7 L2(F, ), U( f )|MU = M f , f L(F, )

    (iii) - C-.

  • 3.4. VON NEUMANN 33

    . (i) ,- .

    . B(F). 1 B(F), (41) = 0 1 W(F), () = E(1). (41) = 0, E() = E(1), () = E(). E() M, B(F) M norm- , (L(F, )) M., f , g W(F), f = g -.., ( f ) = (g) ( f ) = (g).

    - C-.

    .

    f =n

    k=1

    ckk . :

    ( f ) = max1kn{|ck| : E(k) , 0} = max

    1kn{|ck| : (k) , 0} = f .

    , .

    (ii) H0 ={( f ) : f L(F, )

    }.

    H0 H . , -, M = {Et : t R}

    S OT= {((,t]) : t R}

    S OT.

    M M (n)n H0, n M., H0 M. U0 : H0 L2(F, ) : ( f ) 7 f . U0 . f =

    , . :

    f 22 = n

    k=1

    2d = n

    k=1

    ||2() =n

    k=1

    ||2E(), =

    nk=1

    ||2E()2..=

    nk=1

    E()2 = ( f )2

  • 34 3. VON NEUMANN

    U0 U :M L2(F, )., U , :

    U(M) = U(H0)U

    = U0(H0)2

    = { f : f }2 = L2(F, ) f L(F, ), U( f )|MU = M f . f . g L2(F, ). :(U( f )|MU)g = (U( f )|M)(Ug) = (U( f )|M)((g)) = U(( f )(g)) =U(( f g)) = f g = M f g.

    (iii) T M, T ( f ) UT |MU

    M. , M masa, UT |MU

    M, f L(F, ) :UT |MU = M f T |M = UM f U = ( f ) T = ( f )

    (T ( f )) = 0 T = ( f )

    3.4. () P M = {Et : t R}, Borel B(F), P = E(). , P M. 3.8 f L(F, ), E(),

    f dE = P,

    f dE =

    f 2dE =

    f dE., f = f 2 = f -.., f (x) {0, 1}, x. = {x F : f (x) = 1}, f = , P = E().

    () A B(H) {Et : t R} A. , 3.8 A, f (A) = ( f ), f L((A), ).

    3.3. 3.6 vonNeumann M, Hilbert H , A M, {A} = M. {Et : t R}

  • 3.4. VON NEUMANN 35

    A, {Et : t R} = {A}. 3.8, :

    1. M - L((A), ), , A.

    2. H M. unitary :

    U :M 7 L2((A), ), U f (A)|MU = M f , f L((A), )

    3.4. 3.8(ii), M, M = H , unitary

    U : H L2(F, ), U( f )U = M f , f L(F, ), M unitarily M . 3.2, M masa. - , . Radon-Nikodym ([15]) -.. f L1(F, ),

    () =

    f d, B(F).

    W : L2(F, ) L2(F, ) : g 7

    f g , g L2((AR), ), :

    Wg22 =

    f |g|2dxi =|g|2dx1 = g22.

    , W unitary, W : L2(F, ) L2(F, ) : g 7 1

    fg.

    , f L(F, ) : UWM f WU = ( f ). M unitarily M, - , {Et : t R}.

    3.2. M von Neumann, Hilbert H A M, {A} = M. :

    (i) M masa.

    (ii) M .

    (iii) M unitarily M = {M f : f L(((A), )}, - - A.

  • 36 3. VON NEUMANN

    . (i) (ii) 3.3

    (ii) (iii) {Et : t R} A, {Et : t R} =M. 3.4, M unitarily M.

    (iii) (i) 3.7

    : , - .

  • 4

    4.1

    4.1. von NeumannM n, n {+, 1, 2, ...}, P M unitary U : P(H) `2(n) H , M|P(H) masa, :

    adU((A|P(H))(n)) = U(A|P(H))(n)U = A, A M

    M unitarily (M|P(H))(n).

    4.1. () M masa 1.

    () 3.5, adU [(M|P(H))(n)

    ] M,

    [(M|P(H))(n)

    ]= (M|P(H)) B(`2(n)) =

    (M|P(H)) Mn(C) n < +(M|P(H)) B(`2) n = + 4.1. J B(H) F(H).

    . J 1 . J , T J, T x = 0, x H . P 1, y H ,

    37

  • 38 4.

    Pz = (y y)z = z, yy, z H . , S B(H), S (T x) = y. ,

    Pz = z, yy = z, S T xS T x = S T T S z, xx = S T (x x)T S z

    P J, .

    4.2. M von Neumann n, .

    . n < +. M - n, P M unitary U : (P(H))n H , , adU (M|P(H)) Mn(C) M . masa M|P(H). :

    idn :M|P(H) Mn(C) Mn(C) : (Mi j)nn 7 ((Mi j))nn

    Mi j M|P(H). M m. - :

    () m < +, Q M unitary V : (Q(H))m H , adV (M|Q(H)) Mm(C) M

    J = I|Q(H). CJ M|Q(H). A = CJ Mm(C) (M|Q(H)) Mm(C). : Mm(C) A : (i j) 7 (i jJ) -. :

    Mm(C) A adV |A M

    adU (M|P(H)) Mn(C)idn Mn(C)

    : idn adU adV |A : Mm(C) Mn(C) - . ker Mm(C), ker , Mm(C), (1mm) = 1nn. Mm(C) , , ker = 0, n = m.

    () m = +, (), , ker F(H), .

  • 4.1. 39

    4.1. {Et : t R} F, Hilbert H M = {Et : t R}. M n, unitary U : H (L2(F, )) `2(n), {Et}t, U( f )U = (M f )(n), f L(F, ). , M unitarily (M)(n).

    . M n, P M, unitary

    U1 : P(H)`2(n) H , U(( f )|P(H))(n)U = ( f ), f L(F, ). M|P(H) masa, 3.4, unitary

    U2 : P(H) (L2(F, )), U2 M f U2 = ( f )|P(H), f L(F, ). unitary

    U = (U2)(n) U1 : H L2(F, ) `2(n), U( f )U = (M f )(n), f L(F, ).

    4.2. M von Neumann. P M n, n {+, 1, 2, . . . } M|P(H) n.

    4.3. M von Neumann, Hilbert H .

    (i) P M n, P M P P, P n

    (ii) Pa M n, P = Pa - n.

    . (i) von Neumann M n P M, P n. 3.6, A M, {A} = M, , P M, B((A)), P = E(), E() . M n, unitary

  • 40 4.

    U : H L2((A), ) `2(n) U f (A)U = (M f )(n), f L((A), ). U |P(H) : P(H) (M L2((A), )) `2(n). U |P(H) , . , :

    UP(H) = UE()(H) = U(A)(H) = U(A)U(L2((A), ) `2(n)) == (M)

    (n)(L2((A), ) `2(n)) = (M L2((A), )) `2(n), U |P(H) unitary :U |P(H) f (A)|P(H)

    (U |P(H)

    )= (M f )

    (n), f L((A), ), M|P(H) unitarily vonNeumann MM = {M f , f L((A), )}., MM masa. | : B() R+ : B 7 |(B) = (B). J = M L2((A), ) L2(, |) : f 7 f |. J unitary, JMMJ =M| . M| masa, MM masa.

    (ii) P =aA

    Pa, Pa M. F A, -

    aF

    Pa M. F = {F A},

    {

    aFPa : F F

    } , -

    (1.1),

    FF

    {aF

    Pa : F F}

    =aA

    Pa M. M. P M, B((A)), P = E()., k N Pak , Pak P < 1k . Pak P < 1k , k N, P =

    kN

    Pak , M. Pak M, Bk B((A)), Pak = E(Bk). k N, Borel k = Bk \

    k1i=1

    Bi. i j = ,

    i , j,

    k=1k =

    k=1

    Bk = . Pk = E(k) M,

    , x H , Px = Pak x = Pkx =

    k=1

    Pkx.

  • 4.1. 41

    k N Pk Pak , (i) Pk n. 4.1, unitariesUk : Pk(H) L2(k, |k) `2(n), UkM|Pk(H)Uk (M|k )

    (n)

    unitary :

    U :

    k=1

    Pk(H) = P(H)

    k=1

    (L2(k, |k) `2(n)

    ),

    UM|P(H)U =

    k=1

    (M|k

    )(n).

    k=1

    (M|k

    )(n) unitarily

    (M|

    )(n). unitary

    W :

    k=1

    (L2(k, |k) `2(n)

    ) L2(, |) `2(n) :

    k=1

    fk|(n)k 7

    k=1

    fkk

    (n), WM(n)f W =

    k=1

    (M f |k

    )(n), f L(, |),

    .

    4.4. H Hilbert . von Neu-mannH P M , M|P(H) .

    . x1 H M, Q1 - Mx1, Q1 M. x2 H M|Q1 (H), Q2 - M|Q1 (H)x2 =Mx2, Q2 M

    . , xn H Qn = pro j(Mxn), xn+1 M|( n

    i=1Qi

    )(H)

    . n N, Qn = {P M : P Qn},

    M. : I = Q1 Qn Qn+1,n N. P M, P Q1, Px1 = x1 (I P)x1 = 0, x1 M, I = P, Q1 = I.

  • 42 4.

    P M, P Qn, Pxn = xn (I P)xn = 0. xn

    M|(n1i=1

    Qi

    )(H)

    , (I P) n1

    i=1

    Qi

    = 0 Qn+1

    ni=1

    Qi

    n1i=1

    Qi

    , (I P)Qn+1 = 0, PQn+1 = Qn+1 P Qn+1, Qn Qn+1, . :

    (i) n0 N, Qn0 = I > Qn0+1.

    (ii) Qn = I, n 1.

    (i), R = I Qn0+1, R M. MR = M|R(H) AR = A|R(H). xn0+1 M|(n0

    i=1Qi

    )(H)

    ,

    R(

    n0i=1

    Qi

    )xn0+1 = Rxn0+1 = (I Qn0+1)xn0+1 = 0,

    R(

    n0i=1

    Qi

    )= 0, R R

    (n0i=1

    Qi

    ). R(H) =

    n0i=1

    RQi(H)

    : Rxi MR. M M, M|R(H)(Rxi) = 0. :M|R(H)(Rxi) = 0 MRxi = 0 MRQi(xi) = 0 MR|Qi(H)(xi) = 0. xi M|Qi(H), (M|Qi(H)) M|Qi(H). :

    MR|Qi(H) = 0 MRQi = 0 (I MR)Qi = Qi I MR Qi, i {1, ..., n0}. MR = 0 M|R(H) = 0. : RQi(H) =MR(Rxi)() RQix RQi(H). Qix Mxi, (Mn)nN M, Qix lim

    nMnxi. RQix = lim

    nRMnxi = lim

    nMnRxi = lim

    nMn|R(H)Rxi.

    RQi(H) MR(Rxi).() M M. :RQi (M|R(H)(Rxi)) = RQiMRxi = MRQixi = MRxi = M|R(H)(Rxi) RQi(H) MR(Rxi). 3.3, unitary

    Ui : L2((AR), Rxi) RQi(H), : UiM f Ui = f (AR)|RQi(H), f L((AR), Rxi).

  • 4.1. 43

    xi i {1, 2, . . . , n0} , Radon - Nikodym fi =

    dx1dxi L1((AR), xi).

    Wi : L2((AR), x1) L2((AR), xi) : g 7

    fig unitary.

    W = n0

    i=1

    UiWi(U1 )(n0) : RQ1(H)n0 R(H),

    :

    W(f (AR)|RQ1(H)

    )(n0) W = n0i=1

    UiWi (U1 )(n0) ( f (AR)|RQ1(H))(n0) (U1)(n0) n0

    i=1

    Wi Ui

    ==

    n0i=1

    UiWi (U1 ( f (AR)|RQ1(H)U1)(n0) n0

    i=1

    Wi Ui

    = n0i=1

    (UiWiM f Wi U

    i

    )=

    =

    n0i=1

    f (AR)|RQi(H) = f (AR), f L((AR), )

    , MR n0. (ii), Qn = I, n N. xn

    M,n N. 3.2, P M, Pxn = 0, P = 0. P M Pxn = 0. :

    PQn = 0 I P Qn I P Qn = I P = 0., Qn, QiQ j = 0, i , j.

    Zorn {xn : n N}, {yn : n N} M, {Qn : Q

    n = pro j(Myn)}, Q

    nQ

    m, n , m.

    Q =

    n

    Q

    n

    , Q M, Q = {P : P M, P Q}. : Q , I. Q = I. von Neumann M|Qn y Q(H). y M, , P M Py = 0, :

    Py = 0 PQ = 0 I P Q I P Q = I P = 0., QMy = MQy = My, M M. My Q(H) pro j(My)Qn, n N, , {Qn : n N}.

  • 44 4.

    R = I Q, (i),

    W =

    i=1

    Ui (U1 )() : RQ1(H) `2(N) H , W ( f (AR)|R1(H))() W = f (AR),

    f L((AR), )

    4.1 ( ). M von Neumann, Hilbert H , {Et}t - F, {Et : t R} =M.

    () Pn M, 1 n +, , Pn n

    1n

  • 4.1. 45

    Pm

    1n

  • 46 4.

  • 5

    Weyl

    5.1 Stone

    5.1. SOT- unitary , V : R B(H) : t 7 Vt , :

    (i) Vts = VtV1s

    (ii) t R, Vt unitary.

    (iii) SOT-limtt0

    Vt = Vt0 , t0 R

    5.1. {E : R}, Hilbert H . Vt = (ut) =

    R

    eitdE, ut() = eit. {Vt}t SOT- unitary .

    . {Vt : t R} 5.1.

    (i) uts = utu1s ,s, t R Vts = VtV1s , .

    (ii) , Vt ut = 1, t R, Vt . , x H , t R : x = VtVtx Vtx x, . Vt unitary, .

    47

  • 48 5. WEYL

    (iii) x H , :Vtx x22 = (ut 1)(ut 1)x, x =

    |ut() 1|2dx().

    limt0|ut() 1| 0 |ut() 1| 2,

    : R R : 7 2 L1(R, x). , : Vtx x22 0, t 0., Vt , :Vtx Vsx = Vs(Vtsx x) = Vtsx x 0, t s. {Vt : t R} SOT-.

    5.2 (Stone). SOT- unitary {Vt}tR Hilbert H , {F : R} H , Vt =

    R

    eitdF.

    . Banach L1(R,+, , ), f f , f (t) = f (t), -. sesquilinear . f L1(R). , H , : (, ) =

    RVt, f (t)dt.

    |(, )| = R

    Vt, f (t)dt

    R

    |Vt, f (t)|dt

    R

    Vt | f (t)|dt =R

    | f (t)|dt = f 1.

    , sesquilinear , ( f ) B(H), ( f ), = (, ) ( f ) f 1. : - H .

    f , g L1(R), :

    ( f g), =R

    Vt, (R

    f (t s)g(s)ds)

    dt =R

    (R

    Vt, f (t s)g(s)ds)

    dt =

    Fubini=

    R

    (R

    Vt, f (t s)dt)

    g(s)ds t=ts=

    R

    (R

    Vt+s, f (t)dt)

    g(s)ds =

  • 5.1. STONE 49

    =

    R

    (R

    Vt(Vs), f (t)dt)

    g(s)ds =R

    ( f )(Vs), g(s)ds =R

    (Vs), ( f )g(s)ds = (g), ( f ) = ( f )(g),

    -. , f L1(R), , H :

    ( f ), = ( f ), =R

    Vt, f (t)dt =R

    Vt, f (t)dtt=t=

    =

    R

    Vt, f (t)dt = ( f ),

    : f L1(R) ( f ) f . Fourier Gelfand. Fourier F : L1(R) C0(R) 1-1, ,- ( g(t) = g(t),g C0(R)) ([7]). , Stone-Weierstrass ([16]), F (L1(R)) C0(R). F (L1(R)) R, t R, f L1(R), f (t) , 0. , t , s, f L1(R), f (x) = e|x|eitx, :

    f (s) =R

    e|x|+i(ts)xdx = 0 e

    x+i(ts)xdx + +0 e

    x+i(ts)xdx = 1i(ts)+1 +1

    i(ts)1 =2

    (ts)2+1

    : f (t) = 2 , 0. Beurling - Gelfand1

    f L1(R) ([4]), :

    limn f f f 1/n1 = sup {|| : ( f )}

    1/n = sup{|( f )| : M(L1(R))},

    M(L1(R)) L1(R). , L1(R) s : s( f ) = f (s) ([5]). :

    limn f f f 1/n1 = sup{| f (s)| : s R} = f

    f L1(R), f = f , ( f ) , ( f ), = ( f ), = ( f ), , -

    1 A Banach. limnxn1/n1 = sup{|| : (x)}, x A.

  • 50 5. WEYL

    :

    ( f )n = sup {|| : (( f ))}n = sup {|n| : (( f ))} = ( f )n = ( f n)

    f f f 1 ( f ) limn f f f 1/n1 = f .

    f L1(R) f1, f2 L1(R) , f = f1 + i f2.

    ( f ) ( f1) + ( f2) f1 + f2 2 f , . 0( f ) = ( f ), f F (L1(R)). f L1(R), fi = f i , i = 1, 2, f = f1 + i f2, :

    0( f )) ( f1) + ( f2) f1 + f2 2 f , 0 , -, F (L1(R)), C0(R), : C0(R) B(H). 2.3, {E : R}, H , :

    ( f ), =R

    f d, f C0(R)

    ((, ]) = E, . : Vt, =

    R

    eitd(). f L1(R), :R

    Vt, f (t)dt = ( f ), = ( f ), =R

    f d (5.1)

    t0 R, fn(t) = n2en|tt0 |

    R

    fndm = 1, n N. Fourier fn:

    fn() =R

    fn(t)eitdt =R

    n2

    en|tt0 |eitdttt0=x

    =

    =n2

    R

    en|x|ei(x+t0)dx =n2

    eit0R

    en|x|eixdx =

    =n2

    eit0( 0

    e(ni)xdx + +

    0e(n+i)xdx

    )=

    n2

    eit0(

    1n i +

    1n + i

    )=

    =n2

    eit0(n + i + n i

    n2 + 2

    )=

    n2 2ne

    it0

    n2(1 + ( n )2

    ) = eit01 + ( n )2

  • 5.1. STONE 51

    :R

    Vt, fn(t)dt =R

    Vt+t0, fn(t + t0)dt Vt0, , n +

    (5.2)

    g(t) = Vt+t0, , |g(t)| 2. g , > 0, > 0, |g(t)g(0)| < , |t| < . , n0 N,

    |t| fn(t + t0)dt < , n n0.

    |t|fn(t + t0)dt = lim

    M+2 M

    n2

    en|t|dt = limM+

    n M

    en|t|dt =

    limM+

    nent

    nM

    = limM+

    (en enM

    )= en <

    n. : R

    g(t) fn(t + t0)dt g(0) =

    R

    g(t) fn(t + t0)dt R

    g(0) fn(t + t0)dt =

    = R

    (g(t) g(0)) fn(t + t0)dt

    |t||g(t) g(0)| fn(t + t0)dt +

    |t|

  • 52 5. WEYL

    5.2 Stone-Von Neumann

    unitary :

    S : L2(R,m) L2(R,m) : f 7 u f , u(x) = eix (5.5)

    Tt : L2(R,m) L2(R,m) : f 7 Tt f , (Tt f )(x) = f (x t) (5.6)

    SOT-. , > 0 g CC(R), [M,M], M > 0. g , 1 > > 0 |g(x) g(y)| 0 .., T f = Thh f , f L2(R,m). q = Thh , q L(R,m), T f = Mq f .

    5.2. {Vt}tR SOT- unitary -, Hilbert L2(R,m)`2(n), n {+, 1, 2, . . . }, Weyl {S (n) }R. unitary Q (Mm) B(`2(n)), t R Vt = QT (n)t Q.

    . At = VtT (n)t . {S (n) } Weyl {Vt}t {T (n)t }t, :

    AtS (n) = VtT(n)t S

    (n) = e

    itVtS(n) T

    (n)t = e

    iteitS (n) VtT(n)t = S

    (n) VtT

    (n)t = S

    (n) At

    At(S I) = (S I)At, , t R

    As+t = Vs+tT (n)st = VsVtT (n)t T (n)s = VtAsT (n)s = AsT (n)s AtT (n)s As+t = AsT (n)s AtT (n)s

    , `2(n). :, : L2(R) L2(R) C : ( f , g) 7 At( f ), (g ).

    , sesquilinear , :

    |,( f , g)| = |At( f ), (g )| At f g At f g

    At B(L2(R)), At At ,

    At f , g = ,( f , g). t At

    WOT-. , t t, f , g L2(R) :(At A

    t ) f , g = (At At)( f ), (g ) 0

  • 58 5. WEYL

    At(S I) = (S I)At, , t R, , `2(n), :

    At S f , g = At(S f ), (g ) = At(S I)( f ), (g ) =

    = (S I)At( f ), (g ) = At( f ), (S I)(g ) =

    = At( f ), (S g ) = At f , S

    g = S A

    t f , g, f , g L2(R)

    S , At , t R. t L(R,m), Mt = A

    t .

    :

    t = Mt = At At ,

    At unitary , t R.

    x R :

    `2(n) `2(n) C : (, ) = t (x) sesquilinear , :

    |t (x)| sup{|at (x)| : 1, 1} At

    t(x) B(`2(n)), , `2(n) t(x), = t (x). t R , `2(n)

    R C : x 7 t(x), , .

    : t L(R,m), - , : (t, x) 7 t(x), , . (t, x)

    B : R2 C : B(t, x) 7

    [0,x]t(y), dy. t, B x, y 7 t(y), [0, x].

  • 5.2. STONE-VON NEUMANN 59

    t t0, :

    B(t, x) B(t0, x) =

    [0,x]t(y), dy

    [0,x]t0(y), dy =

    =

    [0,x]

    (t(y), t0(y), )dy ==

    R

    (t(y), [0,x](y) t0(y), [0,x](y)) [0,x](y)dy == (t

    t0 )[0,x], [0,x] 0

    AtWOT At0 , t t0. B t.

    f n : R2 C : (t, x) 7 n[B(t, x + 1n ) B(t, x)] , R2. Borel R2 , (t, x) 3 lim

    nf n (t, x).

    f : C : (t, x) 7 lim

    nf n (t, x).

    f ,t : x 7 f (t, x), Lebesgue t R, f t ,

    t .- .

    At , M f t , . As+t = AsT (n)s AtT (n)s t, s R, Nt,s Borel -

    R Lebesgue, x < Nt,s s+t(x) = s(x)t(x s). , x < Nt,s. m N, x [m,m], [m,m] e1 L2(R) `2(n)

    as+t(x)e1 = as+t(x)([m,m] e1)(x) = (As+t([m,m] e1))(x) =

    = (AsT (n)s AtT(n)s ([m,m] e1))(x) = as(x)[T (n)s (AtT (n)s ([m,m] e1))](x) =

    = as(x)at(x s)([m,m] e1)(x) = as(x)at(x s)e1.

    {ei : i N, i n} `2(n). i, j

    3 fn : (X,A, ) R . lim inf fn, lim sup fn X, R .

    = {x X : lim inf fn(x) = lim sup fn(x) , } = {x X : lim fn(x) , } A, f = lim fn : R .

  • 60 5. WEYL

    gi j(x, t, s) = |(s+t(x) s(x)t(x s))e j, ei|. , t, s R Borel Nt,s , gi j(x, t, s) = 0 x < Nt,s,

    R

    gi j(x, t, s)dx = 0, t, s. :

    R2

    R

    gi j(x, t, s)dxd(t, s) = 0Fubini

    R

    R2

    gi j(x, t, s)d(t, s)dx = 0

    R2

    gi j(x, t, s)d(t, s), x < N i j, m(N i j) = 0

    N =i, j

    N i, j

    x 1. {Vt}tR,

  • 5.2. STONE-VON NEUMANN 61

    Vt = WVtW. 5.2, unitary Q (Mm) B(`2(n)), Vt = QT (n)t Q. S

    (n) Q = QS

    (n) , S

    (n) = QS

    (n) Q

    . , {ei : i N i n} `2(n), {Vt}, {S (n) } Q(L2(R,m)e1), ., Hilbert L2(R,m), unitary QW : H L2(R,m).

    5.2. : {U}R, {Vt}tR SOT- unitary Hilbert H , Weyl. n {+, 1, 2, . . . } unitary R : L2(R,m) `2(n) H , U = RS (n) R

    Vt = RT (n)t R, , t R .

  • 62 5. WEYL

  • [1] H. Helson : The Spectral Theorem, Springer-Verlag, New York , (1986).

    [2] H. Helson : Lectures on Invariant Subspaces, Academic Press, New York , (1964).

    [3] K.R. Davidson : Nest Algebras : Triangular Forms for Operator Algebras on Hilbertspace , Longman Scientific & Technical, (1988).

    [4] G. J. Murphy : C-algebras and operator theory, Academic Press, Boston, (1990).

    [5] R.V. Kadison, J.R.Ringrose : Fundamentals of the theory of Operator Algebras,Academic Press, New York, (1983).

    [6] R.Beals : Topics in Operator Theory, University of Chicago Press, Chicago, (1971).

    [7] U.Katznelson : An introduction to harmonic analysis, Cambridge University Press,Cambridge, (2004).

    [8] M. Anoussis, A. Katavolos : Unitary actions on nests and the Weyl relations, Bull.London Math. Soc. 27 (1995), 265-272.

    [9] . : : , (2010).

    [10] . : , , (2008).

    [11] . : : , (2006).

    [12] . : von Neumann - , , , (1997), 147-199.

    63

  • 64

    [13] . : von Neumann nests, (2000).

    [14] . : : 2, (2009).

    [15] . , . : , , ,(2005).

    [16] . , . , . , . : , , , (1997).

  • von Neumann, 21SOT-

    unitary , 47unitary , 31

    , 9

    , 30, 30

    , 7 , 16

    Stone - von Neumann, 60Stone, 48 , 25, 15 , 44

    (masa), 29

    , 21

    , 37

    , 39

    Weyl, 52

    (WOT), 2 (SOT), 1

    , 21

    65

    Eisagwg'hTopolog'ies ston B(H)

    Fasmatik'h Jewr'iaTo Fasmatik'o J'ewrhmaFasmatik'a M'etra kai Anaparast'aseis

    'Algebres von NeumannEisagwg'hParag'omenes 'Algebres von NeumannTo Je'wrhma tou De'uterou Metaj'ethAbelian'es 'Algebres von Neumann

    Fasmatik'h Pollapl'othtaEisagwg'h sthn Jewr'ia Pollapl'othtas

    Oi sq'eseis tou WeylTo Je'wrhma tou StoneTo je'wrhma monadik'othtas Stone-Von Neumann