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. |
CW1.dvi CW1.htm CW1.ps CW1.pdf |
CW2. Two element x,y in a C*-algebra A such that sp(xy) sp(yx). |
CW2.dvi CW2.htm CW2.ps CW2.pdf |
CW3. An involutive Banach algebra A which isn't a C*-algebra. |
CW3.dvi CW3.htm CW3.ps CW3.pdf |
CW4. An involution # on Banach algebra M_{4}(C), two normal matrix T and S such that TS=ST but TS^{#} S^{#}T, S+T isn't normal and ||SS^{#}|| ||S||^{2}. |
CW4.dvi CW4.htm CW4.ps CW4.pdf |
CW5. A Banach algebra with a unique C*-involution. |
CW5.dvi CW5.htm CW5.ps CW5.pdf |
CW6. A C*-algebra in which invertible elements are dense. |
CW6.dvi CW6.htm CW6.ps CW6.pdf |
CW7. A liminal C*-algebra which isn't postliminal. |
CW7.dvi CW7.htm CW7.ps CW7.pdf |
CW8. A closed subalgebra of a C*-algebra that isn't self-adjoint. |
CW8.dvi CW8.htm CW8.ps CW8.pdf |
CW9. A closed left ideal of a C*-algebra without any left approximate identity. |
CW9.dvi CW9.htm CW9.ps CW9.pdf |
CW10. A nonclosed ideal that is not self-adjoint in a commutative C*-algebra. |
CW10.dvi CW10.htm CW10.ps CW10.pdf |
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. |
CW11.dvi CW11.htm CW11.ps CW11.pdf |
CW12. A C*-algebra A where every unitary element is of the form exp(ih) for a self-adjoint h e A. |
CW12.dvi CW12.htm CW12.ps CW12.pdf |
CW13. A C*-algebra that isn't a von Neumann algebra. |
CW13.dvi CW13.htm CW13.ps CW13.pdf |
CW14. A C*-algebra A in which the closed unit ball of A^{+} isn't the closed convex hull of the projections of A. |
CW14.dvi CW14.htm CW14.ps CW14.pdf |
CW15. A primitive C*-algebra with a unique nontrivial closed bi-ideal (and so that it is not simple). |
CW15.dvi CW15.htm CW15.ps CW15.pdf |
CW16. A non-separable von Neumann algebra with a (unique) separable closed *-bi-ideal. |
CW16.dvi CW16.htm CW16.ps CW16.pdf |
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)). |
CW17.dvi CW17.htm CW17.ps CW17.pdf |
CW18. A non-primitive C*-algebra. |
CW18.dvi CW18.htm CW18.ps CW18.pdf |
CW19. A simple C*-algebra. |
CW19.dvi CW19.htm CW19.ps CW19.pdf |
CW20. A non-unital C*-algebra with compact primitive ideal space. |
CW20.dvi CW20.htm CW20.ps CW20.pdf |
CW21. A non-liminal (CCR) C*-algebra. |
CW21.dvi CW21.htm CW21.ps CW21.pdf |
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. |
CW22.dvi CW22.htm CW22.ps CW22.pdf |