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 ex.ey ex+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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