本論文應用函數演算(functional calculus)來探討C*-代數之交換性的兩種判別法，主要結果為：
（１）　C*-代數A具交換性的充分必要條件為對於所有A 中的自伴隨(self-adjoint)元素，x,y,
e^(ix)e^(iy)=e^(iy)e^(ix)
（２）　C*-代數A具交換性的充分必要條件為對於所有A 中的正元素，x,y,
e^(x)e^(y)=e^(y)e^(x).
我們將把（２）作一個推廣如下：
如果f:[a,b]-->[c,d] 是任何一個連續嚴格單調函數，a,b,c,d in R, a<b,c<d,則C*-代數A具交換性的充分必要條件為對於所有A 中的自伴隨元素，x,y,只要spec(x) in [a,b], spec(y) in [a,b],則
　　　　f(x)f(y)=f(y)f(x).

In this thesis, We investigate the problem of when a C*-algebra is commutative through continuous functional calculus, The principal results are that:
(1) A C*-algebra A is commutative if and only if
e^(ix)e^(iy)=e^(iy)e^(ix),
for all self-adjoint elements x,y in A.
(2) A C*-algebra A is commutative if and only if
e^(x)e^(y)=e^(y)e^(x)
for all positive elements x,y in A.

We will give an extension of (2) as follows: Let
f:[a,b]-->[c,d] be any continuous strictly monotonic function where a,b,c,d in R, a<b,c<d. Then a C*-algebra A is commutative if and only if
f(x)f(y)=f(y)f(x),
for all self-adjoint elements x,y in A with spec(x) in [a,b] and spec(y) in [a,b].

1 Introduction ........................... 1
2 Notations and Preliminaries ........................... 2
2.1 Some basic definitions and theorems ..................... 2
3 main results ........................... 5
3.1 Main result I ........................................... 5
3.2 Main result II .......................................... 9
3.3 Main result III ........................................ 14

[1] Wei Wu. An order characterization of commutativity for C*-algebras, Proc. Amer. Math. Soc. Volume 129, number 4, pages 983-987, (2000).

[2] J.B. Conway. A Course in Functional Analysis, 2nd edition, Springer-Verlag, New York,(1990)

[3] K. Zhu. An Introduction to Operator Algebras. CRC Press, Inc., Florida(1993).