Representations of positive projections 1 Introduction - Mathematics ...

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$MATH 5b; Solutions to Homework Set 1 January 2009 1. As K is a ...$

Lebesgue measurable subsets. De ne the ideal A in L0 ( ) by ( A = f 2 \ Lp ( ):jfj 1 (01) c1 (01) for some 0 c 2 R 1 p
and, using that Qf is 1-periodic, Z E Qf (x) d (x) = 1X Z k=0 E1+k Qf (x) e ;x dx = In combination with (11) this shows that Z E1 e e ; 1 Qf (x) e;x dx = for all measurable subsets E 1 1X k=0 Z E1 1X k=0 = 1X Z Z k=0 E1 E1 f (x + k) e ;k (0 1). This implies that Qf (x) e ;(x+k) dx e e ; 1 Qf (x) e;x dx: ! e ;x dx f (x + k) e ;k = e Qf (x) (12) e ; 1 -a.e. on (0 1). Now we take in particular the function f 0 2 A de ned by (10). Then 1X k=0 and so it follows from (12) that f 0 (x + k) e ;k = f 0 (x +1)e ;1 = jlog xj e ;1 Qf 0 (x) = e ; 1 e 2 jlog xj -a.e. on (0 1). Hence Qf 0 is not bounded. However, as observed above, all functions in C are bounded, which is a contradiction. This shows that such a projection Q does not exist. Next we will discuss some further notation and properties concerning subalgebras. As above we assume that A is a Dedekind complete f-algebra with unit element 1 and we assume that B is a Dedekind complete regularly embedded f-subalgebra of A with 1 2 B. Hence CB is a complete Boolean subalgebra of CA. For p 2CA we will denote A(p) =fpf : f 2 Ag (13) which is the band in A generated by p. Note that A(p) isanf-algebra in its own right with unit element p. The Boolean algebra of components of p in A(p) is given by CA(p) =fq 2CA : q pg : 27
Page 1 and 2: Representations of positive project
Page 3 and 4: for all f 2 bL. We stress the fact
Page 5 and 6: that K itself is Dedekind ( -) comp
Page 7 and 8: in C (w) and so it follows that f +
Page 9 and 10: Pe (e) =e and Pe (f) =0whenever f 2
Page 11 and 12: (i). L is locally order separable (
Page 13 and 14: this example also shows that the an
Page 15 and 16: 3 Representations Many of the famil
Page 17 and 18: Lemma 3.3 Given a Dedekind -complet
Page 19 and 20: observations that Ker ( ) = bL \ M(
Page 21 and 22: It is readily veri ed that the exis
Page 23 and 24: of 1 by CA = CA (1). Note that CA i
Page 25: for all c 2 C, bytheaveraging prope
Page 29 and 30: As we have seen in Example 2.12, a
Page 31 and 32: Proposition 4.12 Let A beaDedekind
Page 33 and 34: Then there exists a representation
Page 35 and 36: can adjust the values of the functi
Page 37 and 38: so which implies that u P (p) ; 1 1
Page 39 and 40: Proof. First observe that'jK 2 K s
Page 41 and 42: Proof. Let P : Z(L) ! Z(L) be the c
Page 43 and 44: (ii). If e 1e 2 2F such that e 1 ^
Page 45 and 46: Boolean subalgebra such that CAi =
Page 47 and 48: theorem that B 1 K 1 and 1 (g) = (g
Page 49 and 50: A is a Dedekind complete f-algebra
Page 51 and 52: and (34) now follows by induction o
Page 53 and 54: and since P is strictly positive it
Page 55 and 56: Lemma 7.7 If 2 D and 0 < 1, then [q
Page 57 and 58: Since q( ) 2 CB for all 0 1, it is
Page 59 and 60: (ii). there exists a strictly posit
Page 61 and 62: y Proposition 4.2 there exists a st
Page 63 and 64: S (CB [E ). Moreover, there exists
Page 65 and 66: Proposition 8.6 Assume that A, B an
Page 67 and 68: Since B is Dedekind complete, it is
Page 69 and 70: E is a Dedekind complete f-algebra
Page 71 and 72: Then there exists a - nite measure
Page 73: [10] Dorothy Maharam, The represent

Representations of positive projections 1 Introduction - Mathematics ...

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