gelfandtheorem implies ringsdownloads.hindawi.com/journals/ijmms/1995/397684.pdf · 2019-08-01 ·...

Internat. J. Math. & Math. Sci.VOL. 18 NO. 4 (1995) 701-704

701

GELFAND THEOREM IMPLIESSTONE REPRESENTATION THEOREM OF BOOLEAN RINGS

PARFENY P. SAWOROTNOW

Department of MathematicsThe Catholic University of America.

Washington, DC 20064United States of America

(Received June 27, 1994 and in revised form September 9, 1994)

ABSTRACT. Stone Theorem about representing a Boolean algebra in terms of open-closed sub-

sets of a topological space is a consequence of the Gelfand Theorem about representing a B*-

algebra as the algebra of continuous functions on a compact Hausdorff space.

KEY WORDS AND PHRASES. Banach algebra, B*-algebra, Boolean algebra, Boolean ring,

Stone Representation Theorem.

1991 AMS SUBJECT CLASSIFICATION CODES. Primary 06E15, 06E05, 46J25; Secondary

03G05, 46H15, 46J10

1. INTRODUCTION.

Gelf,’md Theorem in the title is the Representation Theorem for commutative normed rings

with an involution, which caa be stated as follows (in American terminology):

Representation Theorem of Gelfand. For each commutative (complex) B*-algebra/3, with iden-

tity, there exists a compact Hausdorff space S such that /3 is isomorphic and isometric to the

algebra C(S) of all continuous complex valued functions on S (see, for example, first corollary to

the Theorem in 26E of Loomis [3] or Theorem A in section 73 of Simmons [7]).The isomorphism preserves also the involution, i.e. a* , where fi is the member of C(S)

corresponding to a (and a"* corresponds t.o a*).

{A B*-algebra (Sec. 72 of Simmons [7]) is a Banach algebra/3 with an involution x z* such

that x*x II=ll x 11 for all z E B}.

2. MAIN RESULT.

In the sequel we shall establish validity of the following proposition:

THEOREM. Representation Theorem of Gelfand implies Stone’s Representation Theorem for

Boolean Rings (see Appendix Three of Simmons [7])"

702 P. P. SANOROTNON

For each Boolean ring .A there exists a totally disconnected compact Hausdorff space S such that

,4 is isomorl)hic to the Boolean ring A(S) of all open-closed subsets of S (the operations of

are the symmetric difference AAB (Aq B U (A qB and intersection A N B, A C S, B C S).PROOF. Assume validity of Gelfand Theorem. Let and - be operations of .A and let

and 0 be, respectively, its multiplicative and additive identities. We shall say that a finite set

{a a, of members of A is a decomposition of identity if aa @ a a, 1 (below we

shall write a, 1) and a, (R) a 0 if j.

For each decomposition {a, a,} of identity and each set {A1 A,} of complex num-

bers consider a formal sum f Z Aa. Let/3 be the class of all such formal sums. Let us use,=1

the following notation: we shall write [.f] {ha a,} and A, A(a,) if f A,a, is any

member of/’.

Define relation ",-" on B’ as follows: f g if A(a) A(b) for any a e If] and b [g] such

that a . b = 0. It is an equivalence relation. In fact, we only need to prove reflexivity.

Let f g and g h, where f A,a,, g -]pb. and h u,ck. Assume that

a, )c y 0, then from the fact that @b we conclude that there is some integer j such that

a, c, (:_ b # 0. Then both a, (R) b # 0 and b )c # 0 (note that b , b b), which implies

A Pl Uk.

Now let us define addition, nmltiplication, multiplication with complex numbers and involu-

tion on B’ as follows: If f A,a,, g #zb and A is a scalar then

and

fg Z A,#.a, ) b. (2.2)

It is easy to see that these operations are invariant under the relation ",-," i.e. "f g" implies

"f + h g + h, fh gh and Af Ag." For example, if f, g d h are above d f g, then

f + h (A, + v)a, c d g + h ( + u)b c. If (a, c) (b c,) # O, then,k

a, b 0 d k k’. This implies that A, , and from this we conclude that f + h g + h}.Dee the semi-norm on ’ by setting f []= max{] A(a) ]" a If], a 0}. Also it is

ey to s that ]] is invariant under the relation "" i.e. f g implies f }]:] g ]}.Let be the collection of a equivalence closes with respect o "". Then B is normed

linear algebra with respect to the operations induced by operations on ’ (we shall use

notation), the ditive identity 0 of is the set of all members of ;’ of the form f

STONE REPRESENTATION THEOREM OF BOOLEAN RINGS 703

vhere {a, a,, is some decomposition ,,f identit.v and A, 0 for 1 n. Also f 0

if and only if f =0. and it is not difficult to show that .fg I1-<11 f II" for all f, 9

Let B be the completion of B with respect to II. Then B is a commutative B*-algebra with

identity.

Apply Representation Theorem of Gelfand to B" There exists a compact Hausdorff space

S such that B is *-isomorphic and isometric to the algebra C(S) of all continuous complex

valued functions on S. For each f B let .t:(s) denote the corresponding image of f under this

isomorphism. Isometry between B and C(S) means that f sup ]/(s) ].s_S

Now note that there is a natural imbedding of the Boolean ring .A into B: for each a .A let

f, 1-a + 0-a’, where a’ 1 + o is the complement of a in A. (Note that A has also a structure

of a Boolean algebra (see Appendix Three of Simmons [7]).) Then fafa fa, from which we

conclude that ]a(s) assumes either 1 or 0 at any ._ S. Let A {s S’fa(s) 1}, then

is the characteristic functim of A i.e. fa .4, and it follows from continuity of fa that A is

both open and closed in S. The correspondence a A is 1-1 and preserves both lattice and

algebraic operations of/3. A simplest way to establish this is to show that this correspondence

preserves multiplication and colnplementation. But both facts follow easily from the identities

"f(R) ff" and "faG’ 0" (a, b A and a’ a 1).

It remains to show that S is totally disconnected. Since S is a Hausdorff space, we need

only to show that for any open neighborhood U of so S there exists an open-closed set O

such that So O C U. Since every compact T2 space is normal, there exists a continuous real

valued function x(s) (a member of C(S)) such that X(So) 0, z(S U) 1 and 0 < z(s) <_ 1

everywhere else (see Urysohn’s Lemnla in Sec. 3, Chap. 8 of Royden [6] or 3C in Loomis [3]).Let x /3 be such that 2(s)= x(s) nd lt f e B’, f a,a, be such that f- I1< . From

" a, and a, (: a 0 if # j" we conclude that sets A, A2 A, (corresponding to

a, a.,..., a,, under above discussed correspondence a A) are disjoint and U A, S. Hence

there exists exactly one index j (1, 2,..., n) such that so A. The set A is both open and

closed and s A implies ](s) ](so), from which we conclude that A C U: if s A,then Ix(s)I<_l z(s)- () / f(o) x(o) / x(o)I< 1, and this implies that s e U.

To see that each open-closed subset A of S corresponds to some a .,4 we use compactness of

S, which inplies compactness of A. As above, for each s A we select an open-closed set O such

that s O C A. Compactness of A implies that there is a finite set {A1, Am of open-closed

subsets of S, each corresponding to sone a, ,4 (i 1... m), such that A I..J,"= A,. This

implies that A corresponds to sone a A (a a U a... U a,).

REFERENCES

1. GELFAND, I.M., RAIKOV, D.A. and ILOV, G.E. Commutative Normed Rings, Moscow1960.

704 P. P. SAWOROTNOW

2. KOPPELBERT, S. Handbok of Boolean Algebras, Vol. 1, North-Holland, 1989.

3. LOOMIS, L.H., An Introduction to Abstract Harmonic Analysis, Van Nostrmxd, 1953.

4. MACKEY, G.W., Commutative Banach Algebras, lecture notes, Harvard University, 1952.

5. NAIMARK, M. A., Normed Rings, Publisher "Nauka", Moscow, 1968.

6. ROYDEN, H.L., ..R.ea! Analysis, MacMillen Publishing Co., New York, 1988.

7. SIMMONS, George F., Introduction to Topology and h,lodern Analysis, McGraw-Hill, 1963.

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