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Two-sided ideals in $C^*$-algebras

Erling Størmer

1967
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Bulletin of the American Mathematical Society
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If SI is a C*-algebra and 3 and % are uniformly closed two-sided ideals in 21 then so is 3 + $-The following problem has been proposed by J. Dixmier [l, Problem 1.9.12]: is (3+g) + = 3 + +g + , where 8+ denotes the set of positive operators in a family g of operators? He suggested to the author that techniques using the duality between invariant faces of the state space 5(2t) of 2t and two-sided ideals in 2t, as shown by E. Effros, might be helpful in studying it. In this note we shall use such
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... arguments to solve the problem to the affirmative. By a face of 5(21) we shall mean a convex subset F such that if pGF, coG5(2t) and aoeSp for some a>0, then oeÇiF. F is an invariant face if pGF implies the state B->p(A*BA) -p(A*A)~l belongs to F whenever p{A *A) T^O and A G2Ï. We denote by F L the set of operators A&BL such that p(A)=0 for all p£F. If 3C2T, 3 X shall denote the set of states p such that p(A)=0 for all ^G3-E. Effros [2] has shown that the map 3->3 X is an order inverting bijection between uniformly closed two-sided ideals of 21 and w*-closed invariant faces of 5(21). Moreover, (Q ± ) ± = ^i and (F ± ) ± = F when F is a w*-closed invariant face. If 3 and g are uniformly closed two-sided ideals in 21 then (3Pig) x = conv(3 ! -L , S" 1 ), the convex hull of 3 X and g\ and (3 + S) x = 3 J *^S J ". If A is a self-adjoint operator in 21 let denote the w*-continuous affine function on 5(21) defined by Â(p)=p(A). It has been shown by R. Kadison, [3] and [4], that the map A-+ is an isometric order-isomorphism of the self-adjoint part of 2Ï onto all w*-continuous real affine functions on 5(21). Moreover, if 3 is a uniformly closed two-sided ideal in 2Ï, and if/ is the canonical homomorphism of 21 onto 21/3» then the map p-^po\f/ is an affine isomorphism of 5(21/3) onto 3 X -Thus the map yp{A)-^A\ 3 X is an orderisomorphic isometry on the self-adjoint operators in 21/3-We shall below make extensive use of these facts. For other references see [i, §i]. THEOREM. Let %be a C*-algebra. If 3 and % are uniformly closed two-sided ideals in 21 then (3 + g) + -3 + + S + -In order to prove the theorem we may assume 21 has an identity, denoted by /. We first prove a 254

doi:10.1090/s0002-9904-1967-11705-1
fatcat:5oqqqprwkzhrvgsmapapbxcywy