Let X be a complete metric space (C*-algebra valued) and CB(X) be nonempty closed and bounded subset of X. Can we define a mapping T:X------->CB(X), i.e. Can we have a C*-algebra valued multi-valued mapping? If Yes, please can someone provide me with an example of such a mapping?
Because I know if X=R(set of real numbers), any interval I be it open, closed, half-open or half-closed then I is a subset of R and we can define such a mapping T by for instance Tx = [0, x/2), thus we can be able to get a fixed point depending on some conditions either on T, X or both.
I hope my question is well constructed and understandable.
If I am not mistaken, you gave an example yourself:
Take ℂ as Hilbert space and define for every ω ε ℝ the continuous linear mulitplication operators A_ω :ℂ → ℂ, z → A_ω(z) = ωz, and set X:= {A_ω | ω ε ℝ}. Then X ⊂ L(ℂ) is a C*-Algebra and is isometrically canonically isomorphic to ℝ via isomorphisms φ : X → ℝ , A_ω → ω and ψ : ℝ → X, ω → A_ω, that is ψ = φ^{-1}. X is the algebra of real 1x1 matrices. Then, the mapping Π :=ψ∘T∘φ : X → CB(X), where for I ⊂ ℝ φ(I):= {φ(x) | x ε I} and T is defined like in your question is an example for the sought for type of mapping.
Usually qestions of the type: „Can I define a map from one set to another?“ can usually be answered with a yes: As long as a mapping has no requirements to satisfy (other than being a mapping) you can define anything. For example I can define a map M that maps yourself to the set of all the questions you have asked on researchgate, that is
Muhammad →M(Muhammad) := {question | question was asked by Muhammad on RG}
That is a perfect set valued mapping.
:-P
Cheers
p.s.: The composition was wrong, I corrected the mistake
Hi Muhammad
If I have anderstood your question, it is always possible to construct such a mapping. Just assign to any x from X the ball B(x,r) for some given radius r>0. If you do not want to have a fixed point, consider a mapping g :X-----> X without any fixed point and assign to each x the ball B(g(x), d(x, g(x)/2) or even the closure of
B(g(x), d(x, g(x)/2) .
Now, if you want a C*-algebra-valued mapping, just assign to x the C*-algebra C_b( cl( B( g(x),d(x,g(x)) /2 ) ) ) of all bounded continuous complex functions on cl( B( g(x) , d(x , g(x))/2 ) ) instead of B( g(x), d(x,g(x))/2).
can definr , but what is the intension or question in your mind for seeking such function?
- Badges
- Science topic
- Similar topics
- Geoscience
- Cartography
- Mapping