COUNTEREXAMPLES IN C*-ALGEBRAS AND W*-ALGEBRAS |
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PROBLEMS | SOLUTIONS |
CW1. A construction of a bounded approximate identity for a commutative C*-algebra A. |
| CW2.
Two element x,y in a C*-algebra A such that sp(xy) |
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CW3. An involutive Banach algebra A which isn't a C*-algebra. |
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CW4.
An involution # on Banach algebra M4(C), two normal matrix T and S such that TS=ST
but TS# |
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CW5. A Banach algebra with a unique C*-involution. |
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CW6. A C*-algebra in which invertible elements are dense. |
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CW7. A liminal C*-algebra which isn't postliminal. |
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CW8. A closed subalgebra of a C*-algebra that isn't self-adjoint. |
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CW9. A closed left ideal of a C*-algebra without any left approximate identity. |
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CW10. A nonclosed ideal that is not self-adjoint in a commutative C*-algebra. |
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CW11. A closed ideal I of a commutative C*-algebra A and a closed ideal J of I such that J isn't an ideal of A. |
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CW12. A C*-algebra A where every unitary element is of the form exp(ih) for a self-adjoint h e A. |
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CW13. A C*-algebra that isn't a von Neumann algebra. |
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CW14. A C*-algebra A in which the closed unit ball of A+ isn't the closed convex hull of the projections of A. |
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CW15. A primitive C*-algebra with a unique nontrivial closed bi-ideal (and so that it is not simple). |
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CW16. A non-separable von Neumann algebra with a (unique) separable closed *-bi-ideal. |
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CW17. A primitive C*-algebra A acting on a Hilbert space H such that the intersection of A and A' is {0}. (A' is the commutant of A in B(H)). |
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CW18. A non-primitive C*-algebra. |
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CW19. A simple C*-algebra. |
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CW20. A non-unital C*-algebra with compact primitive ideal space. |
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CW21. A non-liminal (CCR) C*-algebra. |
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CW22. A C*-algebra A and a closed bi-ideal I of A such that A/I and I are liminal, but A is not limnial. |
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