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A number of conditions can cause this:
Decreased blood supply: any cause which leads to decreased blood supply in one arm.This is the most common cause.This could be simply due to atherosclerosis which has occurred more in one arm. You will need doppler study of the blood supply in your hands to rule this out. Any growth, etc. in the arm of the underarm which impacts the blood supply could also lead to this.
Reflex Sympathetic Dystrophy syndrome: This is a disorder of the autonomic nervous system which leads to coldness in one arm, and is accompanied by pain in the arm.http://ww2.arthritis.org/conditions/DiseaseCenter/rsds.asp
Harlequin Syndrome: This is a rare condition in which coldness of one arm is accompanied by flushing of face of the opposite side.http://jkms.kams.or.kr/2005/pdf/04329.pdf
Stroke: stroke also leads to coldness of the limb of the affected side but is accompanied by weakness of the limb also.
The most common cause is atherosclerosis of the vessels of the arm. You should consult a doctor for this. You should discuss these conditions with him. You are likely to need Doppler study of the blood supply in your arm.