ALL COUNTEREXAMPLES

ALL COUNTEREXAMPLES

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

COUNTEREXAMPLES IN BANACH AND HILBERT SPACES

BH1. A Banach space with a non-complemented closed subspace.	BH1.dvi BH1.htm BH1.ps BH1.pdf
BH2. A complete metrizable linear space whose metric cannot be obtained from a norm.	BH2.dvi BH2.htm BH2.ps BH2.pdf
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.	BH3.dvi BH3.htm BH3.ps BH3.pdf
BH4. A Banach space X such that all its closed subspaces are complemented.	BH4.dvi BH4.htm BH4.ps BH4.pdf
BH5. A Banach space which isn't metrizable in weak topology.	BH5.dvi BH5.htm BH5.ps BH5.pdf
BH6. A Banach space which is not an inner product space.	BH6.dvi BH6.htm BH6.ps BH6.pdf
BH7. An incomplete inner product space.	BH7.dvi BH7.htm BH7.ps BH7.pdf
BH8. Two closed densely defined operators T and S on a Hilbert space such that T + S isn't closable.	BH8.dvi BH8.htm BH8.ps BH8.pdf
BH9. A Hilbert space whose Hamel dimension and Hilbert dimension are different.	BH9.dvi BH9.htm BH9.ps BH9.pdf
BH10. A nonclosable unbounded operator on a Hilbert space.	BH10.dvi BH10.htm BH10.ps BH10.pdf
BH11. On a separable infinite dimensional Banach space X there exists another norm under which A isn't separable.	BH11.dvi BH11.htm BH11.ps BH11.pdf

*COUNTEREXAMPLES IN C-ALGEBRAS AND W-ALGEBRAS*

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 M₄(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

COUNTEREXAMPLES IN OPERATOR THEORY

OT1. An operator of index zero which isn't invertible.	OT1.dvi OT1.htm OT1.ps OT1.pdf
OT2. A compact operator with no eigenvalues.	OT2.dvi OT2.htm OT2.ps OT2.pdf
OT3. A week-operator closed subalgebra B of bounded operators on a Hilbert space H such that B B", where B" denotes the doubel commutant of B.	OT3.dvi OT3.htm OT3.ps OT3.pdf
OT4. A unitary operator U acting on a Hilbert space whose spectrum is C = {z e C; \|z\| = 1 }.	OT4.dvi OT4.htm OT4.ps OT4.pdf
OT5. An unbounded symmetric operator on an inner product space.	OT5.dvi OT5.htm OT5.ps OT5.pdf
OT6. Two selfadjoint operators T and S on a Hilbert space such that sp(ST) is not a subset of R.	OT6.dvi OT6.htm OT6.ps OT6.pdf
OT7. Two Hermetian operators T and S on a Hilbert space such that S 0 and -S T S but not \|T\| S.	OT7.dvi OT7.htm OT7.ps OT7.pdf
OT8. A selfadjoint operator T 0 on a Hilbert space such that T is neither positive nor negative.	OT8.dvi OT8.htm OT8.ps OT8.pdf
OT9. A bounded operator on a Hilbert space which has no square root.	OT9.dvi OT9.htm OT9.ps OT9.pdf
OT10. A bounded increasing sequence of self-adjoint operators on a Hilbert space which is not uniformly convergent.	OT10.dvi OT10.htm OT10.ps OT10.pdf
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.	OT11.dvi OT11.htm OT11.ps OT11.pdf
OT12. Operators of arbitrary large norms that are bounded by 1 on a given basis of a separable infinite dimensional Hilbert space H.	OT12.dvi OT12.htm OT12.ps OT12.pdf
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.	OT13.dvi OT13.htm OT13.ps OT13.pdf
OT14. An operator T on a Hilbert space such that the set eig(T) of all eigenvalues of T is empty but sp(T) f.	OT14.dvi OT14.htm OT14.ps OT14.pdf
OT15. A sequence of quasi-nilpotent operators acting on a Hilbert space with a norm limit whose spectral radius is 1.	OT15.dvi OT15.htm OT15.ps OT15.pdf
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.	OT16.dvi OT16.htm OT16.ps OT16.pdf
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.	OT17.dvi OT17.htm OT17.ps OT17.pdf
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) {0}.	OT18.dvi OT81.htm OT18.ps OT18.pdf
OT19. Two positive operators T S acting on a Hilbert sace such that S² does not majorize T².	OT19.dvi OT19.htm OT19.ps OT19.pdf
OT20. An unbounded operator on a Hilbert space H annihilating an orthonormal basis of H.	OT20.dvi OT20.htm OT20.ps OT20.pdf
OT21. An operator U on a Hilbert space, other than I, such that sp(U) = {1} and \|\|U\|\| = 1.	OT21.dvi OT21.htm OT21.ps OT21.pdf
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.	OT22.dvi OT22.htm OT22.ps OT22.pdf

COUNTEREXAMPLES IN TOPOLOGICAL HOMOLOGY

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) 0.	TH1.dvi TH1.htm TH1.ps TH1.pdf
TH2. A non-split short complex of Banach spaces whose dual splits.	TH2.dvi TH2.htm TH2.ps TH2.pdf
TH3. A weakly amenable commutative Banach algebra which is not amenable.	TH3.dvi TH3.htm TH3.ps TH3.pdf
TH4. A derivation on an algebra which is not inner.	TH4.dvi TH4.htm TH4.ps TH4.pdf
TH5. A closed unbounded -derivation on a C-algebra A.	TH5.dvi TH5.htm TH5.ps TH5.pdf
TH6. A Banach algebra for which every linear operator is a derivation.	TH6.dvi TH6.htm TH6.ps TH6.pdf
TH7. A non-closable unbounded *-derivation.	TH7.dvi TH7.htm TH7.ps TH7.pdf

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