Banach algebras - regular and singular elements, topological divisors of zero, spectrum of an element, spectral radius formula, radical and semisimplicity - One Line Questions
1.
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) =: —
(lambda*e - x)^(-1)
2.
The spectrum of the identity element 'e' in any Banach algebra A is: —
{1}
3.
What is the spectrum of the zero element '0' in any Banach algebra A? —
{0}
4.
The spectral radius r(x) is the smallest non-negative number r such that: —
||x^n||^(1/n) -> r as n -> infinity
5.
If x is a topological divisor of zero in a Banach algebra A, what can be said about its norm? —
||x|| can be any positive real number
6.
Let A be a Banach algebra. If sigma(x) = {lambda}, then the spectral radius r(x) is equal to: —
|lambda|
7.
If x is a nilpotent element in a Banach algebra A, then its spectral radius r(x) is: —
0
8.
If A is a Banach algebra, then the set of regular elements G(A) is: —
An open set
9.
Which of the following conditions implies that a Banach algebra A is semisimple? —
A is a C*-algebra
10.
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: —
All elements x in A
11.
The set of all singular elements in a Banach algebra A is: —
Always closed
12.
For a Banach algebra A, the spectral radius formula r(x) = lim_{n->inf} ||x^n||^(1/n) implies that r(x) is: —
Less than or equal to ||x||
13.
In a Banach algebra A, what is a regular element? —
An element x for which there exists an element y such that xy = yx = e
14.
The set of all regular elements in a Banach algebra A forms: —
A multiplicative group
15.
What are topological divisors of zero in a Banach algebra A? —
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
16.
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: —
There exists a point x_0 in X such that f(x_0) = 0
17.
The spectral radius r(x) satisfies r(x*x) = r(x)^2 for any element x in a: —
C*-algebra
18.
The spectrum of an element x in a Banach algebra A is the set of scalars lambda for which x - lambda*e is: —
Singular
19.
A Banach algebra A is semisimple if and only if it contains no: —
Non-zero nilpotent ideals
20.
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? —
It is an ideal
21.
The spectral radius formula for an element x in a Banach algebra A states that r(x) is equal to: —
lim_{n->inf} ||x^n||^(1/n)
22.
The radical of a commutative Banach algebra is the intersection of all: —
Maximal ideals
23.
Which property is NOT necessarily true for the spectrum of an element x in a Banach algebra? —
Connected
24.
In a Banach algebra A, if x is invertible, then sigma(x) contains: —
Only non-zero elements
25.
Which of the following is NOT a property of the spectral radius r(x)? —
r(x) = ||x||
26.
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? —
27.
A Banach algebra A is called semisimple if its radical is trivial. What does a trivial radical mean? —
rad(A) = {0}
28.
If x is a topological divisor of zero in a Banach algebra A, then x is also: —
Singular
29.
In a Banach algebra, if x is a topological divisor of zero, then x is also: —
Singular
30.
Which of the following is always true for the spectrum sigma(x) of an element x in a Banach algebra A? —
sigma(x) is a non-empty compact subset of the complex plane
31.
If x is an element of a Banach algebra A and ||x|| < 1, then x is: —
Regular and has an inverse x(e - x)^(-1)
32.
The radical of a Banach algebra A is the set of elements x in A such that x is: —
Quasinilpotent (spectral radius is zero)
33.
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: —
Spectral mapping theorem
34.
Which type of Banach algebra is always semisimple? —
The algebra of all bounded linear operators on a Hilbert space
35.
Which property of the spectral radius formula is crucial for its validity in Banach algebras? —
The completeness of the algebra
36.
For a matrix A in M_n(C), its spectral radius r(A) is: —
The maximum absolute value of its eigenvalues
37.
The set of singularities of an element x in a Banach algebra A is also known as: —
The spectrum
38.
Let A be a Banach algebra. The spectrum of an element x in A, denoted by sigma(x), is defined as: —
The set of all complex numbers lambda such that x - lambda*e is singular
39.
In a Banach algebra A, the radical of A, denoted by rad(A), is defined as: —
The largest two-sided ideal J of A such that A/J is semisimple
40.
If A is a commutative Banach algebra with identity, then the maximal ideals of A are in one-to-one correspondence with: —
The set of all non-zero homomorphisms from A to C
41.
Consider the Banach algebra C(X) of continuous complex-valued functions on a compact Hausdorff space X. Its radical is: —
The zero ideal {0}
42.
Consider the algebra of complex n x n matrices, M_n(C), with the operator norm. What is the spectrum of a matrix A? —
The set of eigenvalues of A
43.
If A is a commutative Banach algebra, its radical is the intersection of all maximal modular left ideals. This is a characterization of: —
The Jacobson radical
44.
What is the spectral radius of an element x in a Banach algebra A, denoted by r(x)? —
The supremum of the absolute values of elements in sigma(x)
45.
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: —
x - lambda*e is not invertible
46.
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? —
The set of elements with multiplicative inverses does not contain x
47.
Let A be a Banach algebra. An element x is a topological divisor of zero if and only if: —
0 is in the closure of the set {xy | ||y||=1}
48.
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: —
xy = 0
49.
A Banach algebra A is semisimple if and only if its Jacobson radical is {0}. This is equivalent to stating that A has no: —
Non-zero quasinilpotent elements