Noncommutative Distance Theory Note 2 C*-Algebra

 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 <  }

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

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