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