Hi,
The Subclavian artery which brings blood to the left arm from the heart may have a problem. This can be atherosclerosis (hardening of the artery),blockage or partial blockage inside of the artery due to blood clot or pressure on the artery in the neck-shoulder area by an extra rib on the top of first rib so called Thoracic outlet syndrome. So you need an Angogram of this left Subclavian artery to see what is going on and why you can not get blood pressure reading on the left arm. A Radiologist can do the Angiogram for you.The problem can be minor or serious depending to Angiogram result.
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Regards,
Dr.Yetimyan