COUNTEREXAMPLES IN BANACH ALGEBRAS

COUNTEREXAMPLES IN BANACH ALGEBRAS

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.	BA1.dvi BA1.htm BA1.ps BA1.pdf
BA2. A Banach algebra generated by idempotents i.e. elements x such that x² = x.	BA2.dvi BA2.htm BA2.ps BA2.pdf
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.	BA3.dvi BA3.htm BA3.ps BA3.pdf
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.	BA4.dvi BA4.htm BA4.ps BA4.pdf
BA5. An algebrically semisimple non-commutative Banach algebra.	BA5.dvi BA5.htm BA5.ps BA5.pdf
BA6. A semisimple commutative Banach algebra with a closed two-sided ideal I such that A/I isn't semisimple.	BA6.dvi BA6.htm BA6.ps BA6.pdf
BA7. A non-maximal primary ideal in a unital commutative Banach algebra A.	BA7.dvi BA7.htm BA7.ps BA7.pdf
BA8. An (algebrically) simple Banach algebra.	BA8.dvi BA8.htm BA8.ps BA8.pdf
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).	BA9.dvi BA9.htm BA9.ps BA9.pdf
BA10. (a) A reflexive Banach algebra. (b) A non-reflexive Banach algebra.	BA10.dvi BA10.htm BA10.ps BA10.pdf
BA11. An element of a Banach algebra which has no logarithm.	BA11.dvi BA11.htm BA11.ps BA11.pdf
BA12. An algebra can not be normed so that it becomes a Banach algebra.	BA12.dvi BA12.htm BA12.ps BA12.pdf
BA13. A commutative radical Banach algebra.	BA13.dvi BA13.htm BA13.ps BA13.pdf
BA14. An element x of a Banach algebra such r(x) < \|\|x\|\|.	BA14.dvi BA14.htm BA14.ps BA14.pdf
BA15. A commutative Banach algebra A with a unique ideal; i.e. Rad(A).	BA15.dvi BA15.htm BA15.ps BA15.pdf
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.	BA16.dvi BA16.htm BA16.ps BA16.pdf
BA17. A Banach algebra with a proper dense two-sided ideal.	BA17.dvi BA17.htm BA17.ps BA17.pdf
BA18. A Banach algebra A in which every singular element is a left or right topological divisor of zero.	BA18.dvi BA18.htm BA18.ps BA18.pdf
BA19. Two element a, b of a Banach algebra such that neither r(ab) r(a)r(b) nor r(a+b) r(a)r(b).	BA19.dvi BA19.htm BA19.ps BA19.pdf
BA20. A normed algebra with non-open group of invertibles (and so the algebra is not Banach).	BA20.dvi BA20.htm BA20.ps BA20.pdf
BA21. A commutative Banach algebra whose unit ball isn't norm compact.	BA21.dvi BA21.htm BA21.ps BA21.pdf
BA22. A normed algebra A whose radical is isomorphic to C.	BA22.dvi BA22.htm BA22.ps BA22.pdf
BA23. (a) A separable Banach algebra. (b) A non-separable Banach algebra.	BA23.dvi BA23.htm BA23.ps BA23.pdf
BA24. Two non-isomorphic Banach algebras with homeomorphically isomorphic invertible groups.	BA24.dvi BA24.htm BA24.ps BA24.pdf
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 ).	BA25.dvi BA25.htm BA25.ps BA25.pdf
BA26. (i) A singly generated Banach algebra. (ii) A Banach algebra can not be singly generated.	BA26.dvi BA26.htm BA26.ps BA26.pdf
BA27. A Banach algebra without any topological divisor of zero.	BA27.dvi BA27.htm BA27.ps BA27.pdf
BA28. A commutative Banach algebra A without any minimal ideals.	BA28.dvi BA28.htm BA28.ps BA28.pdf
BA29. Two elements x,y (xy yx) of a Banach algebra A such that e^x.e^y e^x+y.	BA29.dvi BA29.htm BA29.ps BA29.pdf
BA30. A reflexive Banach algebra whose dual is also a Banach algebra.	BA30.dvi BA30.htm BA30.ps BA30.pdf
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).	BA31.dvi BA31.htm BA31.ps BA31.pdf
BA32. A commutative Banach algebra where 0 is the only nilpotent.	BA32.dvi BA32.htm BA32.ps BA32.pdf
BA33. A non-commutative Banach algebra in which 0 is the only quasi-nilpotent.	BA33.dvi BA33.htm BA33.ps BA33.pdf
BA34. A non-commutative radical Banach algebra which is an integral domain.	BA34.dvi BA34.htm BA34.ps BA34.pdf
BA35. A non-reflexive Banach space isometric with its second conjugate space.	BA35.dvi BA35.htm BA35.ps BA35.pdf
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).	BA36.dvi BA36.htm BA36.ps BA36.pdf
BA37. A Banach algebra with an unbounded approximate identity.	BA37.dvi BA37.htm BA37.ps BA37.pdf
BA38. A topologically nilpotent Banach algebra.	BA38.dvi BA38.htm BA38.ps BA38.pdf
BA39. A non-topologically nilpotent Banach algebra.	BA39.dvi BA39.htm BA39.ps BA39.pdf
BA40. A finite dimensional commutative algebra with nilpotent radical, an identity modulo the radical, but no global identity.	BA40.dvi BA40.htm BA40.ps BA40.pdf
BA41. A Banach algebra having no bounded approximate identity.	BA41.dvi BA41.htm BA41.ps BA41.pdf

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