ALL COUNTEREXAMPLES |
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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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COUNTEREXAMPLES IN BANACH AND HILBERT SPACES |
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BH1. A Banach space with a non-complemented closed subspace. |
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BH2. A complete metrizable linear space whose metric cannot be obtained from a norm. |
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BH3. Two non-isometrically isomorphic spaces with the same duals. So that a such dual space could not be a W*-algebra under any multiplication and involution. |
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BH4. A Banach space X such that all its closed subspaces are complemented. |
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BH5. A Banach space which isn't metrizable in weak topology. |
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BH6. A Banach space which is not an inner product space. |
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BH7. An incomplete inner product space. |
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BH8. Two closed densely defined operators T and S on a Hilbert space such that T + S isn't closable. |
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BH9. A Hilbert space whose Hamel dimension and Hilbert dimension are different. |
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BH10. A nonclosable unbounded operator on a Hilbert space. |
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BH11. On a separable infinite dimensional Banach space X there exists another norm under which A isn't separable. |
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COUNTEREXAMPLES IN C*-ALGEBRAS AND W*-ALGEBRAS |
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CW1. A construction of a bounded approximate identity for a commutative C*-algebra A. |
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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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COUNTEREXAMPLES IN OPERATOR THEORY |
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OT1. An operator of index zero which isn't invertible. |
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OT2. A compact operator with no eigenvalues. |
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OT3.
A week-operator closed subalgebra B of bounded operators on a Hilbert space
H such that B |
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OT4. A unitary operator U acting on a Hilbert space whose spectrum is C = {z e C; |z| = 1 }. |
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OT5. An unbounded symmetric operator on an inner product space. |
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OT6. Two selfadjoint operators T and S on a Hilbert space such that sp(ST) is not a subset of R. |
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OT7.
Two Hermetian operators T and S on a Hilbert space such that S |
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OT8.
A selfadjoint operator T |
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OT9. A bounded operator on a Hilbert space which has no square root. |
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OT10. A bounded increasing sequence of self-adjoint operators on a Hilbert space which is not uniformly convergent. |
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OT11. Given a compact subset K of C, there exists a bounded operator T on a Hilbert space such that sp(T) = K and the set of eigenvalues of T is dense in K. |
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OT12. Operators of arbitrary large norms that are bounded by 1 on a given basis of a separable infinite dimensional Hilbert space H. |
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OT13. Given a compact subset K of C being the closure of its interior, there exists an operator T acting on a Hilbert space H such that sp(T) = K and T has no eigenvalue. |
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OT14.
An operator T on a Hilbert space such that the set eig(T) of all eigenvalues
of T is empty but sp(T) |
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OT15. A sequence of quasi-nilpotent operators acting on a Hilbert space with a norm limit whose spectral radius is 1. |
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OT16. A sequence of nilpotent operators on H which converges with respect to the norm topology on B(H) to an operator which is not topologically nilpotent. |
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OT17.
(a) A Banach space X and an operator T e B(X) having no nontrivial invariant subspace. (b) A Banach space X and an operator T e B(X) having a nontrivial invariant subspace. |
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OT18.
(a) An injective operator on a Hilbert space H such that the range of T, R(T), isn't dense in H. (b) An operator S that is surjective but Ker(S) |
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OT19.
Two positive operators T |
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OT20. An unbounded operator on a Hilbert space H annihilating an orthonormal basis of H. |
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OT21. An operator U on a Hilbert space, other than I, such that sp(U) = {1} and ||U|| = 1. |
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OT22.
A Hilbert space H such that on B(H) (i) the involution isn't continuous with respect to the strong operator topology; (ii) the weak operator topology and the strong operator topology are different; (iii) the operator norm is not continuous with respect to the strong operator topology and so the weak operator topology; (iv) the weak operator topology and the strong operator topology aren't metrizable; (v) the operation multiplication is continuous in neither weak nor strong operator topology. |
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COUNTEREXAMPLES IN TOPOLOGICAL HOMOLOGY |
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TH1.
A unital commutative Banach algebra with a maximal ideal M of codimension 1 and a Banach A-module X such that H2(A,X) = 0 but H2(M,X) |
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TH2. A non-split short complex of Banach spaces whose dual splits. |
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TH3. A weakly amenable commutative Banach algebra which is not amenable. |
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TH4. A derivation on an algebra which is not inner. | |
TH5. A closed unbounded *-derivation on a C*-algebra A. |
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TH6. A Banach algebra for which every linear operator is a derivation. |
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TH7. A non-closable unbounded *-derivation. |
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