Noncommutative Distance Theory Note 2 C*-Algebra

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 = (a_n) : |a_n|² <  }

Operator U_θ     U (a)_n = a_n-1

Operator V_θ       V(a)_n = λⁿa_n-1  λ= exp (2πiθ)

V_θU_θ = λU_θV_θ

A_θ = C*( U_θ , V_θ ) is called noncommutative 2-dimensional torus.

When θ = 0, VU = UV , C(T²) is made again.

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