A primitive A primitive C*-algebra A acting on a Hilbert space H such that AÇA¢ = {0} (A¢ is the commutant of A in B(H)).

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Let H be an infinite dimensional Hilbert space, then K(H) is primitive, since the identity representation
K(H)
®
B(H)
T
\longmapsto
T
is faithful irreducible (if T Î K(H)¢, then for each x in H, T(x[`(Ä)] x) = (x[`(Ä)] x)T. So Tx[`(Ä)] x = x[`(Ä)] T*(x). Hence < x,x > Tx = < x,T*x > x. So Tx = l(x)x for some l(x) Î C. For linearly independent vectors x and y, l(x+y)(x+y) = T(x+y) = Tx+Ty = l(x)x+l(y)y. So l(x+y) = l(x) = l(y). Hence for each e in an orthonormal basis E of H, l(e) = l(e0), where e0 is an arbitrary fixed element of E. Therefore Tx = T(åe Î Eme e) = åe Î Eme l(e0)e = l(e0)x. So T = l(e0) IH. Thus K(H)¢ Í CIH. Obviously CIH Í K(H)¢. So K(H)¢ = CIH). But IH Ï K(H). So K(H)Ç(K(H))¢ = {0}.
Note that for x and y in H the rank one operator x[`(Ä)] y is defined by
(x
Ä
 
y)(z) = < z,y > x.


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On 22 Feb 2001, 00:19.