Appendix 1

A.l Elements of Hilbert Spaces

In this Section we explain the fundamental facts about Hilbert spaces to facilitate the understanding of this book.

DEFINITION A.l.l. Let lR (rcsp. C) be a field constst111g of all rea.) numbers (resp. complex n1nnbers), denoted by o·, ;3, .. . , and X be a. real (resp. complex) linear spa('e with elements denoted by :c, y, .... Then, an inner pmd-uct dellot<'d by(·,-) on X x.Y with val uPs in lR (resp. C) is defined by the mapping satisfying

(y, :r)

(x + y,z) (x, ny)

(x,:r) >

(:c,y),

(:c, z) + (y, z), n(:c, y), 0, (x,x)=O<=>x=O.

X with a.n inner product defined in it is called an inner· pmduct space.

Let Jlxll := ...j(x, :r). Then, the Schwar:: inequality is expressed by the following statement:

THEOREM A.1.2.

l(:c, Y)l ~ IJ:rJI·IIyJI, .c, !/ E X

holds, where the left hand side lenn is exactly equal to the 1·ight hand side if and only if there exists a real (resp. complex) constant >. 8alisfying y = >.x or x = 0.

This inequality shows that the functional 11·11 satisfies the following conditions of a non11:

ll:rll > II a:r II

llx+vll <

0, llxll = 0 ~ :r

Jal·llx II, llxll + llvll-

263

0,

264 APPENDIX 1

If an inner product. space .Y is a. Banach swu:c, that is, a. complete uormed space whose norm arises fro111 the inucr product, this space is said to be a Hilbert space H.

THEOREM A.1.3 (Pa.ra.llelogra.m law).

THEOREM A.1.4 (Polar identity).

(x, y) = ~ [llx + Yll 2 - 11-'C- Yll 2 + illx + iyll 2 - illx- iyll 2] , x, Y E 1l.

Let X be a Banach space with the norm 11·11, and the functional~·,·~ on XxX be defined by the following equation:

If the not·m 11·11 satisfies the parallelogram law, it can be proved that~·,·» satisfies the conditions for inner product, a.ud (X,~-,·») has the same metric structure as II illwrt spac:C's.

Example 1: IR" c'quipped with the iuner product defined by

is a real Hilbert space.

n

(:c, y) = L:l:i1Ji· i=I

Example 2: en equipped with the inner product dcfiued by

71

(x, y) = LXiYi· i=l

is a complex Hilbert space.

Example 3: Let f 2 be the linear space consisting of the square summable complex sequences, and (-, ·) be the fuctiona.l on e2 xt2 defined by

00

({:t'n}, {Yn}) = LXn?Jn, {:en}. {Yn} E e2 •

n=l

Then, (f2, (·,·))is a. llilhert. span•.

:\ PPENDIX I 265

Example 4: Let (n, :F, Jt) be a a-finite measure space, L2 (Q) be the linear space coJtsisting of all equivalence classes of square integrable complex-valued functions on n, and (·,-)be the functional on L2 {Q) xL2 (Q) defined by

(x(·),y(·)) =In x(w)y(w)dw, :r,y E L2 (Q).

Then, (L2 (Q, (·,·))is a. llilbert space.

DEFINITION A.l.!). Le11i he' a Hilbert spare with the inner product(·,·), and x and y be elements of 1£. Then x is said to be orthogonal to y, if (x,y) = 0 holds. A subset {.1:j : jEJ} is said to be a11 orthogonal system if it satisfies (xi,Xj) = 0 for any distinct i,j E./. A11 orthogonal system {xj : jEJ} is said to be a.u orlhorwrmal system, in short ONS, if it satisfies llxill = 1 for all i E J. Moreover, a.n orthonormal system {xj; jEJ} is said to be complete, in short CONS, if (y, Xi) = 0 for all j E J implies y = 0.

Zorn's lemma assures that any Hilbert space ha .. .;; a. complete orthonormal system. In particular, a Hilbert space having a countable orthonormal syste111 is said to be sepamblc.

THEOREM A.1.6 (Bessel's i1wqualit.y). If {xj} is rm orthonormal system, then

L j(:r,:rj)l.2 :::; ll:rjj 2 , :1: E 1£, jEJ

wher-e the equality holds £jJ { :r j} is complete.

THEOREM A.1.7 (FouriPr's ('Xpansion formula). If {:rj} ts a complete orthonormal 8yslf:1n. then

.r = 2:) :r, :r j) :r .i , :r E 1£. JE·I

{(x,xj): j E J} is said to be a sd of Fourier cor.fllcicnl.o; of:z: with respect to{xi: iEJ}.

THEOREM A.l.S (Parseval's equality). {xj} is a CONS ij]

(;r,y) = L(:z:,xj)(:rj,y), :r,y E 1{. jEJ

DEFINITION A.1.9. Lc't. [ be a. su bspa.ce of 1{. Tlwn, [is said to be closed if it is a closed subset of 1i. The o1'ihogonal subspace of[, denoted by [l., is defined by {x E 'H.: (:r, y) = 0, y E £}.

266 APPENDIX 1

It is clear that £.1. is also a closed subspace. For any two subspa.ces £ and IC, £ is said to be odhogonal to K if for any x E £ and any y E IC, (x, y) = 0 holds. For such a.n orthogonal pair£ and K, the direct sum .Cff)/C is a closed subspace of 11.. For any subset £ of 11., ,C.l..l. defined by (.C.l. ).1. is the smallest closed subspace including £. Moreover, if£ is a complete orthonormal systern of 11., then £.1. is equal to {0} and £.1..1. is equal to .C.

THEOREM A.l.lO (Least distance theorem). Let£ be a closed subspace. Then, for any x E 11., there uniquely exists a11 clement z of .C satisfying the following eq·uation:

llx- zil = inf{IJ;c- vii: y E £}.

Let Pc. be the mappin~?; on 11. \\'ith va.lut>s in £ defined by Pc.x = z, where x belongs to 11. a.nd z is the unique elenwnt of £ satisfying the above equality. Then, it is known that the mapping Pc. is a bounded linear operator. This operator is usJJa.lly udl<'d a. projcclion. We have the following two important results:

THEOREM A.l.ll (Projection theorem). Let£ be a closed subspace of11.. Then,

THEOREM A.l.12 (Ri<'sz's theorem). For any bounded linear functional f, there uniquely cxisls y E 11. .<;uch that

f(a:) = (y, x), :c E 11.,

and the following equality holds:

sup{l.f(~r)l: :c E 11., llxll = 1} = IIYII·

DEFINITION A.J.J:t l~'or an~' 1.\\'o Hilbert span's 11. and K, 11. is said to be isomorphic t.o K, denol<'d 11.'::::..1\.:, if there <'xists a bounded linear operator U on 11. with valn<'s in A.: sa1isfying

( x, !J) H = ( {/ :r, U y )K, :c, y E Ji.

It is known tha.t if dim 11. = n, then 11. is isomorphic to then-dimensional vector space !Rn or en' a.nd moreover, if 11. is a separable infinite dimensional Hilbert space, then 11. is isomorphic to C2 .

APPENDIX I 267

Let x (resp. y) be an clenwnt of 1i (resp. K) and ~rOy be the conjugate bilinear functional 011 7-l x A.~ ddi ned by

:~:Oy(n, v) = (:r, ·u)H(Y, v)K, u E 7i, v E K.

Let t: denote the linear space consisting of conjugate bilinear functionals constructed by forming all finite linear combinations of the elements of { x®y : x E 7i, y E K}. It is clear that the sesquilinear functional (from Latin sesqui= 'one and a half') (-, ·) defined by

(x0y, nOv)= (x, u)H(Y, v)JC, x, u E 7-l, y, v E K

satisfies the conditions for the inner product. Now, the tensor product Hilbert space, denoted by 7iOK, is defined as the completion oft: in the norm constructed by the inner product stated above. It is known that if {xj} (resp. {yk}) is a complete orthonorrnal system of 7-l (rcsp. K), then {xj®yk} is also a complete orthonormal system of 1iGK.

Let (X, It) and (Y, v) be two cr-finite measure spaces, 1i be a separable Hilbert space and t:(X, 11; 7i) denote the linear space consisting of the Hilbert space valued functions constructed by taking finite linear combinations of the clements of {f(·)x: J(·) E L2 (.Y,p), x E 7i}. It is clear that the scsqui-linca.r functional«·,·» defin('(l by

nL "'L

«L J(·):rj. L ,r;(·):tfk» j=I k=l

satisfies the conditions for the inner product. Now, the llilbcrt space consisting of all square surnmable Hilbert space valued functions, denoted by L 2(X, J.li 11.), is dcfiucd by the completio11 of t:(X, flj 7-l) in the norm constructed by the inner product «·, ·».

THEOREM A.l.l4. U(.Y, Jt)OL 2 (V, v) is isvrnorphic to U(XxY, ft0v), and L 2 (X,ft)01i is i:wmo17Jhic to L 2 (X,Jt;tl).

Let D be a subspace of 1i and A be a mapping on D with values in K. If A(O:'x + j3y) = nA:z: + {3J3y holds for all o·, j] E C and :r, y E 11., A is called a linem· operator, D is called the donwin of A and denoted by dom A and {Ax : x E domA} is called the range of A and denoted by ranA. The graph of A, which is denoted by 9(A) is defined by {(x, Ax): x E dom A}. If 9(A) is a. closed subset of 7itDK, then A is called a closed operator.

THEOREM A.l.l5. ~I(A) is closed if]' ll:rn- :rll-+0 and IIA.1:n- vll-+0 imply that both x E dornA and .<1:r = y hold.

268 APPENDIX 1

Let A and B be linear operatms. Then, A is said to incl-ude B, denoted by A C B, if 9(A)C9(B) holds. A is called a closable operator if there exists a closed operator including A. Especially, the smallest operator including A is called the closure of A, wIt ich is denoted by A.

Let A be an operator on 1l with values in 1l satisfying domA = 1l. Then, by Riesz's theorem, for any y E dmnA, there uniquely exists z E 1l satisfying

(y, A:t) = (z, x), x E 1l.

With the use of this result, the adjoint opemlor A* of A is defined by A*y = z, where y E 1i and z is the unique vector satisfying the above equality.

DEFINITION A.l.16. Let A be an operator with domA = 1i. A is said to be symmetric if A C A* holds. If a symmetric operator A satisfies domA = domA*, then A is said to be selfadjiont, and if the closure of a symmetric operator A is self adjoint, then A is said to be essmtially selfadjoint.

It can be easily proved that the closure of a symmetric operator A is equal to A**. The kcmcl of a. linear operator A on 1l, denoted by kerA, is the linear subspace of 1i def-ined as {:r E 1i; Ax= 0}.

THEOREM A .1.17. The necessary and su.fJicient conditions which a symmetric operator A should satisfy to be a selfadjoint operator is:

ker(A'" +if) ran(.'l +if)

ker(A"'- if)= {0}, ra.n(A- if)= 1l.

DEFINITION A.l.18. Let :1 be a linear operator on 1l. If there exists a positive constant AI satisfying IIAxll s; AIII.1:II for all x E 1i, then A is said to be bounded. The set of all bounded linear operators on 1l is denoted by B(1l). The norm of A, denoted by 11.411, is defined by

IIAII = sup{IIA1:II: ll:rll = 1}.

It can be proved that 13(71.) equipped with the norm 11·11 is a Banach space.

A bounded operator A is positive if it satisfies

(.1:, A:t) 2: 0, :t E dornA.

It follows from applying Gelfand's representation theorem (Appendix 2) to the smallest commutative algebra generated by {A, I} that there uniquely

APPENDIX 1 269

exists a positive operator 13 satisfying A = 8 2 . According to the unique existence of B, B is denoted by JA and is ca.lled the square mot of A. Especially, IAI denotes JA· A and is called the absolute val-ue of A.

It should be rnentioncd that not all positive operators on a real Hilbert space are selfadjoint, but all positive operators on a complex llilbert space are selfadjoint.

Let A be a bou11ded operator. Then, A is called a. normal operator if A* A = AA *. A is called a. pmjection if A = A* = A 2 • A is called an isometry if IIAxll = jj;cll holds for all x E 'H.. A is called a unitary operator if ranA = 11. and (Ax, Ay) = (x, y) holds for all x, y E 11.. A is a partial isometry if A is an isometry on (kcrA)l.. (kerA)l. is called the initial space asociated with A and ranA is called the final space associated with A.

For any partial isometry A, it is known that A* A is the projection whose range is (kerA)l. and :1.1* is the projection whosP range is ra.n.4.

THEOREM A .1.1 !J. F'or any bounded opemlor A. there uniquely exists a partial isometry II' .wll isfiJin.,r; thr fol!otflilly cqualiolls:

A lVI AI, IAI IV* A, A"' lV*IA"'j,

IA*I lVIAIW*.

Moreover, the initial space associated with the pm·tial isometry l1' is raniAI and the final space as.<;ociatcd with lV iB rau /l. If either A is normal or 'H. is finite-dimensional, /hen JV is ·tmitm·y.

DEFINITION A.l.20. Let A he a liuear operator on domA with the values in ran A. Then, A is said to be invedible if A is i njectivc. A -J denotes the inverse operator of A on ra.uA with the values in domA. Here, the set of all resolvents of A is dcfi ned by

Re(A) = p. E <C: (A- >.l)- 1 is bounded}.

The set of all spectral points, called the 8peclnun, is dcfi ned by

Sp(/l) = <C- He(A).

Moreover, the spectru nt can be devided into tlH' followiug three sets:

(1) The point spcclrurn of :'I is ddi ued by

Sp( 7'>(;1) = {.\ E <C: (A- >.1)- 1 docs not exist}.

(2) The continuous spectrum of A is defined by

Sp(c) = {>. E <C: (A- >.1)-1 is unbounded, ran(A- >.I)= 1l}.

270 APPENDIX I

{3) The r·esidual spectrum of A is dcfi ned by

Sp(r) = p. E C: (.4- >..!)- 1 is uubounded, ran{A- >..I)=/:-7-l}.

It is clear that the following equalities hold:

ra.n(.4- >..!) = dom(!l- >..1)- 1,

Sp(A) = Sp(Pl(A) USp(cl(A) USp(r)(.4).

DEFINITION A.l.2l. Let A be a.n operator on 1{. Then, a. complex number >..is called an appm:rinwlc SJJCclml point if there exists a. sequence {xn} such that

n EN, lim II(A- >..f)xull = 0. n-+oo

Let Sp(r!p)(A) deuotcs the set of all approximate spectral points. Then, it is easily proved that

Sp(Pl(/1) USp(cl(A) C Sp(ap)(/1) C Sp(A),

and if Sp(,·)(A) is empty, tiH'll Sp(ap)(A) = Sp(A) holds.

An elernent of Sp(p) (.~1) is called a proper value (cigcrwalue) of A, and a non-zero vector :c sa.tisf~,in~ /h =>.:cis caiiPd a proper· vector (eigenvector) associated with >.. The sn hspace sparllt('(l by all proper vectors associated with >.. is called the proper 8JH1Cf (eigenspace) associated with >.., and the dimension of this space is called tlte rnultiplicily of>...

DEFINITION A.l.22. Let 1l be a Hilbert space, and { E,, : -oo ::; >. ::; oo} be a family of projections on 1{ indicated by real numbers. Then, {E,\ -oo ::; >.. ::; oo} is called a. 8pectnJ.l rneasw·c if it satisfies

~~;\ ]?;Jl. == B,,E_, EJ,, J.l ~ >.., I i nt II 1~\:r II

,\-+-•:x; 0, :r E 1-l,

lim II E_,:l: - J: II 0, X E 1{. .\-+•X•

THEOREM A.l.2:~. A spectr·al rneaslt1'e is slmn.fJ/y r·ighl continuous and has a left stmng limit if E,\+etE_,, as E tends to 0 decn:asirzgly, and there exists a projecliort denoted byE,\- sati.r;!IJing E,\-cil:-',\-o::;E,\, as E tends to 0 decreasingly. M onovl'T. /('{

APPENDIX l 271

is satisfied. If .4 ·is the OJJO'alm· defined by

(x, Ay) = /_: >.d(x, E,,y), x, y E 1£,

then A ·is selfadjoint.

THEOREM A.l.24 (spectra.! decomposition theorem). Po1· any selfadjoint operator A, th.e1'e uuiqudy o:i.<;ts a spcclml m.ca.<;ure E.\ satisfying

(x, Ay) = ;~: >.d (x, E_w), :t:, y E 1£.

THEOREM A.1.25. >. E Sp(Pl(A) holds if and only if if E,,_0=f.E>.. holds, and ran ( E,\ - E.\-o) i.<; c:J.:actly equal to the proper space of>.. AI oreo-ver, if Sp(A) = Sp(Pl(A) ho/d,r,;, lhcu Sp(A) is countable and A can be represented by

A= L ..\1\, .\eSp(.·1J

where P,, is the p7'0jcdion whose range is ran (e., - B.\-o).

DEFINITION A.l.2G. For a.ny :c, y E 1£, h't x()y be the operator defined by

(:rt_;>Y)z = (y, z):t, :; E 1£.

Then, :rQfj is bonndc'd and ran(.1:C•Y} = {..\:r: ..\ E C} l1olds. l'dorcovcr, the following equalities hold:

ll:~:c·JJII ( .r:::.fj)'"

( n:r )0(/Jy)

(:1: + y)O~ ( xQy)( :;(}w)

A(:rC)Y)

(.7>:·:·Y)A

II-1·IIIIYII. yC.)x,

nfJ(xCy), xO~+ y:,)~,

( 11 z)xnw /:1' -· ,

(Ax )Oy, .1 E /3(1£),

:r () ;1 * y·. :1 E 13 ( 1l) .

THEOREM A.l.27. (I) Ld .tl be the operator dc.fincr!IJ.tJ Lie.! ..\Ft:iOYi· Then the followin.fJ cqua.lilic.'i hold:

.4~ = LAiY/()Xj, jEJ

272

A" A

IAI

II All

APPENDIX 1

~ 2 L I.-\JI YJ0JJj, jEJ

L IAJIYJ0Yj, jEJ

IIA"II = IIA"AII 112 = IIIAI "· {2} For A of (1), lei c and d be positive twmbc1·s satisfying 0 < c~d. If for any j E J c ~ l.-\JI ~ d holds, then A is inuertible and A -I can be represented by

1-1 ~ \ -1 0-f =L"'i YJ'·1:j.

jEJ

THEOREM A.1.28. For an operator A on 1l,

{1} A is a projeclimt opn·atm· ¢:> V CONS{:tj} C ranA, A= LXj 0ij. J

{2} A is a unitm'y opu·ator ¢:> 3 CONS {:cj}. {:11y} C 1l, .t\ = LXj 0 YJ· J

{3} A is an isom.dric opcmlor ¢:> 3 ONS { :r1 }, CONS {:l}j} C 1l, A = L:xJ 0fJJ· j

(4) Aisasemi-isonwlricopemtor¢:>3 ONS{:rj}, {JJj} C 1l,A=L:xJ0 J

An operator A E B(1l) satisfying dim(ranA)< oo is called finite rank. We denote the set of all finite rank operators by F(1l): A E F(1l) ¢:> 3n E

n N, {.-\J} C IR, CONS {.?:J}, {yj}, .t\ = L: Aj:tj c) Yi·

j=l

An operator A : 1£ 1 ---+ 'H 2 between two Hilbert spaces 1£ 1 and 1l2

is completely contin1wus (c.c. for short) or compact if the irnagc A£ of a bounded set £ C 'H through A is totally bounded. We denote a set of all c.c. operators by C(1l1, 1-l2). Then the followi11g statements are satisfied.

1° A is c.c. ¢:>if :tn ~ :r (i.<•., (y. :r 11 )---+ (y, :~:), Vy E 1l, weak converyence),

implies Axn 4 A:c (i.e .. ll:b:,, - ;hjj---+ 0. stron.fJ convcl:fJence).

2° {An} C C (1l1, 1l2) and /\ 11 ~A (i.e., operator (uniform) nMm IIQII = sup{IIQ:rll ;:c E 7-lJ, !1:1:11 = 1}, IIAn- All--+ 0, unifonn converycnce, or norm. conver:ryencc) and if' ;1 E 13 (1l 1 , 1l2 ) (i.P., the set of all bounded linear operators fro111 7-1. 1 to 'fl 2 ) ,the11 /\ E C (1-1. 1 , 1l2 ).

3° C (1l 1 , 1l2 ) is a closed subspace in Ban<u·h space lJ ('fl 1 , 1-1. 2 ) w.r.t. the operator norm, therefore C (71_ 1 , 1£2 ) is a Banach space itself.

In the following sta.t<'ments, we assunte 1{1 = 11. 2 = 1l, hence, C(H1, 1l2)= C(1l, 11.)= C'(1l).

APPENDIX 1 273

4° C('H.) is a. two-sided ideal of H('H.) {i.e., VA E C(1l), VB E 8(1£) => AB, BA E C'(7i).

5° If one of A, A*, IAI and IA*I for A E /3 ('H) is c.c., then the other ones are also c.c.

6° For any sequence of points { Aj} C C satisfying Aj -+ 0 and two ONS{xj}, {yj}, .4 = l:Aixi ®iii is c.c.

7° If A E C('H.), then there exists x E 1l such that

l(x, Ax) I= max {l(y, Ay)l: y E 11., IIYII = 1}.

Moreover, if A E C(1l) is selfadjoint, then there exists x E 'H. such that IIAII = l(x, Ax)l. Here xis the eigenvector of .4. Tha.t is, Ax= Ax (A= IIAII' -11.411).

THEOREM A.l.29 (Eigenvalue expansion theorem). A selfadjoint operatm· A E C('H.) satisfies the following properties:

{1) Sp(A) = Sp(P)(A) U {0} C IR and the set Sp(.4) is at most countable. Eigenvalues exr-ept 0 /l(we finite multiplicity and can be arrrmged as follows:

AJ: lAd= JIAII (111ultiplif'ily= nd => ...\1 = .-\2 = · · · = An1;

An 1+J: 1-',,.+11 = max{l...\1: ...\ E Sp(.:\)- {,\I}} (multiplicity= n2)

:::;.. .A,,. +I = Ant +2 = · · · = .,\.,11 +n2

Then we dr:tenninf' A11 rc:cm·sivcly. 11n· ,._Hruence {...\ 1, A2, ••• } obtained is finite or I Ani ~ 0.

(2} Ifwe can choost the eigcnveclorsxn (llxll = l}forever·yA,1 E Sp(P) (A) such that Xn.l..X 711 (m, :f. n), then A== l:A11 X11 0.in. If every eigen·value is nondegenemtc, llntt thh; decompo8ifiolt ;,.. ·auique.

(3} If£= spa.u{x 71 : u = I, 1, · · ·} and A.~= {z E 11.; Az = 0}, then

£ = K.l., !J = L (:t:n, y):c11 + PJ(y, Ay == L A11 (x 11 , y)xn (Vy E 'H.).

THEOREM A.1.30 (~liui-rna.x theorem). When A E C (11.) is positive, the eigenvalues { A,J of 11 can be calc·ulated irt the follo·wing mini-max fonn:

An= min {max {(A:c, x): llxll = 1, x.l.C}: dim .C = n- 1}.

Here £ is an ( n - 1) -dimensional closed subsJiace (A 1 is the eigenvalue calculated in 7°).

8° When A, B E C (11.) ar£' selfadjoint OfJer·alors, the follo-wing relation is satisfied:

:1H = H/1 ¢:? 30NS {:ru} C 1l.such tha.t.

274 APPENDIX 1

go For any A E C (1£) lhe1·e exists a unique point sequence {An} such that 0 $ An ,J.. 0 (An: eifjCHIJalue of IAIJ. Tlten A = I: AnXn 0 Yn for ONS

n {xn}, {Yn}·

10° If a selfadjoint opaalm· A satisfies An E C (1£) for a natural number n E N, then A is c.c. il.r:;e/f.

DEFINITION .A.L~l. For any two CONS {xi}, {xi} C 1£, if one of the following three infinite sPries

J J IJ

converges, then the other ones also do and they have the same limit value. Then A E B (1£) is called a Schmidt operator· { llilbert-Sclnnidt operator) and the set of all Schmidt operators is denoted by S (1£).

For any CONS {.1:i} and A E S (1-l), !lA IIz = 2:;: I!Axjll 2 is a norm J

(Schmidt norm ( Jlilbcri-Schmidt norm)) a)l(l S (1£) is a. Banach space w.r.t. ll·lb·

The elements .4, BE S (1£) ltave the following properties :

lo llx 0 Vll2 = llxi!IIYII, Vx, Y E 1£. 2° IIAII $ IIAib' IIA"II2 = IIAIIz, IIABIIl $ IIAIIIIBII2 I IIBAib <

IIAIIIIBib => S (1£) is a two-sided ideal of B (1£). 3° A E s (1£) => !AI E S' (1l)' IIIAIII2 = 11.4112.

For any .4, BE .S (1l) <utd any CONS {:t:j}, ((.4, /3)) = Lj (Axi, Bxi) absolutely convNges, wlticlt is independent of a choice of {:t:j}. Then S (1£) is a Hilbert space w.r.t. this innN product((·,·)).

This inner product.((-.·)) ha.o;.; tltc following propNt.ies: 4° ((x 0 y, u (..:) v)) = (:~:, u) (y, '1.')' V:z;, y, II, 'U E 1{, 5° For A,B E S(1l).C E H(1l), ((A*,/3")) = ((B,!l)), ((CA,B))

((A, C* B)) , ((.'IC, /3)) = ((A, HC*) ). 6° S (1l) C C (1l), so that A = Lj AjXj (' l/j for any A E S (1l). Moreover,

c.c. operator A = Lj AjXj 0 Yi E S (1l) ~ L.i 1Ail 2 < oo.

DEFINITION A. 1.:32. If a.n infinite sNies L.i (.1: j, A:r J) absolutely converges for any CONS { .'L' j} C 1{, tltPn the limit va.hw does not depend on the choice of CONS {xj}· Then A E 13 ('H) is called a trace class opn·alor and the set of all trace class operators is <1<-noU•d by T (1l). Not<' that a positive trace class operator is oftc•n called a. density opcmlo1· in physics.

Lj (xj, A:1:j) is ndiPd a lntcc and denoted IJy t.r A.

A trace class OfH'ra.tor ltas the following properties:

APPENDIX 1 275

1° The following st.atcmu'nts are equivalent: (i) /\ E T (1l), (ii) lA I E T(1l), (iii) 1.41 1/ 2 E S(/1}, (iv) triAl< oo.

2° For any .4, /3 E T (1l) aud a:,/3 E C, a:A + !111 E T (1l), A* E T (1£). If 11.411 1 = tr lA I ('v'A E T (1l)), then 11·11 1 becomes a nann (trace norm)

and T (1l) becomes a Banach space w.r.t. 11·11 1 •

3° trx 0 y = (y, x), llx 0 flll 1 = llxllllvll, 'v'x, Y E 1l. 4° T (1l) is a two sided ideal of B (1l). 5° IIAII ~ 11.4112 ~ !lAIIt· 6° A E T (1l) <=> 38, C E S (1l) such that A= JJC. 7° A = L AjXj 0 Yj E C (1l) (Aj 2:: 0). A E T(1l) <=> L Aj < oo. Then

j j

IIAIII = r= Aj. J

4° and 5° imply T (1l) C ,..,. (1l)

THEOREM A.l.:~:3. F (H) c T (H) c S (H) C C (1l) c l3 (1l).

1° F (1l) is the minimal irlf;'(tl in other spaces. C (1£) is an ideal closed by a uniform nonn in B (1l). JlfoT"eotJ(:T, if 1l is sepam.ble, then C (1l) is the maximal ideal.

2° {i) F (11.) ll·llt = T (H) (ii) F (1l) ll·lb = S (1£) (iii) F (1£) IHI = C (1l). 3° Each F (1l), T (1l), S (1l), C (1l) i.<> dcw;e w. r·.t. the weak topology (An~ A} iu H (11.).

Let X* be the set of all bounded linear functionals <p (i.e., <p: X -t C is linear and I'P (A)I ~ i\/ll.illl, 0 < M < +oo) on a Banach space X. Then the following sta.tesmentl-) a.r<' held:

(1) For any o.p E C (1/.r then' exsits a unique 7~; E 7' (1l) such that <p (A)= trT~PA ('v'.4 E C (1l)).

(2) If <pr (.4) = tr1'.4 ('v'.4 E C (1l)) for any T E 1' (1l), then 'PT E C (1l)*. (3) IITII 1 = llcprll =sup {I'PT (.I\) I :A E C (IT), IIAII = 1} (4) For any 1/.• E T (11.)* thPre exsits a uuique Hr1. E 13 (1l) such that

1/'(.4) = trAB,1. ('v'A E T(1l)). (5) If V'B (A) = t.r:l /J ('v'.'l E 'J' (11.)) for any H E 13 (1l), then li-'B E T (1l)*

a.nd 111311 = II·~''HII· T E T (11.) and IJ E 13 (H) Cillt be idPnt.iliNI \\'ith <;'T E C (1l)* and

1/JB E T (1lr' resp<•ctiVPiy. so t.lta.t \\'(' !taw tit<~ rollowing theorem:

THEORSM A.l.3.·1. ( 1) C (11.)"' = T (11.). {2} T (1l)* = B (1l). {3} s (1l)* = s (1l). {4) B (1l)* = T (1l) tfl C (1l)J_, where

C {1l)J_ = {;p E /3 (11.)"': <p (A)= 0, 'v'A E C (1l)}.

Appendix 2

A.2 Elements of Operator Algebras

In Appendix 1, we discussed the Hilbert space method to describe quantum systems. Presently, a more general method, the operator algebraic approach is considered. Namely, we explain son1e of fundamental facts of C*-algebras and Von Neumann algebras. ThP operator algebraic approach to quantum systems is particularly important when a quantum system has infinitely many degrees of freedom. In physics, one co11sidcreds several systems with an infinite number of degrees of freedom, such as quantum fields, quantum statistical systems with symnretry breaking, etc.

Let A be a linear space on the set of complex numbers C. A is said to be an a.lgebmifit satisfies: (i) ABE A for any :1, BE A; (ii) (A+B)+C = A+ (B +C), (AB)C = A(BC) for any A, 13, C EA.

An algebra. A having the involution* frorn A to A such that (i) (A*)* = A, (ii) (A+ >..13)* = A• +Xu·, (iii) (A/Jt = J3•A· for any A, l3 E A and >.. E C, is called a. *-al.fJclnn 011 C.

Moreover, the lto1"111 11·11 011 A is a rnappi11g from A to JR+ satisfying (i) IIAII ~ 0, 11.111 = 0 ¢} /\ = 0, (ii) 11,\AII = 1,\IIIAII, (iii) IIA + Bll ~ II All+ liB II for II, B E A a11d ,\ E C.

DEFINITION A.2.1. Let A be a.11 algebra with nonn 11·11·

(1) A is a 1w1·med algebra if 11/1/311 ~ IIAIIIIHII for any A, /3 EA. (2) A is a Banach alrJcbra if A is a complete normcd algebra w.r.t. 11·11· (3) A is a Banach *-algcu1Yt (f3•-afgcbra) if A is a. Ba11aclr algebra with

IIA*II = IIAII for any A EA. (4) A is a C*-algebm if A is a. !3'"-algebra. with 11.-1•AII = IIAII 2 for any

A EA.

Let us give :some exanr pies for the above algebras.

(1) The set Aln(C) of n x n matrices with complex entries is a C*-a.lgebra. (2) The set B(1i) of all bounckcl linear operators on a Ililbert ::;pace 1i

and the set C(1i) or all cornpa.ct operators on 1i arc C*-algcbras. (3) The set T(1i) of all trace class operators on 1i and the set S(1i) of all

Hilbert-Schmidt operators are !3*-a.lgcbras, but not C*-algebras. (4) The set C(X) of all continuous functions 011 a locally compact

Hausdorff space X is a C* -algebra.

277

278 APPENDIX 2

Now take a subset A of /3(1l) and define the commltlcmt of A by

A'=: {A E /3 ('H) : AB = BA, BE A},

and the double col/1'11/llla'lll /\ 11 by

A"= (A')'

DEFINITION A.2.2. A is a Von Neumann algebra ( VN-algebra for short) if A"= A.

Remark that a VN-algebra is a C*-algcbra.. It is easily shown that

{1) A" ::) A, (2) A'= A(3 l = A(·5l = ... , (3) A"= A('1l = A(nl = ... .

Let A be a c~-algebra witl1 an identity I (i.e.,/\/= Iii= A, VA E A). Any C*-algebra. without I can be always exteltdf'd to a C'*-a.lgrbra with /.

DEFINITION A.2.3. Let il, 13 be elements of A and A, ft be el(•ments of C. {1) A is regular (invertible) if there exists .4-I E A satisfying AA- 1

A- 1A =I. (2) Resolvent set: Re(.1) ==: {A E C : :3 (A- Al)- 1 E A}.

(3) Spectral set: Sp(A) ==: C- Re(A). {4) The mapping \ fron1 A toC is a clunnclcT if y is a homomorphism not

identically 0. The set Sp(A) of all charact.ers of A is called a character space or a. speclml spocc.

(5) The mapping .1 f'rotn Sp(A) to C is defined by A (y) = y (.4), and the correspondence 1\ : A -t A is called a Grlfwul repre.'>entalion.

Note that >. E Sp(A) iff there exists a character \ such that y(A) = >., which is a reason that we call Sp(A) a spectral space.

THEOREM A .2.4. The followin.fJ slrt.lenwnls an NJilivalcnl:

{1} A is an abelian (i.e. _,\ 13 = 13A, VA. lJ E A) C*-algcbra. {2} The HHifJ]Jin.r; A frolll A to C'(.S'tJ(A)) (IIIC sri of all CU!tlinuous functions

on Sp(A)) is r1 '"-isoiJirlry (i.e.,.\·= (.1)*, 11/111 = 11.<111). By the above theor('f!l, A and C(Sp(A)) an• idf'ntilied. Sp(A) is a weak *-compact I·lausdorfr span>, and it becOII\('S local!~· compact when IE A.

The mapping ..p from A to Cis a. linear fuw·liotwl 011 A if' <p(AA+pB) = >.<p(A)+Jli.P(B), and a linear functiona.l tp fro111 A to Cis said to be positive if <p(A* A) 2: 0 for any A EA. Tl1e norm 11·11 of <pis defined by

II if II = s 11 p { I y ( /1 ) I : A E A' II A II ~ 1 } .

For a linear functional cp, the following th(•orcms arc satisfied:

;\PPENDfX 2

1° Schwarz's inequality: j<p(A*/3)j 2 :::; j<.p(/1"/l)jj<.p(B*B)j. 2° A= I=> II'PII = <.,?(/).

279

<p is faithful i[ ';?(A" A) = 0 => A = 0. Let A"', called the dual, be the set of all continuous linear functionals <p on A (i.e., II An - Am II ---7 0 => jcp(An)- <p(Am)l ---7 0), and A+ be the set of all positive cp in A*.

DEFINITION A.2.5. :

(1) <p E A+ is a slate on A if !l'Pll = 1. The set of all states on A is denoted by 6(A) or 6 for simplicity.

(2) For if!I, l.f!2 E A+, 'PI dominates t.p2 (denoted by 'PI ~ 'P2) if l.f!l - l.f!2 E A+.

(3) 'P E 6 is a mi:red stale if there exist A E JR+ and V' E 6 satisfying A<p~'l/J, and 'P is a pare slate if <p is not mixed state.

When A contains /, t.p is a. mixed state iff" there exists A E (0, 1) and

7/Jt, 'I/J2 E 6 such that ~'I f lh, r..p = A~'J + (1- A)lh. The topology on A* is usnally defined by one of the follO\ving two£­

neighborhoods N,(.p) of t.p for an~' cp E A":

i\'; ( y) = { •t:'• E A"' : II t.p - t!• II < E} '

and

N, ( y) '= { ~· E A" : j 'P {. I J.-) - 1/' ( /1 k) j < E, :11 , · · · , ,1\ n E A} .

The first topology is callc~d t.h{' uniform. lopoloqy and the second is the weak *-topology or a(A", A) -lopolom;.

Below we list up some fundal!wntal propc'rLies of' tlu' sla.t.e space 6 as 1° and 2° above.

3° \Vhen I E A, 6 is \\'('ak *-colllpact. 4° By using the 1\r<'in-~·lilrnan theorent \V(' obtain 6 = w"' co ex 6, where

ex 6 is the> set of all extr('flte points of 6 (an c:.~.:ln:rnc poiHt of 6 is an elemcut not rcprt•seul.ed by a. convex contbination or two other clements of 6). \Vhen I E A, ex 6 = 6 7,, i.e., the set of all pure states. ~vloreover,

w*cocx6 =: {~nAn'f'n: ~nAn= l, An~O,cp.,1 E ex6}-w*, where -W* means the closure or t.ltt' set { ... }with respect to the weak*-topology.

Let a be a ma.pping front a one-pararneter group G to Aut(A), which is the set of all autolltorpltisms OIL A satisfying the following conditions :

(i) O't+s = O'tl:ts,

(ii) E~6ilat (/1)- /Ill= 0 (sli'OII!J col!tinuily),

{iii) a 1(A*) = n 1{.1)* {ViE G. V.:l E A).

Tlte gerund fJIWnlu/1/ dywtmi('ul syslt/1/ (CQDS) is described by a triple (A, 6, n) or (A, 6, n{(,')). This description should co11tain the usua.lllilbert space dcscripliott as a s1wcial cas<'. This is ass!JI'ed by the f'ollowing GNS (Gelfand- 1\'ai 11w rl.-- Scr;u I) //l(m·en1:

280 APPENDIX 2

THEOREM A.2.6 (GNS). For· any <p E 6, there ·uniq·uely exist the following items up to the unitaty equivalence:

{1} GNS llilbCT't space 1-l.p; {2} GNS repr·esentalion 1L.; (i.e., 'lro.p is a *-ho111011101']Jit.ism., i.e., rro.p(AB) =

rro.p(A)rro.p(/3), n.;(A") = rro.p(A)*) fmm A lo 13(11.); {3} GNS cyclic vector.?: . .; E H.; (i.e., {rr.,(J\).t: . .;: ;1 E A}-= 7-I.'P) satisfying

<p(A) = (x.p, rr.,(A):r"').

THEOREM A.2.7. For an o:-invariant slate <p (i.e., <p(o:t(A)) <p(A), 'v' A E A, Vt E JR), there exists a slmngly continuous unitary one­parameter group { u 1 : t E IR} satisfying t!te following equalities:

{1} UtX<p = Xo.p 1

{2} 11'o.p(o:1(11)) = u1rr,.p(A)u_ 1•

From the abov£' thcor£'nls, the usual llilbert space description is reconstructed if it is needPcl. ~·lorpover, if we take a. sufficiently large Hilbert space, then we have

THEOREM A.2.8. For geuem.l C* -algcbm A. there exists a 1/ilbcrt space 1-l such that a C*-algebrn B (C 13(11.)) on.'}{ is isomo1phic io A.

Next we discuss some topologies on 8(1-l). Let {.4.\} be a net of B(1-l), and "A>. ~ A" means that A.\ converges to A as A ---t '·ex/' with respect to the (operator) topology r:

(1) Uniform (operator) lrJ!Jofogy Tn: ~ IIA.\- Ali-t 0.

(2) Strong (opernlor) lopolorJ.IJ r·': .&b II(A.\- A) .1:1!---t 0, V:r E 1-£.

(3) Weak (opera/or) lopolo.qy T"': ~ (a:, (.'l\- :l) y) ---t 0, V3:, y E 1-l.

(4) Ultmstmng (oJHTolor) topolo.r;y T 11 "': ~ For any {:r11 } C 7-I.F

{ {xn} C 1-l: L:;~l IJ:cnll2 < +oo, L~,;,J II(A.\- .4) Xnll 2 ---t 0}. (5) Ultmwcak (opcmlol'} topology Tu.w:

!de~ Ln J(xn, (.4.\- A) Yn)l ---t 0, V {xn}' {y.,J E 1-lF. (6) Strong* (operator) lOJJology r-~·:

def 2 2 ¢=:::} IJ(A.\- A) :cll + IJ(A~- A*) xll ---t 0, Vx E 7-1..

(7) Ultrastmng* (operator) topology Tus*:

!de~ Ln {II (A,\ - /l) 3: n 11 2 + II (A~ - A •) :r"' 11 2 } ---t 0, V { :t n} C 1-l.

These topologies are n'la t<'d as rollows:

Tu > Tll8* > TII.S > Tu.w

v v v TS* > T' > Tw

("strong" topology > "weak" topology). When dim1-l < +oo, every topology coincides each other.

APPENDIX 2 281

Denote by B a uuif'ormly bounded subset of 13(11.), for example, the unit ball 81 = {A E B(Jl) : IIAII :S 1}. On this ball, T 1L.w = Tw,Tu.s = 7 s, 7 us• = 7 s•.

When A.\ --t A, B.\ --t 13 in a certain topology r, we shall show whether the following propositiolls hold or not: (a.) A~ --t A*; (b) A.\Q --t AQ and QA.\ --t QA, VQ E /3(11.); (c) A,d3.\ --t AB; (d) A.\B.\ --t AB for {A-X}, {B.,} C B.

We write 0 if the proposition holds, a1td write x if it does not.

T (a.) (b) (c) (d) Tu 0 0 0 0 r' X 0 X 0 Tw 0 0 X X

Tli.S X 0 X 0 Tli.W 0 0 X X TS* 0 0 X 0

TUS* 0 0 X 0

Note that r 1 < T'2 implies ~l:W2 (the closure of T2-topology) C 9JF1 • Let 9:11 be a. *-algebra such that 9Jl 11 ··' = 9.J1 for all algebra 9Jl C B(Jl). A linear functional c.p is T-con1.inuorrs if A.\ --t A(T) impliPs .y(A.\- /1) --t 0. When r 1 < r2, r1 -contilluit~' of c.p irnplics r 2-continuity of c.p.

Denote by 9:!1'" the set of all 7 11-continuous linear functionals. Put 9:11+ _ {cp E 9Jl*: c.p(A*!\)20}, 6:::: {.p E 9Jl:t: 11-,;11 = 1},9:11"':::: {c.p E 9:11*: rw-continuous},9Jl. = {.y E 9J1"';r«w_continuous}.

We have the following fH'OJWrtiPs:

5° 9:11"' = 9:11*, where the clmwre is taken for the norm 11·11 on 9Jl*. 6° cp E 9Jl"' {:::} 3 {.7: 11 }n=l, {Yn}n=l C 11., and 3.1V E N such that cp (A) =

I:;';=l (xn, Ay.,). r cp E 9Jl* {:::} 3 {:r"}n=l, {:Yn}n=l C }{., and such that

L~=l ll:z:nll 2 < +oo, I::~= I 11Ynll 2 < +oo, ..P (.1\) = LH (.?:n, Ayn)·

When 9Jl is identical witlr /3(H), we hav£' ~R. = T(H.) (trace cla ... ">s). This identity is particrili1 rly i 111 port ant.

THEOREM A.2.D. 9Jl* is r1. Banach space tCilh respect to the nonn of 9:11*, and (9Jl*)* = 9:11.

DEFINITION A.2.10. c.p E 9Jl* is norma.! if 0 :S A.\ t .4 implies r.p (A.\) t cp (A).

The following two theorer11s an~ essential.

282 APPENDIX 2

THEOREM A.2.11. For 'P E 9J1:;_, the followin.r; slolcmenls arc equivalent: {1) <p E 9JI*. {2) t..p is normal. {3) There exists p E T('Ji)+ s·uch that t..p(A) = trpA.

THEOREl\1 A.2.12 (\'on Ncuma.llll density theorem). lV!ten !!JI ts a *­algebra with I, the following equations are equioalcnl:

{ 1) 9JI" = 9JL {2) 9JF = 9JL {3) illrs* = 9JL

In the sequel, we assume that 9J1 is a Von Neu ma.llll algebra.

DEFINITION A.2.1:3. 9J1 is said to bc a-ji11ilr' if any family of mutual orthogonal projections is countable.

THEOR8M t\.2.11. '/'/!( full01nin_r; 8latcmrnls rm utllil'(l/cnl:

{1) 9J1 is a-finilr. {2) There c:J:i81s a faithful IWT"IIIal .'i/alr on 9.ll. {3) 9J1 is isommpltir· to a Von Netwumrt algdnn 9J1 1 !tat>ing a sepamting

and cyclic ucclor :r (sec below).

For a. Von Ncurna.Hn algebra 9J1 over a. llilbcrt space K, a vector x E K is cyclic if 9Jix = K, and is sepm·aling if A:r = 0 implies A = 0.

Now one can easily prove the following two propositions:

S0 If 'Ji is separa.hl<', t.lwn 'Ji is a-finitc>. 9° If t..p is a fait.hf'ul nor111al st.ate, then rr.;(9JI)" = 7!".;(9JI) ::= 9JI, 'Ji,.,!::::! H.

S0 implies t.hat alrnost all Von Neumann algebras w;cd in physics are a-finite.

Let P(9JI) be a spt. of all proj<'ctions 011 ~.rt. and E, FE P(9JI). E domirwlts F (dP!!Olc'd by F'<!:E) if ranF CranE, awl 1~ is equivalent

to F (denoted by E ::= F) if there exists <I partial isonwtry such as E = W*l¥, F = lVW*.

DEFINITION A.2.l!J. (I) 1:: is finite if E:::: F <!: E implies E = F. (2) E is scnnJinile if' ar1~' F <!: /:' lras 110 finit<' subprojection. {3) B is in/illite if"/~' is rrot firritc. (4) B is pnnly ill.fiuil< if tlr<'n' <·xists 110 non-zc•ro finite projc•ct.iorr F such

that F < g

DEFINITION J\.~.J(i. L<'t 9JI be· a. Vorr Ncrrrll<~lllr algebra..

APPENDIX 2

{1) 9Jt is finite¢:? I is finite. {2) 9Jt is semifinite ¢:? I is semi-fi·nite. {3) 9Jt is infinite¢:? I is infinite. (4) 9Jt is purely infinite¢:? I is purely infinite.

283

Let us introduce the "trace" on a. Von Ncuma.nn algebra 9J1. A tracer over 9Jt is a map frorn 9J1+ to IR+ = [0, oo] satisfying:

(i) r(,\A +B) = -\r(A) + r( B), \fA, 13 E 9J1+, >.:2:0. (ii) r(A* A) = r(AA"'), \fA E 9JL

This T is not a linear functional on a. VN-algebra. 9Jt in general, but it can be extended to a linear functional T on 9Jt in the following sense: Put 9Jt~ :::: {A E 9J1+ : r(A) < oo} and 9Jtf :::: {A E 9J1 : A* A E 9Jt~}, 9J1t = {~i= 1 AiAiBi; Ai, I3i E 9J1f, n < oo}.

THEOREM A .2.17. ( 1) 9.H 1 is a two sided idcal of 9J1, namely, 9J1 1 9J1 C 9J1 and 9J1 9J1 1 C 9JL

(2) Ther·c exists a unique lincm· fu~tctional r on 9:11 1 such that r t 9J1 1 n 9J1+ = T and r(AB) = r( BA) (\fA E 9:11 1, \f 13 E 9J1).

We denote r by the same symbol T for simplicity.

DEFINITION A.2. U:~. Let r be a. trace,

(1) Tis faithful if r(/1) = 0, .:\ E 9J1+ impli('S /\ = 0. (2) Tis normal if 0 :2: :1.\ t .·1 irnpli('s r(A.\) t r(-'1). (3) Tis finite if r(A) ~ x, \f.'1 E 9:11+. (4) T is sc1ni.fini!c if for a11~1 :l(:f: 0) E 9J1+, tlrerc exists 13(# 0) E 9J1+

such tha.t 0 ~ /3 :S .'\ a11d r(/3) < +oo. (Note' that if Tis normal and semifinite, thc11 r(.·l) = srrp{r(/3): 0 ~ 13 :S /\, r(l3) < +oo})

(.5) A family of tracC's {T.J : j E ./} is su.f]icirut if' f'or a.11y A(:f: 0) E 9J1+, there exists j E J such that TJ(A) =/= 0.

THEOREM A.2.19. ( 1} fl)l is finite 1j9JI has r1. family of sujjicicnt normal fin£le trace::;.

(2) When 9J1 is CJ-finilr:, 9J1 is .finite if 9)1 ha8 r1. faithful nomud finite trace. (3) 9J1 is .sewi-jinifr if 9.H has a foilltful IWI'/1/(/I sc-rnifinite trace.

Examples of tran's:

(1) VVhen 9JI = Aln(C), r(A) = trA = Li~ 1 rlii, aii =diagonal elements of A is a faithful 11orrnal finite trace.

(2) \Vhen 9JI 13(H) and di11r H oo, r(i\) tr A (= L:n(X 11 , A:r 71 ) for a CONS {.7:n} of H) rs a faithful uorma.l semifinite trace.

284 APPENDIX 2

(3) When rot= L00 (0.,p) (a finite Von Neumann algebra on 1l = L 2 (0.,J.L)), f E M(0.) (the set of a.ll p-measura.ble functions) and /?:..0, JL-a.e., put

r (A) =In A (w) f (w) dp (w).

Then

(i) f > 0, p-a.e. => r is a faithful normal trace. (ii) f E £1(0.,Jt) => r is a finite trace.

(iii) f E L 00 (0., JL) => r is a. semi finite trace.

We now explain sonw rcprc:sentations of cOilltllUtative C*-algebras. The fact that A is commutative allCI 1 E A implies A= C(O) with 0. = Sp (A). So, for any r.p E A+.t = C(H)+.t, there exists a. Baire measure JL such that

r.p (A) = j~ A (w) djt (w)

by the Riesz-Ma.rkov-l(a.kllt.a.ni theorem.

Moreovet·, put 7-l'P = 1l11 = L2 (0., p) aml ( rr1, (.f)g)(t) = f(t)g(t) for any g E 1lw Then

THEOREM A.2.20. Let 1l uc a separalJ/e llilberl space and rot be a commutative \'on Nettmann algcum. Then ther·e exists a locally compact Hausdorff space 0. satisfying the second countable axiom (i.e., it has a countable basis of open sd.c;) and a pmbabilily measure JL such that rot = L=(n, JL).

From these facts, we rt'cognize t.hat the forn11dation of QPT (quantum probability theory) by Von Nc~tllll<lllll algebras or c~-a.lgPbra.<.; is indeed an extension of CPT ( cla.s:;ical probability theory).

As explained above, there exist various Von Neu111ann algebras, and purely infinite Von Neumann algebras are Heeded in some special situations. Although it is sufficient to treat semifinite Von Neumann algebras over separable Hilbert spaces, such as B(7-l), a. purely infinite Vo11 Neumann algebraic (C'"-a.lgebraic) description is fundamental for the case when a physical system has infi11il<' 11111nber of degrees of freedom and some kind of symmetry breaking.

We summarize the description of quantlltll systems by Von Neumann algebras and C* -algebras.

.\PPENDI.\ '1 285

(1) Description by Von Neumann algebras: The Von N0urna.nn algebraic description of quantum systerns is given by 4-tuplc (1i, P, 9.Jl, <p):

CPT QPT n {::} 1l F {::} PH f {::} A JL {::} <pE6

£= (f2) {::} 9J1

(2) Description by C*-a.lgebras: The C*-algcbra.ic description of quantum systems is given by (A,<p):

CPT QPT n {::} Does not exist F {::} Does not exist p {::} <pE6

ex (n) ¢::? A

Once <pis given, we ca11 construct the llilbert space 1l.., through theGNS construction thPOr<'lll. TIH•rpfor<' th<• dPsniption (1) is essentially the same as that of (2).

Finally, we briefly disclJss the l~l\1S (Kubo-l\1artin-Schwinger) state and the Tomita- Takesal.:i theory.

Let (A, 6, a) be a C'"-dynarnica.l system. A state <p E 6 is a IO..JS state with respect to a constant /3 and a 1 if for any .4, B E A, there exists a complex function FA.a(z) such that (1) FA.a(z) is analytic for any z E D{3 = {z E C; -{3 < lm :: < 0} (if /3 < 0, De:={:: E C; 0 < lm z < -/3}), (2) FA,B(z) is bounded and continuous for any z E IJr1 = {z E C; -/3:::; Im z:::; 0}, (3) FA,B(z) satisfies the following boundary conditions: (i) l~,B(t) = <p( a 1 (A) B) a.nd (ii) C.1,B (l- i;3) = <p( Bo·t( A)). The Kl\1S state with respect to the constant ;3 and o 1 is calkd (/3, o·1)-ld1S state. \Ve denote the set of all (/3, nt)-KMS states by ](J3(Cl').

One ca.n easily provP the following two propositions:

1° <pis (0, ot)-KMS stale {=:::> <p is a tracial state. 2° <pis (/3, nt)-I\:MS state {=:::> rp is ( -1, o/3t)-l\:i\'IS state.

THEOREM A.2.21. (13, nt)-1\"J\IS :,;late <p is n-hwariant.

THEOREM A.2.22. The following slatcJnf"nls ore equivolent: (1) <p E ]{ fj ( n) · {2} <p(AB) = rp(/3nta(il)), \f/1, 13 E A, 1 •

{3) For any I wfw.w l'ottf'i(/' /mnsfor/11 is in c;jx'(IR.) and for any A, 13 E A, the followill.rJ Ufttal ion is sol is.fiu/:

j'f(t.)<.p(nt(.t1)13)dl = l f(t- i:J)<.p(/3n 1(A))dt,

286 APPENDIX 2

where f(t- i/3) = J /ciw(t-ifi)dw.

THEOREIVI A.2.:n. If 'P is faithful, then for a cedain ;3, there uniquely exists a one-pammclcr r111iowmphism group n 1 such that 'P is ({3, a.t)-1\JvfS state.

Let {7i<P, Jr<P, :r,.p, uf} be theGNS represPntation of'{) E 6. r.p is defined by lj)(Q) = (x<P, Q:r,.;) for any Q E 1r<P(A)", which is called the natural extension of'{) to 1r"'(A)". The natuml c.rlcnc;ion l'V1 of n 1 is defined by iit(Q) = u'fQu':.t for any Q E 7rcp(A)".

3° For any 'P E J(ll(n), a1.(Q) = Q, VQ E Z"'. 4° K13 (a:) = {'P} :=::;. Z<.p = CI. 5° <.p E ex /(n) :=::;. Jr . .p(A)' n u'f(IR.)' = CI. 6° <.p E Ka(o:) n ex l(n) :=::;. 'P E ex f(;1 (o) {=:::::} 2..p =Ct. 7° Since Ko(o) is cotnpact. in wPa.k*-topology and convex /\'13(n) = w *

coex Ko(n).

Let us consider a Von N<'IJ Jli<Utn algebra 91 having a cyclic and separating vector x in a Hilbert space 7{. Define conjugate linear operators 50 and F0

by 5. \ 1* . o-' :r = .' :r (VAE91),

t>l':z: = :·\''":r (V.il' E 91).

Their domains contain ':ll.r and 91':r, rr'sjwctivc•ly, and tl1e~~ are closable operators: So = s·, f.;, = r ( 1) s; = F, r:; = s·. (2) S = Jt1 112 polar dl'CDillposit.ion.

t1 is called the modular OfH m.lor w.r.t {91, :r }, \\'liicli is unbounded positive selfadjoint. J is call<•d the rnodular conju.r~alc opuolor, which is conjugate unitary (i.e., (J:c, Jy) = (y. a:), J2 = 1 for any :r, y E 1£).

(3) Ll = FS. C:l.- 1 =SF. ( 4) p = J S J = C:l.t /2./ = ./ t::,. -I /'2.

Remark: These OjH'rators dc'IH'Ild 011 :c, so that til<'~' should he dcHoted by Sx, Fx, etc., but W<' OIIIitt<•d :r h<'l'<'.

THEORElVI A.2.2l (To111ita). (I} .191.! = ')1' . .J9l'.J = 91. (2) J AJ =A', (V:\ E 91 n ')1' ). (3} Lli191C:l.-i1 = 91. VIE IR..

Take ¢(A)= (:t,.l:l:), A E 91 and defin<' aj'(.'1) = _6.i 1.'\.6.-i1 , A E 91.

THEORE~11 A.2.2.1. (Ta/,·csaki) (!J satisfies the I\'AfS-crmdilioll w.1·.t. af at !3=-l.

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Index

(m, V)-Markov process, 210 B"-algebra, 277 c" -algebra, 277 c" -system containing all above, 107 F-equivalent rings, 27 F-homogeneous rings, 27 K-multiplier, 209 L*-space, 26 N-representability problem, 176 Q-entropy, 96 Q-entropy =A-entropy, 135 S-entropy, 68 a-finite, 282 a-( A·. A)-topology, 279 £-entropy in GQDS, 245 £-entropy of a state, 244 d-dimensional entropy, 31 m-representation, 209 n-positive map, 198 A-measurement, 129 A-iliermodynamics, 129 • -algebra, 277 *-isometry, 278 "dressed" state, 106 "naked" state, 106

abelian algebra, 278 absolute continuity of distribution

function, 31 absolute value of an operator, 269 adjoint operator, 268 algebra, 277 algebra, B*, 277 algebra, c*, 277

301

algebra, * -, 277 algebra, abelian, 278 algebra, Banach, 277 algebra, Banach • -, 277 algebra, dual, 279 algebra, norm, 277 algebra, Von Neumann (VN-), 278 amount of information, 16 amplification (lifting), 220 amplitude of probability, 134 approximate spectral point, 270 Araki relative entropy, 71

Baire measure, 12 Baire subsets, 12 Banach • -algebra, 277 Banach algebra, 277 Banach space, 264 Bessel's inequality, 265 big system, 9 bit, 16 Boltzmann's constant, 15 Boolean ladder, 26 Borel a-algebra, 205 Borel measure, 12 Borel subsets, 12 bounded operator, 268 Brownian motion, 205

canonical distribution, 133 canonical state, 133 canonical transformation, 187 Cantor set, 243 capacity dimension, 239

302

capacity dimension of a continuous state in CDS, 252

causal system, 106 CD entropy, 23 CDS, 65 channel capacity, 49 channel, chaotic, 76 channel, chaotic for a subset, 76 channel, deterministic, 76 channel, ergodic, 76 channel, normal, 78 channel, orthogonal, 76 channel, quantum, 73 channel, stationary, 76 chaotic channel, 76 chaotic for a subset, 76 character, 278 character space, 278 Choquet simplex, 66 classical KS-entropy, 114 closable operator, 268 closed operator, 267 closed subset, 265 closure of an operator, 268 CNT-entropy, 68 coarse graining, 42 communication channel, 48 communication theory, 16 commutant, 278 compact operator, 272 complete set of observables, 124 completely continuous operator, 272 completely positive (CP) channel, 73 completely positive map, 198 completely positive maps, 197 complex system, 6 complexities of entropy type, 113 complexity C (f), 110 complexity H1 (a), 112 complexity of description, 112 complexity of state, 107 compound state, 74, 77 computable fuction, 112 conditional entropy, 23 conditional probabilities, 23

INDEX

conjugate bilinear functional, 267 CONS, complete orthonormal system,

265 contimuous system, 107 continuous dependence of CD entropy

on probabilities, 24 continuous spectrum, 269 contraction, 196 convergence, norm, 272 convergence, strong, 272 convergence, uniform, 272 convergence, weak, 272 convolution, 205 cosmoscopic domain, 9 CPT, classical probability theory, 284 cyclic vector, 282

de dicto, 4 de re, 4 decomposable linear map, 227 decomposable maps, 197 degree of chaos, 108 density operator, 13, 274 density operator, regular, 179 det, 16 detailed balance (classical), 216 detailed balkance condition (quantum),

217 deterministic channel, 76 diffusion process, 205 dimension of information, 30 dimension of probability distribution,

31 discrete observable, 23 discrete random variable, 23 discrete system, 107 dissipative part, 199 distribution function (distribuant), 31 divergence, 41 domain of a linear operator, 267 domination, 279, 282 double commutant, 278 dual algebra, 279 dual time evolution, 195 dynamical entropy, 115

dynamical entropy by the complexities, 115

dynamical entropy through quantum Markov chain, 117

egenvector (proper vector), 270 eigenvalue (proper value), 270 eigenvalue expansion theorem, 273 eignespace (proper space), 270 elementary events, 12 entropy, 14 entropy defect, 17 entropy gain, 18 entropy of a general state, 66 entropy of measurement, 136 entropy production, 37, 100, 101 equilibrium A-macrostate, 132 equilibrium state, 204 equivalence, 282 equivalent w.r.t. an operator, 127 ergodic channel, 76 essentially selfadjoint operator, 268 estimated Hamiltonian, 193 estimated operator, 169 estimated value, 128 estimation of an operator, 169 estimator, 6 events, 12 extreme point, 279

faithful KMS state, 96 faithful linear functional, 279 faithful trace, 283 final space, 269 finite poartition, 85 finite projection, 282 finite rank operator, 272 finite trace, 283 finite VN-algebra, 283 Fourier coefficients, 265 Fourier's expansion formula, 265 fractal, 229 free energy, 17

Gaussian channel, 253 Gaussian convolution semigroup, 207

INDEX

Gaussian measure, 253 Gelfand representation, 278

303

general linear group GL(n, C), 208 general quantum communication

process, 75 general quantum dynamical system

(GQDS), 279 generalized E-entropy, 245 generalized exponential distribution

(semi-Gaussian or Laplace), 154 generalized gamma functions, 144 generalized Gauss function, 46 generalized Gauss function of order n,

153 generalized Hilbert space, 136 generalized inverse temperatures, 132 generalized Markov chain, 82 generalized measurement, 136 generalized scaling dimension, 234 generalized semi-Gaussian densities,

46 generalized temperatures, 126 genetic sequences, 246 genome, 8 Gibbs state, 133 GNS (Gelfand-Naimark-Segal)

theorem, 279 GNS cyclic vector, 280 GNS Hilbert space, 280 GNS representaton, 280 GQDS, 65 graph of a linear operator, 267 great canonical distribution (state,

ensemble), 133 group, general linear GL(n, C), 208 group, special unitary SU(n), 206

Hamiltonian part, 199 hartley, 16 Hartree-Fock type macrostate, 174 Hausdorff dimension, 235 Hausdorff measure, 235 hierarchical observable, 24 higher-order thermodynamics, 138 higher-order thermometers, 138 Hilbert space, 264

304

Hilbert-Schmidt norm, 274 Hilbert-Schmidt operator, 274 homogeneous state, 83

inclusion of operators, 268 indistinguishibility of information, 28 infinite projection, 282 infinite VN-algebra, 283 information, 16, 28 information A-thermodynamics, 130 information dimension of a state, 246 information dynamics (ID), 105, 108 information energy, 57 information gain, 34 information gain (relative

information), 34 information theory, 16 information without probability, 25 information-theoretic decision rule,

128 Ingarden-Urbanik (IU)-entropy, 135 initial space, 289 inner product, 263 inner product space, 263 input system, 106 invariant (stationary) state, 204 invertible operator, 269 involution, 277 isoentropic motion, 169 isolated system, 195 isometric lifting, 74 isometric operator, 272 isometry, 269 isomorphic Hilbert spaces, 266

Jensen's inequality, 57 joint entropy, 23 joint probabilities, 23

kernel of a linear operator, 268 Klein's inequality, 63

INDEX

KMS (Kubo-Martin-Schwinger) state, 285

KMS states, 66 Koch curve, 231 Kolmogorov conditions, 197

Kolmogorov's £-entropy, 244, 249 Kullback information (entropy), 41

lattice sphere, 239 law of the broken choice, 24, 28 least distance theorem, 266 lifting, 73, 220 lifting, linear, 73 lifting, nondemolition, 73 linear channel, 73 linear functional, 278 linear functional, faithful, 279 linear functional, positive, 278 linear lifting, 73 linear operator, 267 linear response perturbed state, 97 linear response state, 97 local character of information, 28

macroscopic measurement, 129 macrostate, 127 macrostate invariant in time, 131 macrostate, reglar, 188 macrostate, regular, 180, 189 Markov kernel, 251 Markovian evolution, 222 master equation, 221 maximum mutual entropy, 252 measure, probability, 204 measure-preserving transformation,

114 measurement, 74 mesoscopic case, 152 mesoscopic domain, 8 microscopic measurement, 129 middle state, 124 mini-max theorem, 273 minimum programme, 112 mixed state, 279 mixing entropy, 68 modalities, 2 modular conjugate operator, 286 modular operator, 286 modulo A., 42 monotonicity of information, 28 multiplicity, 270

multiplier, 209 multivariate distribution, 49 mutual entropy, 77 mutual information, 35

nat, 15 natural extension of a one-parameter

automorphism group, 286 natural extension of a state, 286 negentropy, 17 noise power, 49 non-trivial Boolean ring, 26 nondemolition lifting, 73 norm, 263,277 norm convergence, 272

INDEX

norm of a linear functional, 278 norm, Schmidt (Hilbert-Schmidt), 274 norm, trace, 275 normal channel, 78 normal linear map, 196 normal operator, 269 normal trace, 283 normed algebra, 277

objective-subjective entropy, 57 Ohya E-entropy, 252 one-parameter convolution semigroup,

205 one-particle observable, 177 ONS, orthonormal system, 265 open system, 74, 195 operator (uniform) norm, 272 operator of finite rank, 272 operator, adjoint, 268 operator, bounded, 268 operator, closable, 268 operator, closed, 267 operator, compact, 272 operator, completely continuous, 272 operator, density, 274 operator, essentially selfadjoint, 268 operator, estimated, 169 operator, invertible, 269 operator, isometric, 272 operator, modular, 286 operator, modular conjugate, 286

305

operator, normal, 269 operator, positive bounded, 268 operator, projection, 269, 272 operator, Schmidt (Hilbert-Schmidt),

274 operator, selfadjoint, 268 operator, semi-isometric, 272 operator, symmetric, 268 operator, trace class, 274 operator, unitary, 269, 272 optical fiber communication, 118 orthogonal channel, 76 orthogonal states, 78 orthogonal subspace of a subspace,

265 orthogonalsubspaces, 266 orthogonal system, 265 outer measure, 234, 235 output system, 106

pair of strong complexity, 108 parallelogram law, 264 Parse val's equality, 265 partial function, 112 partial isometry, 269 Peierls' inequality, 63 point spectrum, 269 polar identity, 264 positive bounded operator, 268 positive cone, 195 positive dynamical semigroup, 195,

196 positive linear functional, 278 positive linear map, 196 power set, 234 principle of maximum entropy, 126 probability measure, 204 probability space, 12 projection, 266 projection operator, 269, 272 projection theorem, 266 projection, finite, 282 projection, infinite, 282 projection, purely infinite, 282 projection, semifinite, 282 proper space (eigenspace), 270

306

proper value (eigenvalue), 270 proper vector (eigenvector), 270 pulse amplitude modulation, 119 pulse position modulation, 119 pure state, 279 purely infinite projection, 282 purely infinite VN-algebra, 283

QDS, 61 QMC, quantum Markov chain, 81 QPT, quantum probability theory, 284 quantum channel, 73 quantum Markovian master equations,

196 quantum relative entropy, 70 quantum system, 107 quantum system, 107 quasicompound state, 77 quasimutual entropy, 78, 245

Renyi entropies, 51 Renyi entropy of order a, 54 Renyi information gain of order a, 54 random variable, 12 random variable norm, 254 range of a linear operator, 267 reduced p-fermion density operator,

176 reduced dynamics, 220 regular (invertible) element, 278 regular density operator, 179 regular macrostate, 180, 188, 189 relative entropy, 18 relative entropy functional, 71 relative information (information

gain), 34 relaxing state, 204 rescaling time, 223 resolvent set, 278 resolvents, 269 Riesz's theorem, 266

sample, 6 scaling dimension, 232 scaling invariant set, 231 Schatten decomposition, 62

INDEX

Schmidt norm, 274 Schmidt operator, 274 Schwarz inequality, 263 Schwarz's inequality, 279 Second Principle of Thermodynamics,

14 self-similar set, 231, 237 selfadjoint opretator, 268 semantic space, 22 semi-isometric operator, 272 sernifinite projection, 282 sernifinite trace, 283 sernifinite VN-algebra, 283 sernigroup evolution, 74 semigroup, Gaussian convolution, 207 separable Hilbert space, 265 separating, 282 sesquilinear functional, 267 Sierpinski gasket, 231 sigleton, 66 signal power, 49 signal transmission, 107 similar contractive mapping, 237 singular reservoir, 226 small system, 9 special unitary group SU(n), 206 spectral decomposition theorem, 271 spectral measure, 121, 270 spectral set, 278 spectral space, 278 spectrum, 269 square root of a positive operator, 269 squeezed states, 135 standard deviation, 43 state, 279 state, equilibrium, 204 state, invariant (stationary), 204 state, mixed, 279 state, pure, 279 state, relaxing, 204 stationary (invariant) state, 204 stationary channel, 76 statistic, 6, 42 statistical hypotheses, 19 stochastically independent, 24 strong continuity, 279

strong (operator) topology, 280 strong • (operator) topology, 280 strong continuity, 195 strong convergence, 272 sufficient family of traces, 283 sufficient partitioning( sufficient

statistic), 42 sufficient transformations, 43 support, 78 symmetric operator, 268

temperature coefficients, 126 tensor product Hilbert space, 267 theorem, GNS

(Gelfand-Naimark-Segal), 279 theorem, Von Neumann density, 282 thermodynamically regular, 127 Third Principle of Thermodynamics,

18 Tomita-Takesaki theory, 285 topological dimension, 239 topology, strong (operator), 280 topology, strong • (operator), 280 topology, ultrastrong (operator), 280 topology, ultrastrong • (operator), 280 topology, ultraweak (operator), 280 topology, uniform (operator), 280 topology, weak (operator), 280 trace, 274 trace class operator, 274 trace norm, 275 trace over a VN-algebra, 283 trace, faithful, 283 trace, finite, 283 trace, normal, 283 trace, semifinite, 283 transformation (channel) system, 106 transinformation, 35 transition expectation, 82 transmitted complexity, 107

INDEX 307

Uhlmann relative entropy, 71 ultrastrong (operator) topology, 280 ultrastrong • (operator) topology, 280 ultraweak (operator) topology, 280 ultraweak continuity, 196 uncertainty, 16 uniform (operator) topology, 280 uniform convergence, 272 uniform topology, 279 unit element of a Boolean ring, 26 unitary evolution, 74 unitary implemented transformation,

187 unitary operator, 269, 272 universal computer, 113

value of an observable, 127 variance, 43 variation of information, 41 Vlasov-type equations, 183 VN-algebra, finite, 283 VN-algebra, infinite, 283 VN-algebra, purely infinite, 283 VN-algebra, semifinite, 283 Von Neumann (VN-) algebra, 278 Von Neumann density theorem, 282 Von Neumann entropy, 61

weak (operator) topology, 280 weak ·-topology, 279 weak convergence, 272 weak coupling limit, 223 weight, 56 weighted entropy, 56 weighted entropy of a random

variable, 57

Fundamental Theories of Physics

22. A.O. Barut and A. van der Merwe (eds.): Selected Scientific Papers of Alfred Lande. [1888-1975]. 1988 ISBN 90-277-2594-2

23. W.T. Grandy, Jr.: Foundations of Statistical Mechanics. Vol. II: Nonequilibrium Phenomena. 1988 ISBN 90-277-2649-3

24. E.l. Bitsakis and C.A. Nicolaides (eds.): The Concept of Probability. Proceedings of the Delphi Conference (Delphi, Greece, 1987). 1989 ISBN 90-277-2679-5

25. A. van der Merwe, F. Selleri and G. Tarozzi (eds.): Microphysical Reality and Quantum Formalism, Vol. 1. Proceedings of the International Conference (Urbino, Italy, 1985). 1988 ISBN 90-277-2683-3

26. A. van der Merwe, F. Selleri and G. Tarozzi (eds.): Microphysical Reality and Quantum Formalism, Vol. 2. Proceedings of the International Conference (Urbino, Italy, 1985). 1988 ISBN 90-277-2684-1

27. I.D. Novikov and V.P. Frolov: Physics of Black Holes. 1989 ISBN 90-277-2685-X 28. G. Tarozzi and A. van der Merwe (eds.): The Nature of Quantum Paradoxes. Italian

Studies in the Foundations and Philosophy of Modem Physics. 1988 ISBN 90-277-2703-1

29. B.R. lyer, N. Mukunda and C.V. Vishveshwara (eds.): Gravitation, Gauge Theories and the Early Universe. 1989 ISBN 90-277-2710-4

30. H. Mark and L. Wood (eds.): Energy in Physics, War and Peace. A Festschrift celebrating Edward Teller's 80th Birthday. 1988 ISBN 90-277-2775-9

31. G.J. Erickson and C.R. Smith (eds.): Maximum-Entropy and Bayesian Methods in Science and Engineering. Vol. 1: Foundations. 1988 ISBN 90-277-2793-7

32. G.J. Erickson and C.R. Smith (eds.): Maximum-Entropy and Bayesian Methods in Science and Engineering. Vol. II: Applications. 1988 ISBN 90-277-2794-5

33. M.E. Noz and Y.S. Kim (eds.): Special Relativity and Quantum Theory. A Collection of Papers on the Poincare Group. 1988 ISBN 90-277-2799-6

34. I.Yu. Kobzarev and Yu.I. Manin: Elementary Particles. Mathematics, Physics and Philosophy. 1989 ISBN 0-7923-0098-X

35. F. Selleri: Quantum Paradoxes and Physical Reality. 1990 ISBN 0-7923-0253-2 36. J. Skilling (ed.): Maximum-Entropy and Bayesian Methods. Proceedings of the 8th

International Workshop (Cambridge, UK, 1988). 1989 ISBN 0-7923-0224-9 37. M. Kafatos (ed.): Bell's Theorem, Quantum Theory and Conceptions of the Universe.

1989 ISBN 0-7923-0496-9 38. Yu.A. Izyumov and V.N. Syromyatnikov: Phase Transitions and Crystal Symmetry.

1990 ISBN 0-7923-0542-6 39. P.F. Fougere (ed.): Maximum-Entropy and Bayesian Methods. Proceedings of the 9th

International Workshop (Dartmouth, Massachusetts, USA, 1989). 1990 ISBN 0-7923-0928-6

40. L. de Broglie: Heisenberg's Uncertainties and the Probabilistic Interpretation ofWave Mechanics. With Critical Notes of the Author. 1990 ISBN 0-7923-0929-4

41. W.T. Grandy, Jr.: Relativistic Quantum Mechanics of Leptons and Fields. 1991 ISBN 0-7923-1049-7

42. Yu.L. Klimontovich: Turbulent Motion and the Structure of Chaos. A New Approach to the Statistical Theory of Open Systems. 1991 ISBN 0-7923-1114-0

Fundamental Theories of Physics

43. W.T. Grandy, Jr. and L.H. Schick (eds.): Maximum-Entropy and Bayesian Methods. Proceedings of the lOth International Workshop (Laramie, Wyoming, USA, 1990). 1991 ISBN 0-7923-1140-X

44. P.Ptak and S. Pulmannova: Orthomodular Structures as Quantum Logics. Intrinsic Properties, State Space and Probabilistic Topics. 1991 ISBN 0-7923-1207-4

45. D. Hestenes and A. Weingartshofer (eds.): The Electron. New Theory and Experiment. 1991 ISBN 0-7923-1356-9

46. P.P.J.M. Schram: Kinetic Theory of Gases and Plasmas. 1991 ISBN 0-7923-1392-5 47. A. Micali, R. Boudet and J. Helmstetter (eds.): Clifford Algebras and their Applications

in Mathematical Physics. 1992 ISBN 0-7923-1623-1 48. E. PrugoveCki: Quantum Geometry. A Framework for Quantum General Relativity.

1992 ISBN 0-7923-1640-1 49. M.H. Mac Gregor: The Enigmatic Electron. 1992 ISBN 0-7923-1982-6 50. C.R. Smith, G.J. Erickson and P.O. Neudorfer (eds.): Maximum Entropy and Bayesian

Methods. Proceedings of the lith International Workshop (Seattle, 1991). 1993 ISBN 0-7923-2031-X

51. D.J. Hoekzema: The Quantum Labyrinth. 1993 ISBN 0-7923-2066-2 52. Z. Oziewicz, B. Jancewicz and A. Borowiec (eds.): Spinors, Twistors, Clifford Algebras

and Quantum Deformations. Proceedings of the Second Max Born Symposium (Wroclaw, Poland, 1992). 1993 ISBN 0-7923-2251-7

53. A. Mohammad-Djafari and G. Demoment (eds.): Maximum Entropy and Bayesian Methods. Proceedings of the 12th International Workshop (Paris, France, 1992). 1993

ISBN 0-7923-2280-0 54. M. Riesz: Clifford Numbers and Spinors with Riesz' Private Lectures to E. Folke

Bolinder and a Historical Review by Pertti Lounesto. E.F. Bolinder and P. Lounesto (eds.). 1993 ISBN 0-7923-2299-1

55. F. Brackx, R. Delanghe and H. Serras (eds.): Clifford Algebras and their Applications in Mathematical Physics. Proceedings of the Third Conference (Deinze, 1993) 1993

ISBN 0-7923-2347-5 56. J .R. Fanchi: Parametrized Relativistic Quantum Theory. 1993 ISBN 0-7923-2376-9 57. A. Peres: Quantum Theory: Concepts and Methods. 1993 ISBN 0-7923-2549-4 58. P.L. Antonelli, R.S. Ingarden and M. Matsumoto: The Theory of Sprays and Finsler

Spaces with Applications in Physics and Biology. 1993 ISBN 0-7923-2577-X 59. R. Miron and M. Anastasiei: The Geometry of Lagrange Spaces: Theory and Applica-

tions. 1994 ISBN 0-7923-2591-5 60. G. Adomian: Solving Frontier Problems of Physics: The Decomposition Method. 1994

ISBN 0-7923-2644-X 61 B.S. Kerner and V.V. Osipov: Autosolitons. A New Approach to Problems of Self-

Organization and Turbulence. 1994 ISBN 0-7923-2816-7 62. G.R. Heidbreder (ed.): Maximum Entropy and Bayesian Methods. Proceedings of the

13th International Workshop (Santa Barbara, USA, 1993) 1996 ISBN 0-7923-2851-5 63. J. Pefina, Z. Hradil and B. Juito: Quantum Optics and Fundamentals of Physics. 1994

ISBN 0-7923-3000-5

Fundamental Theories of Physics

64. M. Evans and J.-P. Vigier: The Enigmatic Photon. Volume 1: The Field B(3>. 1994 ISBN 0-7923-3049-8

65. C.K. Raju: Time: Towards a Constistent Theory. 1994 ISBN 0-7923-3103-6 66. A.K.T. Assis: Weber's Electrodynamics. 1994 ISBN 0-7923-3137-0 67. Yu. L. Klimontovich: Statistical Theory of Open Systems. Volume 1: A Unified

Approach to Kinetic Description of Processes in Active Systems. 1995 ISBN 0-7923-3199-0; Pb: ISBN 0-7923-3242-3

68. M. Evans and J.-P. Vigier: The Enigmatic Photon. Volume 2: Non-Abelian Electro-dynamics. 1995 ISBN 0-7923-3288-1

69. G. Esposito: Complex General Relativity. 1995 ISBN 0-7923-3340-3 70. J. Sld1ling and S. Sibisi (eds.): Maximum Entropy and Bayesian Methods. Proceedings

of the Fourteenth International Workshop on Maximum Entropy and Bayesian Methods. 1996 ISBN 0-7923-3452-3

71. C. Garola and A. Rossi (eds.): The Foundations of Quantum Mechanics - Historical Analysis and Open Questions. 1995 ISBN 0-7923-3480-9

72. A. Peres: Quantum Theory: Concepts and Methods. 1995 (see for hardback edition, Vol. 57) ISBN Pb 0-7923-3632-1

73. M. Ferrero and A. van der Merwe (eds.): Fundamental Problems in Quantum Physics. 1995 ISBN 0-7923-3670-4

74. F.E. Schroeck, Jr.: Quantum Mechanics on Phase Space. 1996 ISBN 0-7923-3794-8 75. L. de Ia Pefla and A.M. Cetto: The Quantum Dice. An Introduction to Stochastic

Electrodynamics. 1996 ISBN 0-7923-3818-9 76. P.L. Antonelli and R. Miron (eds.): Lagrange and Finsler Geometry. Applications to

Physics and Biology. 1996 ISBN 0-7923-3873-1 77. M.W. Evans, J.-P. Vigier, S. Roy and S. Jeffers: The Enigmatic Photon. Volume 3:

Theory and Practice of the B(3) Field. 1996 ISBN 0-7923-4044-2 78. W.G.V. Rosser: Interpretation ofClassical Electromagnetism. 1996

ISBN 0-7923-4187-2 79. K.M. Hanson and R.N. Silver (eds.): Maximum Entropy and Bayesian Methods. 1996

ISBN 0-7923-4311-5 80. S. Jeffers, S. Roy, J.-P. Vigier and G. Hunter (eds.): The Present Status of the Quantum

Theory of Light. Proceedings of a Symposium in Honour of Jean-Pierre Vigier. 1997 ISBN 0-7923-4337-9

81. Still to be published 82. R. Miron: The Geometry of Higher-Order Lagrange Spaces. Applications to Mechanics

and Physics. 1997 ISBN 0-7923-4393-X 83. T. Haldoglu and A.S. Shumovsky (eds.): Quantum Optics and the Spectroscopy of

Solids. Concepts and Advances. 1997 ISBN 0-7923-4414-6 84. A. Sitenko and V. Tartakovskii: Theory of Nucleus. Nuclear Structure and Nuclear

Interaction. 1997 ISBM 0-7923-4423-5 85. G. Esposito, A.Yu. Kamenshchik and G. Pollifrone: Euclidean Quantum Gravity on

Manifolds with Boundary. 1997 ISBN 0-7923-4472-3 86. R.S. lngarden, A. Kossakowsld and M. Ohya: Information Dynamics and Open

Systems. Classical and Quantum Approach. 1997 ISBN 0-7923-4473-1

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