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

1. 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||
2. 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
3. In a Banach algebra, if x is a topological divisor of zero, then x is also:
A) Regular
B) Invertible
C) Singular
D) Nilpotent
4. 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||
5. The radical of a commutative Banach algebra is the intersection of all:
A) Maximal ideals
B) Minimal ideals
C) Prime ideals
D) Idempotent elements
6. 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
7. 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
8. 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
9. 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
10. 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
11. 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
12. 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}
13. 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
14. 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
15. 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
16. 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
17. 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)
18. 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
19. 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
20. 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
21. 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
22. 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
23. 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
24. 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
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. 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)
27. 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
28. 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
29. 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
30. 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
31. 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
32. 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
33. 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
34. 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}
35. 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
36. 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
37. 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
38. 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
39. 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
40. 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)
41. 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)
42. 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
43. 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
44. 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
45. 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
46. 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
47. 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
48. 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
49. 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
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