TRACE
“Ive, wearing a royal-blue T-shirt, was affable, but there was little trace of English irony.”
1{little trace of English irony}?
Jonathan Ive and the Future of Apple - The New Yorker
http://www.newyorker.com/magazine/2015/02/23/shape-things-come
via Instapaper

Question

TRACE
“Ive, wearing a royal-blue T-shirt, was affable, but there was little trace of English irony.”
1{little trace of English irony}? 
Jonathan Ive and the Future of Apple - The New Yorker
http://www.newyorker.com/magazine/2015/02/23/shape-things-come
via Instapaper

Answer

"Trace" means a small amount of something.  For example, people who support organic farming talk about "traces of pesticides" - small amounts of pesticides that are left behind on food.  So, in your sentence, the interviewer didn't notice any English* irony in Ive's speech or personality.

*Here, "English" means "from England," not the English language.