Monday 30 April 2018

Noncommutative Distance Theory Note 2 C*-Algebra



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, and are homeomorphism as space.
3
<Noncommutative 2 dimensional torus>
2-dimensional torus     T2
Function over Tis identified with double periodic function f(x,y) = f(x+2πy) = f(xy+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.

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