What Do You Know About Herz Schur Multipliers?

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Questions: 10 | Attempts: 123 | Updated: Mar 18, 2023

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What Do You Know About Herz Schur Multipliers? - Quiz

Herz–Schur multipliers were named after two scientists: Carl S. Herz and Issai Schur. The multiplier was introduced by Haagerup and collaborators in 1985 and it is a special kind of mapping used in the field of representation theory, from a group to the complex number field. This quiz will help you know a bit more about Herz–Schur multipliers and if you already know, you can still try it out.

Questions and Answers

1.

What is the S in Carl S. Herz?
- A.
  
  Stanley
- B.
  
  Samuel
- C.
  
  Seth
- D.
  
  Scott
Correct Answer
B. Samuel

Explanation
The S in Carl S. Herz stands for Samuel.

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2.

Which of these is not a category of Herz-Schur multipliers?
- A.
  
  Positive multipliers
- B.
  
  Idempotent multipliers
- C.
  
  Radial multipliers
- D.
  
  Negative multipliers
Correct Answer
D. Negative multipliers

Explanation
The question asks for a category of Herz-Schur multipliers that is not included in the given options. The options provided include positive multipliers, idempotent multipliers, radial multipliers, and negative multipliers. Out of these options, the correct answer is "Negative multipliers," as it is the only option that is not a category of Herz-Schur multipliers.

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3.

Under what are the multipliers categorized?
- A.
  
  Systematic analysis
- B.
  
  Dirichlet analysis
- C.
  
  Transformational analysis
- D.
  
  Harmonic analysis
Correct Answer
D. Harmonic analysis

Explanation
The multipliers are categorized under harmonic analysis. Harmonic analysis is a branch of mathematics that deals with the representation of functions or signals as the superposition of basic waves. It involves the study of the properties and behavior of harmonic functions, Fourier series, and Fourier transforms. Multipliers, in this context, refer to functions that act on the Fourier coefficients of a function, modifying or transforming them in some way. Therefore, harmonic analysis is the appropriate category for the study of multipliers.

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4.

The existence of a semi-group of Herz-Schur multipliers generated by a proper function is equivalent to which property?
- A.
  
  Schur's property
- B.
  
  Herz multipliers
- C.
  
  Haagerup property
- D.
  
  Linearly bounded function
Correct Answer
C. Haagerup property

Explanation
The existence of a semi-group of Herz-Schur multipliers generated by a proper function is equivalent to the Haagerup property.

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5.

A Herz-Schur multiplier is a group of what?
- A.
  
  Arithmetic theory
- B.
  
  Marginal theory
- C.
  
  Representational theory
- D.
  
  Convolution
Correct Answer
C. Representational theory

Explanation
A Herz-Schur multiplier is a group that belongs to the field of representational theory. This theory focuses on studying the different ways in which groups can be represented or expressed through mathematical structures. The Herz-Schur multiplier specifically deals with the representation of groups in terms of their multipliers, which are certain types of functions that provide information about the group's structure and properties. Therefore, the correct answer is representational theory.

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6.

Which is another term that can be used to describe weak topology?
- A.
  
  Initial topology
- B.
  
  Final topology
- C.
  
  Basic topology
- D.
  
  Fixed topology
Correct Answer
A. Initial topology

Explanation
The initial topology is another term that can be used to describe weak topology. It is a topology on a set that is generated by a family of maps from the set into topological spaces. It is the coarsest topology that makes all these maps continuous. The initial topology is often used in the context of topological vector spaces, where it characterizes convergence of sequences and continuity of linear functionals.

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7.

Which of the following is another topology?
- A.
  
  Tough topology
- B.
  
  Great topology
- C.
  
  Strong topology
- D.
  
  Dual topology
Correct Answer
C. Strong topology

Explanation
The correct answer is "Strong topology". This is because "Strong topology" is listed as one of the options for another topology. The other options, "Tough topology", "Great topology", and "Dual topology", are not listed as possible answers.

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8.

What happens if the conditions of Haagerup's theorem are fulfilled?
- A.
  
  ||u||cbm = u(e)
- B.
  
  ||u||cbt = u(e)
- C.
  
  ||u||cbm = u(f)
- D.
  
  ||u||cbn = u(e)
Correct Answer
A. ||u||cbm = u(e)

Explanation
If the conditions of Haagerup's theorem are fulfilled, it means that the norm of the completely bounded multiplier (||u||cbm) is equal to the value of u at the identity element of the group (u(e)).

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9.

The set of all Idempotent Herz-Schur multipliers is what type of algebra?
- A.
  
  Quadratic algebra
- B.
  
  Boolean algebra
- C.
  
  Schur's algebra
- D.
  
  Negative algebra
Correct Answer
B. Boolean algebra

Explanation
The set of all Idempotent Herz-Schur multipliers is a type of algebra known as Boolean algebra. Boolean algebra is a mathematical structure that deals with binary variables and logical operations such as AND, OR, and NOT. It is named after mathematician George Boole and is widely used in computer science and digital electronics for designing and analyzing logic circuits. In Boolean algebra, the concept of idempotence is important as it refers to an operation that, when applied multiple times, produces the same result as if it were applied only once.

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10.

Haagerup was introduced together with another scientist. Which of these is the scientist?
- A.
  
  P. Austin
- B.
  
  L. Schur
- C.
  
  J. Kraus
- D.
  
  H. Henry
Correct Answer
C. J. Kraus

Explanation
The correct answer is J. Kraus because the question states that Haagerup was introduced together with another scientist. Therefore, J. Kraus must be the scientist who was introduced with Haagerup.

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