TY - JOUR

T1 - Ideals of a C*-algebra generated by an operator algebra

AU - Juschenko, Kate

PY - 2010/1/1

Y1 - 2010/1/1

N2 - In this paper, we consider ideals of a C*-algebra C*(B) generated by an operator algebra B. A closed ideal J ⊆ C*(B) is called a K-boundary ideal if the restriction of the quotient map on B has a completely bounded inverse with cb-norm equal to K-1. For K = 1 one gets the notion of boundary ideals introduced by Arveson. We study properties of the K-boundary ideals and characterize them in the case when operator algebra λ-norms itself. Several reformulations of the Kadison similarity problem are given. In particular, the affirmative answer to this problem is equivalent to the statement that every bounded homomorphism from C*(B) onto B which is a projection on B is completely bounded. Moreover, we prove that Kadison's similarity problem is decided on one particular C*-algebra which is a completion of the *-double of M2(ℂ).

AB - In this paper, we consider ideals of a C*-algebra C*(B) generated by an operator algebra B. A closed ideal J ⊆ C*(B) is called a K-boundary ideal if the restriction of the quotient map on B has a completely bounded inverse with cb-norm equal to K-1. For K = 1 one gets the notion of boundary ideals introduced by Arveson. We study properties of the K-boundary ideals and characterize them in the case when operator algebra λ-norms itself. Several reformulations of the Kadison similarity problem are given. In particular, the affirmative answer to this problem is equivalent to the statement that every bounded homomorphism from C*(B) onto B which is a projection on B is completely bounded. Moreover, we prove that Kadison's similarity problem is decided on one particular C*-algebra which is a completion of the *-double of M2(ℂ).

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U2 - 10.1007/s00209-009-0594-8

DO - 10.1007/s00209-009-0594-8

M3 - Article

AN - SCOPUS:84655177791

VL - 266

SP - 693

EP - 705

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 3

ER -