TRB PG Assistant - Banach algebras - regular and singular elements, topological divisors of zero, spectrum of an element, spectral radius formula, radical and semisimplicity Online Test (MCQ Quiz)

1. In a Banach algebra A, what is a regular element?

A) An element x such that x*x = e B) An element x for which there exists an element y such that xy = yx = e C) An element x such that x^n = 0 for some positive integer n D) An element x for which x + a = 0 for some element a in A

2. An element x in a Banach algebra A is called singular if it is not regular. What is an equivalent condition for an element x to be singular?

A) x has a multiplicative inverse B) x is nilpotent C) The set of elements with multiplicative inverses does not contain x D) x is idempotent

3. What are topological divisors of zero in a Banach algebra A?

A) Elements x such that x*y = 0 for some non-zero y B) Elements x such that there exists a sequence (y_n) of elements in A with ||y_n|| = 1 and lim_{n->inf} xy_n = 0 C) Elements x such that x is not invertible D) Elements x such that x^2 = x

4. If x is a topological divisor of zero in a Banach algebra A, what can be said about its norm?

A) ||x|| must be 1 B) ||x|| must be 0 C) ||x|| can be any positive real number D) ||x|| must be strictly positive

5. Consider the Banach algebra C(X) of continuous complex-valued functions on a compact Hausdorff space X. An element f in C(X) is a topological divisor of zero if and only if:

A) f is the zero function B) f(x) = 1 for all x in X C) There exists a point x_0 in X such that f(x_0) = 0 D) f(x) is non-zero for all x in X

6. Let A be a Banach algebra. The spectrum of an element x in A, denoted by sigma(x), is defined as:

A) The set of all complex numbers lambda such that x - lambda*e is nilpotent B) The set of all complex numbers lambda such that x - lambda*e is a topological divisor of zero C) The set of all complex numbers lambda such that x - lambda*e is singular D) The set of all complex numbers lambda such that x - lambda*e has a multiplicative inverse

7. Which of the following is always true for the spectrum sigma(x) of an element x in a Banach algebra A?

A) sigma(x) is an empty set B) sigma(x) is a singleton set C) sigma(x) is a non-empty compact subset of the complex plane D) sigma(x) is a non-empty open subset of the complex plane

8. Let A be a commutative Banach algebra with identity e. If x is an element of A, then lambda is in sigma(x) if and only if:

A) x - lambda*e is invertible B) x - lambda*e is not invertible C) x - lambda*e is a topological divisor of zero D) x - lambda*e is nilpotent

9. What is the spectral radius of an element x in a Banach algebra A, denoted by r(x)?

A) The supremum of the absolute values of elements in sigma(x) B) The infimum of the absolute values of elements in sigma(x) C) The maximum value of the real parts of elements in sigma(x) D) The minimum value of the imaginary parts of elements in sigma(x)

10. The spectral radius formula for an element x in a Banach algebra A states that r(x) is equal to:

A) lim_{n->inf} ||x^n||^(1/n) B) lim_{n->inf} ||x^n|| C) sup_{n} ||x^n||^(1/n) D) inf_{n} ||x^n||^(1/n)

11. Which property of the spectral radius formula is crucial for its validity in Banach algebras?

A) The completeness of the algebra B) The existence of an identity element C) The boundedness of the norm D) The submultiplicative property of the norm

12. If A is a C*-algebra, what is the relationship between the spectral radius r(x) and the norm ||x|| for any element x in A?

A) r(x) = ||x|| B) r(x) <= ||x|| C) r(x) >= ||x|| D) r(x) = 0

13. Consider the algebra of complex n x n matrices, M_n(C), with the operator norm. What is the spectrum of a matrix A?

A) The set of eigenvalues of A B) The set of singular values of A C) The set of diagonal entries of A D) The set of trace of A

14. For a matrix A in M_n(C), its spectral radius r(A) is:

A) The maximum absolute value of its eigenvalues B) The minimum absolute value of its eigenvalues C) The sum of its eigenvalues D) The product of its eigenvalues

15. In a Banach algebra A, the radical of A, denoted by rad(A), is defined as:

A) The set of all nilpotent elements in A B) The set of all elements x in A such that x*x = x C) The largest two-sided ideal J of A such that A/J is semisimple D) The set of all elements x in A such that x + y = 0 for some y in A

16. A Banach algebra A is called semisimple if its radical is trivial. What does a trivial radical mean?

A) rad(A) = {0} B) rad(A) = A C) rad(A) = {e} D) rad(A) = {x | ||x|| < 1}

17. Which of the following is a property of the Jacobson radical (which coincides with the radical defined for semisimplicity in many contexts) of a Banach algebra?

A) It is a multiplicative subset B) It is an ideal C) It is a sub-algebra D) It is a field

18. If A is a commutative Banach algebra, its radical is the intersection of all maximal modular left ideals. This is a characterization of:

A) The set of regular elements B) The set of topological divisors of zero C) The Jacobson radical D) The set of idempotent elements

19. A Banach algebra A is semisimple if and only if it contains no:

A) Invertible elements B) Idempotent elements other than 0 and e C) Non-zero nilpotent ideals D) Topological divisors of zero

20. Consider the Banach algebra C(X) of continuous complex-valued functions on a compact Hausdorff space X. Its radical is:

A) The set of constant functions B) The zero ideal {0} C) The set of functions vanishing at a specific point D) The set of non-zero functions

21. Which type of Banach algebra is always semisimple?

A) The algebra of all bounded linear operators on a Hilbert space B) The algebra of convergent sequences with the supremum norm C) The algebra of continuous functions vanishing at infinity D) The algebra of strictly contractive linear operators

22. The spectrum of the identity element 'e' in any Banach algebra A is:

A) {0} B) {1} C) {0, 1} D) The set of all complex numbers

23. If x is an element of a Banach algebra A and ||x|| < 1, then x is:

A) Singular B) A topological divisor of zero C) Regular and has an inverse x(e - x)^(-1) D) Nilpotent

24. Let A be a Banach algebra. If sigma(x) = {lambda}, then the spectral radius r(x) is equal to:

A) 0 B) 1 C) |lambda| D) Re(lambda)

25. The set of all regular elements in a Banach algebra A forms:

A) An ideal B) A multiplicative group C) A vector subspace D) A commutative ring

26. If x is a topological divisor of zero in a Banach algebra A, then x is also:

A) Regular B) Singular C) Nilpotent D) Idempotent

27. In a Banach algebra A, if x is invertible, then sigma(x) contains:

A) Only 0 B) Only 1 C) No elements D) Only non-zero elements

28. The spectral radius r(x) satisfies r(x*x) = r(x)^2 for any element x in a:

A) General Banach algebra B) Commutative Banach algebra C) C*-algebra D) Normed algebra

29. What is the spectrum of the zero element '0' in any Banach algebra A?

A) {0} B) {1} C) The empty set D) All complex numbers

30. If x is a nilpotent element in a Banach algebra A, then its spectral radius r(x) is:

A) 1 B) Greater than 1 C) 0 D) Undefined

31. The set of all singular elements in a Banach algebra A is:

A) Always an ideal B) Always a sub-algebra C) Not necessarily closed D) Always closed

32. Let A be a Banach algebra. The spectral mapping theorem for polynomials states that for any polynomial p(z), sigma(p(x)) = p(sigma(x)). This holds for:

A) All elements x in A B) Only normal elements x in A C) Only self-adjoint elements x in A D) Only invertible elements x in A

33. If x is an element of a Banach algebra A, and lambda is not in sigma(x), then lambda is in the resolvent set of x. The resolvent map is defined as R(lambda, x) =:

A) (x - lambda*e)^(-1) B) (lambda*e - x)^(-1) C) (x + lambda*e)^(-1) D) (lambda*e + x)^(-1)

34. The radical of a Banach algebra A is the set of elements x in A such that x is:

A) Singular B) A topological divisor of zero C) Quasinilpotent (spectral radius is zero) D) Nilpotent

35. A Banach algebra A is semisimple if and only if its Jacobson radical is {0}. This is equivalent to stating that A has no:

A) Zero divisors B) Non-zero annihilators C) Non-zero quasinilpotent elements D) Non-zero idempotent elements

36. Which of the following conditions implies that a Banach algebra A is semisimple?

A) A is finite dimensional B) A is commutative C) A has a bounded approximate identity D) A is a C*-algebra

37. If x is an element of a Banach algebra A, then sigma(x^n) = {lambda^n | lambda in sigma(x)}. This is an instance of the:

A) Spectral radius formula B) Spectral mapping theorem C) Gelfand-Mazur theorem D) Stone-Weierstrass theorem

38. Let A be a Banach algebra. An element x is a topological divisor of zero if and only if:

A) x is singular B) x is nilpotent C) 0 is in the closure of the set {xy | ||y||=1} D) 0 is in the closure of the set {yx | ||y||=1}

39. The spectrum of an element x in a Banach algebra A is the set of scalars lambda for which x - lambda*e is:

A) Invertible B) Singular C) Nilpotent D) Idempotent

40. If A is a Banach algebra, then the set of regular elements G(A) is:

A) A closed set B) An open set C) A compact set D) A finite set

41. Which property is NOT necessarily true for the spectrum of an element x in a Banach algebra?

A) Non-empty B) Compact C) Connected D) Bounded

42. The spectral radius r(x) is the smallest non-negative number r such that:

A) ||x|| <= r B) ||x^n|| <= r^n for all n C) ||x^n||^(1/n) <= r for all n D) ||x^n||^(1/n) -> r as n -> infinity

43. A Banach algebra A is semisimple if and only if for every non-zero element x in A, there exists an element y in A such that:

A) xy = 0 B) yx = 0 C) x + y = e D) xy = yx = e

44. If A is a commutative Banach algebra with identity, then the maximal ideals of A are in one-to-one correspondence with:

A) The set of all prime numbers B) The set of all invertible elements C) The set of all non-zero homomorphisms from A to C D) The set of all nilpotent elements

45. The radical of a commutative Banach algebra is the intersection of all:

A) Maximal ideals B) Minimal ideals C) Prime ideals D) Idempotent elements

46. Which of the following is NOT a property of the spectral radius r(x)?

A) r(x + y) <= r(x) + r(y) B) r(xy) <= r(x)r(y) C) r(cx) = |c|r(x) for scalar c D) r(x) = ||x||

47. In a Banach algebra, if x is a topological divisor of zero, then x is also:

A) Regular B) Invertible C) Singular D) Nilpotent

48. The set of singularities of an element x in a Banach algebra A is also known as:

A) The resolvent set B) The spectrum C) The set of topological divisors of zero D) The set of nilpotent elements

49. For a Banach algebra A, the spectral radius formula r(x) = lim_{n->inf} ||x^n||^(1/n) implies that r(x) is:

A) Always less than ||x|| B) Always greater than ||x|| C) Less than or equal to ||x|| D) Equal to ||x||

Banach algebras - regular and singular elements, topological divisors of zero, spectrum of an element, spectral radius formula, radical and semisimplicity - Online Test

Test Results