This is primarily a font problem, although there is an unpleasant silence in the HTML specs.
Modern browsers interpret ⟨ and ⟩ as referring to the U + 27E8 MATH LEFT ANGULAR CHAIR "β¨" and U + 27E9 MATHEMATICAL RIGHT ANGLE CHAIR "β©", unofficially known as "bra" and "ket". This interpretation becomes official in the name of symbolic links in HTML5.
These symbols are suitable for use in many mathematical notations, and the ISO 80000-2 standard clearly indicates that they are used, for example. for some scalar product designations. But support for them in fonts is quite limited. On older Windows systems, no fonts contain them. On new Windows systems, starting with Windows Vista, Cambria Math should be available. You may have tested the old version of Windows, but it is also possible that Chrome cannot find the font you want. To give him a helping hand, use the CSS rule that this font offers, for example. with attribute
style="font-family: Cambria Math"
You might want to add some other fonts to the list using fonts that are known to contain characters. See My Guide to Using Special Characters in HTML .
The disgusting change is that in HTML 4.01 in the entities ⟨ and ⟩ defined as relating to U + 2329 ONION ANGLE HOUSING "<" and U + 232A RIGHT ANGLE CORRECTION ">". They are logically less satisfactory (and deprecated from Unicode Standard), but they have slightly broader font support.
Thus, in addition to declaring fonts containing the characters you use, you need to decide which pair of these characters you use or use something else; This is a difficult question. If you use them, it is better to use them as such (in an HTML document encoded with UTF-8) or using numeric characters, such as ⟨ . The reason is that ⟨ and ⟩ should not work sequentially; they probably work with HTML5 in all modern browsers, but there is hardly any reason to take a chance when you can unambiguously specify the characters you need.
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