COUNTEREXAMPLES IN BANACH ALGEBRAS |
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PROBLEMS | SOLUTIONS |
BA1.
(a) A unital Banach algebra, except the algebra of complex numbers, without nontrivial idempotent. (b) A unital Banach algebra with a nontrivial idempotent. |
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BA2. A Banach algebra generated by idempotents i.e. elements x such that x2 = x. |
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BA3.
A compact Hausdorff space X and subalgebras of C(X) satisfying in only three conditions of four following conditions: (a) uniformly closed, (b) separating the points of X, (c) containing constant functions, (d) closed under complex conjugation. |
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BA4. A Banach algebra A such that Rad(A) is a proper subset of the set {x ; r(x) = 0} of all quasi-nilpotent elements. |
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BA5. An algebrically semisimple non-commutative Banach algebra. |
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BA6. A semisimple commutative Banach algebra with a closed two-sided ideal I such that A/I isn't semisimple. |
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BA7. A non-maximal primary ideal in a unital commutative Banach algebra A. |
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BA8. An (algebrically) simple Banach algebra. |
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BA9. A Banach algebra A, a closed subalgebra B of A and an element a e A such that sp(A,a) = sp(B,a). |
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BA10.
(a) A reflexive Banach algebra. (b) A non-reflexive Banach algebra. |
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BA11. An element of a Banach algebra which has no logarithm. |
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BA12. An algebra can not be normed so that it becomes a Banach algebra. |
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BA13. A commutative radical Banach algebra. |
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BA14. An element x of a Banach algebra such r(x) < ||x||. |
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BA15. A commutative Banach algebra A with a unique ideal; i.e. Rad(A). |
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BA16. A Banach algebra A that is a topological direct sum (as a Banach space) of a pair of its Banach subalgebras which are isometrically isomorphic to A. |
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BA17. A Banach algebra with a proper dense two-sided ideal. |
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BA18. A Banach algebra A in which every singular element is a left or right topological divisor of zero. |
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BA19.
Two element a, b of a Banach algebra such that neither r(ab) |
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BA20. A normed algebra with non-open group of invertibles (and so the algebra is not Banach). |
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BA21. A commutative Banach algebra whose unit ball isn't norm compact. |
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BA22. A normed algebra A whose radical is isomorphic to C. |
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BA23.
(a) A separable Banach algebra. (b) A non-separable Banach algebra. |
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BA24. Two non-isomorphic Banach algebras with homeomorphically isomorphic invertible groups. |
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BA25. A commutative Banach algebra whose unit ball has no extreme point (and so it isn't the dual space of any Banach space by the Krein-Milman theorem ). |
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BA26.
(i) A singly generated Banach algebra. (ii) A Banach algebra can not be singly generated. |
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BA27. A Banach algebra without any topological divisor of zero. |
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BA28. A commutative Banach algebra A without any minimal ideals. |
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BA29.
Two elements x,y (xy |
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BA30. A reflexive Banach algebra whose dual is also a Banach algebra. |
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BA31. A Banach algebra A that cannot be a (vector space) direct sum of its radical Rad(A) and a Banach algebra B that is homeomorphically isomorphic with A/Rad(A). |
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BA32. A commutative Banach algebra where 0 is the only nilpotent. |
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BA33. A non-commutative Banach algebra in which 0 is the only quasi-nilpotent. |
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BA34. A non-commutative radical Banach algebra which is an integral domain. |
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BA35. A non-reflexive Banach space isometric with its second conjugate space. |
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BA36. A Banach algebra A with a Banach subalgebra B and an element b e B such that sp(A,b) is a proper subset of sp(B,b). |
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BA37. A Banach algebra with an unbounded approximate identity. |
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BA38. A topologically nilpotent Banach algebra. |
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BA39. A non-topologically nilpotent Banach algebra. |
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BA40. A finite dimensional commutative algebra with nilpotent radical, an identity modulo the radical, but no global identity. |
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BA41. A Banach algebra having no bounded approximate identity. |
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