Noncommutative Distance Theory
Note 2
C*-Algebra
TANAKA Akio
1
<C*-algebra>
Complex field C
Algebra over C A
A algebra satisfies next conditions, it is called *-algebra.
Arbitrary x,y ∈ A
(xy)* = y*x*
(x*)* = x
A ∋ x ↦ x* ∈ A
Norm ||・|| of *-algebra A satisfies next conditions, it is called C*-norm.
Arbitrary x,y ∈ A
||xy|| ≤ ||x|| ||y||
||x*x|| = ||x||2
When algebra A is complete on C*-norm, it is called C*-algebra.
2
<Gel’fand-Naĭmark theorem>
Compact Hausdorf space X
Universal continuous function over X C ( X )
C ( X ) has identity element.
C ( X ) is called commutative C*-algebra.
When C ( X ) and C ( Y ) are equal as C*-algebra, X and Y are homeomorphism as space.
3
<Noncommutative 2 dimensional torus>
2-dimensional torus T2
Function over T2 is identified with double periodic function f(x,y) = f(x+2π, y) = f(x, y+2π).
Measurable function that has inner product makes Hilbert space L2(T2).
Operators that product function exp(ix) and exp(iy) U and V
Sequence space l2(Z) = { a = (an) : 
|an|2 < 
}
|an|2 < }
Operator Uθ U (a)n = an-1
Operator Vθ V(a)n = λnan-1 λ= exp (2πiθ)
VθUθ = λUθVθ
Aθ = C*( Uθ , Vθ ) is called noncommutative 2-dimensional torus.
When θ = 0, VU = UV , C(T2) is made again.
Tokyo December 4, 2007
Sekinan Research Field of Language
www.sekinan.org