spaces of the type gm and gl basic properties

Vol. 12 (1977) REPORTS ON MATHEMATICAL PHYSICS No. 3

SPACES OF THE TYPE GM AND G& BASIC PROPERTIES

WAWRZYNIEC Guz

Institute of Physics, Gdansk University, Gdansk, Poland

(Received April 2, 1976)

This paper is devoted to the study of partially ordered real vector spaces with a norm additive on the positive cone. Such spaces are of great importance for a description of a time evolution in classical and quantum statistical mechanics, and also play an important role in the axiomatical formulation of statistical physical theories. The present paper has a purely mathematical character and has been prepared for later applications in physical axiomatics, which will be the content of a subsequent paper.

1. Definitions and notation

all

Let X be a vector space over the field R of real numbers.

DEF. A subset A s X is said to be convex if x, y E A implies tx+ (1 -t)y E A for scalars t E (0, 1) c R.

DEF. We say that a subset C c X is a cone if (i) C+C -c C,

(ii) tC E C for each t 2 0 (t E R).

We shall call a cone C proper if Cn - C = (0).

Let now < be a partial (semi)ordering’ in X.

DEF. We say that X (or, to be more precise, (X, C)) is a partially (semi)ordered vector

space if the following conditions are satisfied:

(i) x S y (x, y EX) implies x+2 < y+z for all 2 EX,

(ii) x < y (x, y E X) implies tx < ty for all t > 0 (t E R).

Note that in a partially semiordered vector space X the subset C = (x B X: x 2 0} is a cone; moreover, it is a proper cone whenever < is a partial ordering. Elements of C are said to be positive and C is called the positive cone of (X, <).

Conversely, if C is a cone (a proper cone, respectively) in a real vector space X, then

1 In contrast to the partial ordering, the partial semiordering, in general, does not possess the property :

x<y,y<x~x=y.

WI

286 W. GUZ

one can define a partial semiordering (a partial ordering, respectively) in X by

x<:y S-y--XEC,

and C then becomes the positive cone of the space (X, <). Thus for any real vector space X one has the one-to-one correspondence between the

family of all cones (proper cones, respectively) in X and the family of all partial semi- orderings (partial orderings, respectively) in X. The partially semiordered vector space (X, <) may therefore be written as a pair (X, C).

DEF. We shall say that a cone C E X is generating if X = C-C.

DEF. A linear functional f on X is said to be positive if f(x) 2 0 for all x 2 0, i.e. if f takes nonnegative values on the positive cone of X.

The set of all positive functionals on the partially semiordered vector space (X, C) will be denoted by C*. C* is, of course, a cone in the space X* of all linear functionals on X, and therefore (X*, C*) is a partially semiordered vector space.

DEF. The subspace X?’ : = C*- C* of X* (or, to be more precise, the pair (P, C*)) will be called the order dual of (X, C).

It is not difficult to show (see, e.g., [l]) that the partial semiordering in Xp induced by the cone C* is a partial ordering if and only if the positive cone C of the space X is generating.

DEF. A cone C c X is said to be regular if the set C* is total, that is, if for each non-zero x E X there is f E C* such that f(x) # 0.

Note that each regular cone is, obviously, proper.

2. Properties of GM and GLO-spaces

Let (X, C) be an arbitrary partially semiordered vector space.

DEF. An element e E X is said to be an order unit of the space (X, C) if

by0 2 e+txEC. If!<T

Obviously, e E C. Also, it can easily be verified [6] that if (X, C) possesses an order unit, then X = C-C.

DEF. A partially semiordered vector space (X, C) is said to be a GM-space’ (sym- bolically: (X, C) E GM) if

(i) (X, C) possesses an order unit,

(ii) the positive cone C is regular.

Any GM-space is thus, in fact, a triple (X, C, e), where e is an order unit for (X, C).

’ An equivalent definition of a GM-space was given by E. Miles [61.

SPACES OF THE TYPE GM AND GL 287

Remark 1. By the remark at the end of Section 1 we see that the positive cone C of a GM-space (X, c) is always proper; hence the partial semiordering in X induced by C is, in fact, an ordering. Any GM-space is therefore a partially ordered vector space.

It can be shown for a GM-space (X, C) with an order unit e E C that [6]:

(a) Zf f E C* and f(e) = 0, then f = 0.

(b) The set {f E C*: f(e) = l} is total.

(c) The order unit e E C enables us to define the following norm in X:

llxlle := s4.$lf(x)l: f E C*, f(e) = 11.

(d) Every positive functional f E C* is II * I/,-continuous, and

Ilf IV = f(e),

where (( - lie denotes the standard Banuch norm of the Banach dual of (X, II . I I,).

Remark 2 ([6], p. 220). If u is another order unit for (X, C), then the norms I I * I/,,

and I] - ( le are equivalent. From (a)-(d) we find for any GM-space (X, C):

(1) XJ’ := C*- C* E X’, where (Xe, I I * II’) stands for the Banach dual of (X, I I * II&

(2) llxlle = ~~P{lf(x)l: f E c*, Ilfll’ = 11

= s~P{lfol: f EXP, Ilfll’ = 1>

= s~P{lfol: f EXP, Ilfll” G 11.

Remark 3. It can be shown (see [6]) that Xp = X’.

DEF. A partially semiordered vector space (X, C) is called a GLO-space (symboli- cally: (X, C) E GLo) if

(i) X= C-C,

(ii) X admits a norm (1 * II, which is additive on C:

Ilx+ull = II4l+tl~ll for all x,y E C.

Therefore a GL,-space is, in fact, a triple (X, C, ]I - 11); the norm ]I - (1 will be called later on a GL,-norm in X.

Using (d), we come directly to the following result:

(3) If (X, C, e) E GM, then (Xp = Xe, C*, II . II’) E GL,.

Let now (X, C, I] * 11) E GL,, . We put, by definition,

e(x) := llxIII--IIxAI

whenever x = x1 -x, , where x1, x2 E C. Note that e is well-defined, i.e. e(x) does not depend on the selection of x1 and x2.

It can easily be seen that:

(4) The functional e on X is linear and continuous, i.e. e E X’ = the Banach dual of

288 W. GUZ

(X, 11 * 1 I); moreover

and llell = 1 whenever C # (0).

Furthermore, e is positive and e(x) = 11x1 I for x E C.

(5) The cone C’ := C*nX’ is a total set. Hence C is regular, and thus the partial semi-

ordering in X induced by C is a partial ordering.

Proof: Let f E A”. For any x E C one has

I_Wl G llfll * lbll = llfll * 44; hence

llfll * e-f, llfll * e+.fe C’, and therefore

(2.1)

f= $(llfll * e+_f>-+(ll.fll * e-f) E C-c’. This shows that X’ = C-C’, and consequently C’ must be total, since X’ is total (by the Hahn-Banach theorem).

(6) The functional e is an order unit for (X’, C’).

Proof Let f E X’ = C’- C’, f # 0, and let f = fi -fi, where fl, fi E C’. One can assume without any loss of generality that fi, fi # 0. In fact, fi = 0 implies f = -fi

= fi -2f,, where fi # 0, and similarly, fi = 0 implies f = f1 = 2fl -fl, where fl # 0.

We shall now show that e-ktfe C’ (2.2)

for all t e [-llfill-‘, Ilfzll-‘]: hence, by putting T := min(llfill-l,lIfill-l) we conclude immediately that e is indeed an order unit for (X’, C’). Let us first note that for each non-zero g E C’ one has e- tg E C’ whenever t E [0, llgl I-‘]. In fact, for all x E C and t E [0, llgll-‘1 we have

t*!?(x) G t* llgll * llxll G IIXII, hence

(e-Q)(x) = Ilxll-t*g(x) 2 0.

Therefore, t E [0, I ( fi) I-l] implies e - tfi E C’ ; hence

e+tf = (e-tf2)+tfl EC’,

and similarly, t E [- Ilflll-‘, 0] implies e+ tfl E C’, hence also

e+tf = (e+tfl>+(-t)fi EC’. Thus (2.2) is proved.

Let (A’, C, I I - I I) e GLo . Note that the canonical injection x + i of the space X into the bidual x” defined by

S(f) :=fW, f EX’,

transforms, the cone C into the cone C” E x”. Indeed, for each x f C one has

G(f) = f(x) 2 0

for all f E C’, which just means that $ E C”.


Hence, as an immediate consequence we obtain that C” is total, and therefore

of

(7) (X’, C’, e) E GM.

We thus have two natural norms in the Banach dual X’ of the GLO-space X:

(a) the standard Banach norm of the dual space, which will be denoted by II-II,

(b) the norm 1) - Ile induced by the order unit e E C’ of the GM-space (X’, C’). The Banach dual of (X’, II-II) will be denoted by (X”, (1. II), and the Banach dual

(X’, II - IL) by We, II - II?. Then a natural question appears: what is the relation between I I * 1) and I I * lie? The

answer to this question is given by the following theorem:

(8) For every f E X’ one has

Ilflle G Ilfll.

Proof: Let f i A”. For every FE C’* such that F(e) = 1 we find, using (2.1), that

hence If’(f>l G ll$ll;

Ilfll, = s~p{lKf~l: J’EC’*, F(e) = 1) < Ilfll.

As a direct consequence of (8) one obtains:

(9) X’=( = X’p) c X” and llFl( < I IFI le for each FE Xe.

We shall complete the result of (9) by adding the following one.

(10) For each F E c’* c X” one has

IIFII = IIFII”. Proof. In fact, since 1 IelI < 1, we have

l[FjI > IF(e)1 = F(e) =llFJl” (see (d), Section 2),

which, together with the opposite inequality (see (9)) leads to I IF( I = llF11”.

DEF. Let (X, C, II - 11) E GL,,. We shall call (X, C, )I - 11) a GL,-space (and write

(X, C, II * II) E GL) if

A v x=x1 -x2 and ll4l = Il~~ll+ll~~ll. xx X&#2

GL1-space is much more regular than a GL,-space. For instance, result (8) may be now strenghtened as follows :

(11) Let (X, C, I I - I I) E GL, . Then for each f e X’ we have

llflle = Ilfll.

Proof: Let x be an arbitrary element of X, and let x1, x2 E C be such that

Then x = x1-x2 and llxll = II~IIl+ll~zll.

ll4l = 11~111+11~211 = ll~1ll’+llw

290 W. GUZ

(see (10); A denotes the canonical injection X 5 X”); hence

llxll 2 Il~l-~zlle = 11~11”.

But since R = gI -AZ E C’* - C’* = _Yp = Ye, we get, by (9),

llxll = 11~11 G ll~ll”, and therefore 11x1 I = [$I le.

For an arbitrary f E x’ we thus have (see (2))

llf II, = sup{lFcf)l: FEX’~, llFll’< I>

2 sup{l~cf)l: XEX, I$]] G 1)

= s~P(lfol: xex, llxll < 1)

= llfll.

which, together with the opposite inequality (see (8)), leads to I If lie = I If I I. As an immediate consequence of (11) we obtain:

(12) If (X, C, I I * (I) e GL1, then X’=(= X’P) = X” and

IFII’ = IFII for all FE _JP = X”.

3. Examples

(a) (R, C, 1) * II), where R is the field of real numbers, C := R, = {x E R: x 3 0},

and Ilxll : = 1x1 = the absolute value of the real number x. For each x E R one has

x = x+-x- and llxll = .x++x- = IIx+ll+llx-11,

where xf := f(lxl+x) = max(x, 0), x- := :(1x1-x) = max(-x, 0); x+, x- e C. The norm (( - }I is, of course, additive on C; hence (R, C, II - 11) e GL, .

(b) (R”, C, I I * II), where ZP is the nth Cartesian power of R, C := R”, = {x = (x1, . . . . . . . x,)eR”: xiER+ for all i= 1,2 ,..., n}, and

. .

II@ 1, ***, xdll := c lxil. 1=1

For every x = (x1, . . ..x.,)eR we have

x = x+-x- and llxll = IIx+II+IIx-II,

where x* := (x:, . . ..x.‘); x+, x- E C. Obviously, (R”, C, II - 11) E GL, . W (CW bl), C, II * II), w h ere C([a, b]) is the space of continuous real functions

ontheinterval[a,b] E R,C:= C+([a,b])= (foC([a,b]):f>O},andIIf]I :=ilf(x)ldx. (I

For each f E C( [a, b]) one has

f=fLf- and Ilfll = Ilf+ll+llf-II,


where f+ := i(if[+f) = max(f, 0), f - := i(lf/-f) = max(-f, 0), f +, f - E C. Since the norm I[ - 11 is additive on C, we have (C([a, b]), C, II* ll)eGL,.

(d) Let S be a non-empty set and let B(S) be a u-algebra of subsets of S. We call the

pair (S, B(S)) a measurable space. Let V = V(S, B(S)) be the set of all finite o-additive real functions v: B(S) + H.

Such functions are called finite signed measures on B(S). The set V, endowed with usuai operations of addition and multiplication of functions by real numbers and equipped with the norm

llall := MS)9 where 1~1, the so-called total variation of IJJ, is defined by

lpll := v++v-9 where

v+(B) : = sup{p,(B,): B, c B, B, E B(S)},

v-(B) := sup{-y(Bl): B1 E B, B, E B(S))

= -inf{v(B,): B, c B, B1 E B(S)}, BE B(S),

is a real Banach space (see, e.g. [IO], Ch. 1, Sec. 3). The positive and the negative parts of QI, y+ and q-, respectively, are ordinary meas-

ures on B(S). It can also be shown (see, e.g., [2], Appendix) that

IPW = sup l{f(x)v(dx) j (p, E v, B E B(S)) B

where the supremum is taken over all bounded B(S)-measurable functions f: S + R such that sup If(x)1 < 1.

With re?pct to the natural partial ordering defined by

Q, Q q~ :e q(B) < p(B) for all B E B(S),

V becomes a partially ordered real Banach space. Moreover, we have

V= V+-V+,

where V+ = {p E V: p 2 0} = the positive cone of V, since for each p7 E Y the so-called Jordan decomposition (see, e.g., [IO], Ch. 1, Sec. 3) holds:

qJ = qP-q3-. It can also easily be shown that

Ildl = IlP+ll+ll~-Il.

From the definition of the norm in V it easily follows that for v e V,

lldl = v(S);

hence we find the norm I] * 11 to be additive on V+ , and therefore (V, V+ , II * II) E GL .

292 W. GUZ

(e) Let H be a separable complex Hilbert space and let V = V(H) be the set of all self-adjoint trace-class linear operators in H (for the definition, see, for example, [8]

or 171). The set Vendowed with ordinary operations of addition and multiplication of operators

by real numbers is a real vector space which with respect to the norm defined by

IlTll := tr]T], where I TI : = (T*T)‘12,

becomes a Banach space (see, e.g., [S]). With respect to the natural partial ordering defined by

T< W:+(Tx,x)< (Wx,x) for all x E H,

W becomes a partially ordered real Banach space. Moreover, one has

V= V+-V+,

where V+ is the positive cone of V, since each element T E V may be written as

T = T+-T-,

+og where T+ := j sPT (dr), T- := -

0 i sP&ds), PT being the spectral measure -02

T+,T-EV+. We also have

IITII = Il~+ll+llT-II,

and from the definition of the norm in V it follows readily that for T E V+

lITI = trT;

of T,

hence we find the norm I I - j I to be additive on V+ , and thus (V, V+ , II * II) E GL1.

(f) Let (L, < , ‘) be an orthomodular u-orrhoposet, i.e. a partially ordered set L with the least element 0 in it, equipped with an orthocomplementation3 ‘: L + L, and such that

(i) for any sequence {ai}~=~ of pairwise orthogonal elements of L (i.e. such that

ai < aj for i # j) there exists its least upper bound c ai in L; i-1

(ii) a< b (a,bEL) implies b = avc for some CEL, c< a’.

DEF. (Zierler [ll]). By a signed measure on L we shall mean any function x: L + Ru (+ CO}U { - m} satisfying the following conditions:

3 An orthocomplementation of L, L being a partially ordered set with the least element 0, is, by definition, a mapping ‘: L + L such that

(i) a” = a for each a E L, (ii) a < b (a, 6 f L) implies b’ < a’,

(iii) a < b and a < b’ implies a = 0.


(i) x is u-orthoadditive, that is,

for any sequence {Ui}Ci of pairwise orthogonal elements of L,

(ii) x(0) = 0,

(iii) x takes at most one of the values +co, -co.

DEF. When x(u) > 0 for all a E L, we say that x is a measure on L. If, in addition, x(l) = 1, where 1 = 0’ (1 is then the greatest element in L), we call x a probubiIity meus- ure on L.

Let X = X(L, <, ‘) be the set of all bounded signed measures on ~5.~ If we define in X the operations of addition and multiplication by a real number, and the partial ordering in an usual way, then X becomes a partially ordered real vector space. Furthermore, if we put by definition (compare example (d))

/lx/l := sup{x(a): a E L}fsup{--x(a): a E L},

then it can easily be verified (compare [I 11) that JI - 11 is a norm in X, which is additive on the positive cone X+ = {x E X: x > 0). Moreover, X is then a partially ordered normed vector space, that is, the positive cone X+ is norm-closed. Finally, it can be shown that (X, 1) * 11) is complete.5

Let us note that in X one may introduce another norm 11. Ill additive on X+ ; it is defined by

IIXIII := sup{lx(u)l: a f L}.

It can easily be verified that for each x E X one has

IIXIII d llxll G 2llxlll; hence we find the norm I I - I I 1 to be equivalent to [ I-11. Moreover, on the positive cone X+ the norms II - 1 II and 11. II coincide, since for x e X+ we have

llxll = ll~lll = x(U, and thus also I I * ) I1 must be additive on X+ .

If we now take into consideration the subspace V : = X+ -X+ , then (V, X+, II* 11)

eGLo, and also (V,X+,(I.II,)EGL~.

Remark. Example (f) includes examples (d) and (e) as particular cases. To see that (e) is a special case of (f) it suffices to put in example (f)

6% < , ‘) = (wo, G, I ),

4 A signed measure x is said to be bolNIded if there is a constant K E R such that Ix(u)1 6 K for all aGL.

’ The proof of the completeness of (X, 11 * ) I) for the case where L is a lattice was given by Kronfli r51 (see alSO [41).

294 W. GUZ

where L(H) is the lattice of all closed subspaces of a separable complex Hilbert space H with the set inclusion E: taken as partial ordering and with the orthogonal complement J_ as orthocomplementation. Then, the correspondence between example (f) and (e) may be established using the well-known Gleason theorem [3] (see also [9]).

4. GL-spaces and d-spaces

Let (X, C, 11.1 I) be an arbitrary GLO-space. Using the GL,-norm I I - I I, one can define another GL,-norm in X by putting:

llxlll := inf{llx,ll+llx,ll: x = x1-xX,; xl, x2 EC>.

We shall show that:

(13) II - II f II - Ill, that is, llxll < IIxllIjiw aNx EX; moreover, II * II and II - II1 coincide on the positive cone C:

llxll = llxlll for’ each x E C.

Proof: Let x E X. Then, for any pair x1, x2 E C such that x = x1 -x2, one has

lbll = II& -x211 G Il~,ll+ll~2ll;

hence 11x1 I < I lxlll. If x E C, then, obviously,

IIXIII = Il-4l+lloll = llxll.

(14) Zf II - 11’ is any norm in X such that llxll’ < llxlll for x E C, then

IIxIJ’ < llxlll for all xEX.

Proof: In fact, let x be an arbitrary element of X. Then, for each pair 3~1, x2 E C

such that x = x1 -x2, we have

llxll’ = IIx1-&II’ < II~III’+II~2ll’ G IlxIllI+ll~2llI

= Ilxlll+ll~211 (see (13)); hence, of course,

ll4l’ =G Il4lI.

As a direct consequence of (14) and (13) one obtains:

(15) I I * II1 is the greatest element in the set of all GL,-norms in (X, C, II * I I>, which

coincide with 1) . ] J on C.

DEP. The norm II - III will be called the dominant-norm of the C&-norm II * II and

will be denoted by (I * /Id. Note that the operation 1) * I I -_* I I * IId defined in the set of all GL,-norms of a’GL,-

space (X, C) possesses the following obvious property:

(16) Zf I I * II and I I - 11’ are two GL,-norms in (X, C) such that 11x1 I G I MI’ for X E c, then llxlld < Ilxll~ for all x EX.

In particular, for any GL,-norm 11 * I I one has

11’ lldd = II ’ iid*


DEF. We shall call the operation I( + II + II - I(., the closure operation, and the GLo- norm satisfying the condition

II ’ II = II ’ lid

will be called closed.

DEF. A GL,-space (X, C, 11 - 11) is said to be a GL-space (in symbols: (X, C, II * II) E GL) if its GL,-norm )I * 1) is closed.

(17) Let (X, C, I I - (I) E GL,, . Then the folIowing four conditions are equivalent:

(i) II ’ II = II ’ Iid;

(ii) II - II = II * 11; for some GL,-norm II * II’;

(iii) Zf II - 11' is any norm in X such that IIxII’ < llxll for x E C, then IIxII’ < llxll for

all x EX;

(iv) Zf 11 - 11’ is any GL,-norm in (X, C) such that (1x1 I’ = 11x1 I for x E C, then 11x1 I’

< I(xIJ for aN xE:X.

Proof: The implications (i) * (ii) and (iii) =E- (iv) are trivial. The implication (ii) => (iii) was shown in (14). To prove the implication (iv) * (i) it suffices to put I I * I I’ = I I * I Id

in (iv). Then, using (iv), we find I I * IId < I I * II ; hence II . IId = II * I). Note that any GL-space possesses the following property, which, as we already seen

(see (1 l)), holds for GL,-spaces:

(18) 0’3 C, II * II) E GL * b, llflle = llfll.

Proofi Let x E X and let x1, x2 be two arbitrary elements of the positive cone C such that x = x1 -x2. Then 2 = i1 -$2, where gl, j;z E C” ( h denotes the canonical injection X ‘5 X,‘), and

11~111+11~211 = 11~111+11~211 = 11~111’+11~211’ (see (10))

hence a ll+-&ll= = Ilily,

llxll = llxlld = inf{lixlIi+llx211: x = x1-x2; xl, x2 E c>

2 11~11’7 which together with (see (9))

leads to lbll 7 iI31 G 1141’

IMI = IIW The remainder part of the proof is simply a repetition of that of (11).

(19) An example: Any GL,-space (A’, C, II - II) is a GL-space, i.e. any GL,-norm is closed.

In fact, let I I - II’ be any norm in X such that IJyl I’ < ) Iyl) for y E C, and let x be an arbitrary element of X. Let x = x l-x2, where x1, x2 EC and Ilx,ll+llx211 = Ilxll. Then Il4l’ = IIXI -4l’ G Ilxlll’+ll~211’ < II~Ill+llx211 = Ilxll, and therefore, applying (13, we get II ’ II = II ’ lid.

296 W. GUZ

Now we shall be interested in an inner characterization of a GL-space, which does not involve the concept of the norm. However, before doing this, some new concepts should

be introduced. Let (X, C) be any partially semiordered real vector space.

DEF. A positive linear functional d E C* is said to be total if the set

is total. [O,d]:= {fEC*: f< d} =x*

Note that when there is a total functional d on X, then the positive cone C of X is obviously regular, and thus the partial semiordering in X induced by C is a partial ordering.

Notice also that any total functional d E C* has the following property:

d(x) = 0, x E C implies x = 0.

In fact, x E C and d(x) = 0 leads to f(x) = 0 for every fo [0, d]; hence x = 0 by the totality of [0, d].

(20) An example: In any GL,-space (X, C, 1) * II) the functional e E C* defined on page 287 is total.

Indeed, for f E C’ - {0} we get

whenever x E c; hence

(Ilfll-‘f>(x) < lbll = 44,

=+-llfll-lf E C*, that is,

Ilf II-‘f G e. Therefore, f(x) = 0 for all f B [O, e] implies f(x) = 0 for all f E C’, hence x = 0, since C’ is total (see (5)).

(21) If d is u totalfunctional on (X, C), where C is a generating cone in X, then, using d,

one can define a norm in X which is additive on C, i.e. a GL,-norm in (X, C).

Namely, it is then sufficient to put by the analogy to the construction II * II --) II * IId

(remember that when (X, C, ]I * 11) E GLO, then d(x) = e(x) = llxll for x E C):

]]x]] := inf(d(x,)+d(x,): x = x1-x2; x1, x2 E C}.

Let us note that

(22) llx]] = d(x) for x E C.

Proof: In fact, let x E C and let x = x1 -x2, where x1, x2 E C. Then d(x,)+d(xz) > d(x,) = d(x+x2) = d(x)+d(x,) > d(x), and d(x)+d(O) = d(x), hence llxll = 44.

(23) The norm 1 I * 1) in X induced by a total functional d E C* is closed, i.e. (X, C, I I * I I) E GL.

Proof: Let us consider an arbitrary norm )I * I I’ in X such that

Ilvll’ G lbll for Y E C,


and let XEX, x = x1-x2, x,,x~EC. Then

Ilxll’ = II%--&II’ < II~1Il’+ll~ZII’ G Il~lll+ll~zll

= c&x,) + d(x,) (see (22)) ;

hence, of course, Ilxll’ < IlxJl. This proves, according to (17), that II * II = II * IId.

DEF. A triple (X, C, d), where X is a vector space over the field R of real numbers, C is a generating cone in X, and d is a total functional on X, will be called a d-space. The GL-norm in Xinduced by d will be called the d-norm of the d-space (X, C, d), and denoted

by II * IId- (24) The d-norm 11. IId of (X, C, d) is the greatest element in the set of all norms II - II

in X satisfying the condition

XEC, d(x) = 1 =P llxll < 1. (4.1)

Proof: Let x E C, x # 0; then d(x) > 0 and x may be written as x = d(x)x’, where x’ = d(x)-‘xE Cn{yeX: d(y) = 11.

Let now II * II be any norm in X having property (4.1); then

llxll = d(x) * I Ix’ll < d(x) = lIdId (see (22));

hence, by (17), llxll < llxlld for all x EX, i.e. II - 11 < I( - (Id.

Swing up our considerations one can write:

(25) To each GL,-,-norm II * II in (X, C), where X = C-C, there corresponds a unique total functional d = d,,.,, E C* such that

d(x) = llxll for XEC, namely, d = e.‘j

Conversely. with each total functional d on (X, C), X = C-C, one can associate a (closed) GL,-norm 11 - II = I j - lIta, such that

llxll = d(x) for x E c, (4.2)

namely,11 * ) I = ) I * IId, but the GL,-norm 1 I - I I satisfying (4.2) may be not unique.’ How- ever, there is a unique closed GL,-norm (I * 1) in (X, C) with

llxll = 44 for XEC;

this is exactZy the d-norm 1 I - II of the d-space (X, C, d).*

Remark. Notice also that the above-described correspondences (I * II -+ d,,.,, and d + II * (lcdj, when the first is restricted to the set of closed GL,-norms (i.e. GL-norms) only, are one-to-one; moreover, they are order isomorphisms, that is,

6 The uniqueness of d is obvious, since for any other linear functional d’ on X such that d’ = I( . (( onCtheequalityd’=(~~))=donCimplicsd’=d.

‘I There exist, in general, more than one GL,,-norm in (X, C) satisfying (4.2~see examples below. * This follows immediately from (22) and (16).

298 W. GUZ

(i) II * II < II * II’ (where II * II, II - II’ E W * d,,.,, G 4,.1,; (ii) d < d’ (where d and d’ are total functionals) o I j * II(d) < I I - I I(d,). We shall now construct some examples of other GL,-norms in the d-space (X, C, d),

which are different from II* IId and coincide with d on C. By an analogy to example (f), the following GL,-norm may be defined in X:

llxll := s~p{f(x): f E P, dl)+sup{ -f(x): f E P, 4). Moreover, the definition above may be generalized as follows9 Let L be a total subset of [0, d] such that L r=, (0, d} ; we put by definition

llX]lL := sup{f(x): _fEl;)+sup(-f(x): f EL}.

Note that for each x E X the set (if(x)1 : f E L} is bounded. In fact, let x = x1 --x2, x1, x2 E C (xl and xz are fIxed) ; then for any fo L

VW = If&d-f(xz)l G fh>+fW G 44 +4-d; hence, for any f E L,

.and thus llxllL < 2(d(x,)+d(x,)). The verification that I I * I IL possesses all properties of the norm, and that ) \ * ) it is ad-

ditive on C, is straightforward. The second, more conventional norm in X which may be defined by using the total

functional d, or, more generally, by using the total set L c [0, dj (L 2 (0, d}), is the ordinary sup-norm defined by

IW := ~UP(lfW fe L).

Keeping in mind that for all x E X and all f e [0, d]

If(x)l f d(xA +d(xJ

whenever x = x1 -x2, x1, xz E C, we find that

IlxV G I141d*

And here the verification of the norm conditions for II + I JL as well as the proof of ad- ditivity of 1) . ]I: on C follows in a straight course.

It can also easily be verified that for each x E X

Ilxll; G IIXIIL G 211~llL.

Moreover, these two norms are identical on C, since

IIXIIL = lMIL = d(x) = llxlld for x E C. (4.3)

Furthermore, from (4.3) it follows easily that [I - (( d is the closure (i.e. the dominant-norm)

for both norms II - 1 IL and 11. IIL. Indeed, equality (4.3) leads immediately (see (16)) to

II’ IILd = 11 * iiLd = II ’ ildd = II ’ Iid-

’ Such a generalization is important in physical applications-see our subsequent paper.


REFERENCES

[l] Alexiewicz, A.: Functional analysis, Warsaw, 1969 (in Polish). [2] Dyukin, E. B.: Markovian processes, Moscow, 1963 (in Russian). [3] Gleason, A. M.: J. Math. Mech. 6 (1957), 88.5. [4] Gudder, S. P.: Znt. J. Theor. Phys. 7 (1973), 205. [S] Krontli, N.: Znt. J. Theor. Phys. 3 (1970), 191. [6] Miles, P. E.: Trans. Amer. Math. Sot. 107 (1963), 217. [7] Ringrose, J. R.: Compact non-self-anjoint operators, Van Nostrand, London, 1971. [8] Schatten, R.: Norm ideals of completeIy continuous operators, Springer, Berlin, 1960. [9] Varadarajan, V. S.: Geometry of quantum theory, Vol. 1, Van Nostrand, Princeton, 1968.

[lo] Yosida, K.: Functional analysis, Springer, Berlin, 1965.

1111 Zierler, N.: Proc. Amer. Math. Sot. 14 (1963), 345.

spaces of the type gm and gl basic properties

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