Inner product space over a Hilbert C*-module

In summary, the conversation is asking whether a Hilbert A-module with a C^*-algebra has an approximate identity e_\alpha = \langle u_\alpha , v_\alpha \rangle where u_\alpha and v_\alpha are norm bounded by 1. It is mentioned that any C^*-algebra has a bounded approximate identity, but the speaker is unsure of how to prove that the approximate identity in this case is norm bounded by 1. They are seeking help with this proof.
  • #1
CornMuffin
55
5
I am working on this problem, and I am having difficulties with a certain part of a proof:



If [itex]A[/itex] is a [itex]C^*[/itex]-algebra. And [itex]X[/itex] is a Hilbert [itex]A[/itex]-module. Can we say that [itex]\langle X,X \rangle[/itex] has an approximate identity [itex]e_\alpha = \langle u_\alpha , v_\alpha \rangle[/itex] such that [itex]u_\alpha , v_\alpha[/itex] are norm bounded by 1. Where [tex]\langle \cdot , \cdot \rangle [/tex] denotes the inner product.

I know that any [itex]C^*[/itex]-algebra has a bounded approximate identity, and [itex]\langle X,X \rangle[/itex] is a [itex]C^*[/itex]-algebra. So we can have [itex]\| e_\alpha \| = \| \langle u_\alpha , v_\alpha \rangle \| \leq 1[/itex]
 
Physics news on Phys.org
  • #2
. However, I am not sure how to prove that the approximate identity is norm bounded by 1. Any help would be appreciated.
 

Related to Inner product space over a Hilbert C*-module

1. What is an inner product space over a Hilbert C*-module?

An inner product space over a Hilbert C*-module is a mathematical structure that combines the concepts of an inner product space and a Hilbert C*-module. It is a vector space equipped with an inner product that satisfies certain properties and is defined over a Hilbert C*-module, which is a type of Banach space.

2. What are the properties of an inner product in this context?

In an inner product space over a Hilbert C*-module, the inner product must be linear in its first argument, conjugate-linear in its second argument, and satisfy the Hermitian symmetry property and positive-definiteness property. These properties ensure that the inner product behaves similarly to the inner product in a standard Hilbert space.

3. How is an inner product space over a Hilbert C*-module different from a standard Hilbert space?

An inner product space over a Hilbert C*-module is a generalization of a standard Hilbert space. In a standard Hilbert space, the inner product is defined over a field of scalars, usually the real or complex numbers. In contrast, in an inner product space over a Hilbert C*-module, the inner product is defined over a Hilbert C*-module, which can be a more general algebraic structure.

4. What are some examples of inner product spaces over Hilbert C*-modules?

One example is the space of square-integrable functions over a Hilbert C*-module, which is a generalization of the space of square-integrable functions over a standard Hilbert space. Another example is the space of operators on a Hilbert C*-module, which is a generalization of the space of operators on a standard Hilbert space.

5. How are inner product spaces over Hilbert C*-modules used in scientific research?

Inner product spaces over Hilbert C*-modules have applications in many areas of mathematics, physics, and engineering. They are used to study quantum mechanics, functional analysis, and operator algebras, among other topics. They also have practical applications in signal processing, control theory, and data analysis.

Similar threads

What is the formula for the norm of a vector cross product?
    • Calculus and Beyond Homework Help
Replies
5
Views
1K
Bounded operators on Hilbert spaces
    • Calculus and Beyond Homework Help
2
Replies
43
Views
3K
Proving a fact about inner product spaces
    • Calculus and Beyond Homework Help
Replies
9
Views
1K
Center of mass of spherical shell inside of cone (Apostol Problem).
    • Calculus and Beyond Homework Help
Replies
3
Views
601
I Inner and Outer Product of the Wavefunctions
    • Quantum Physics
Replies
8
Views
2K
Homework Help: Quantum Information Theory 1
    • Advanced Physics Homework Help
Replies
0
Views
221
Dimension of orthogonal subspaces sum
    • Calculus and Beyond Homework Help
Replies
4
Views
1K
I Inner product vs dot/scalar product
    • Calculus
Replies
4
Views
594
Proving a function is an inner product in a complex space
    • Calculus and Beyond Homework Help
Replies
17
Views
1K
Determining whether a function is an inner product
    • Calculus and Beyond Homework Help
Replies
9
Views
2K

Hot Threads

Recent Insights

Back
Top