Left-align all columns in equation

Just expanding a bit on @vonbrand's answer ...

The alignat environment is an extension of the align environment. It uses the & alignment symbol to align the blocks in alternating left and right manner.

• If two left-aligned blocks are supposed to follow each other, use && rather than &.

• You can create extra space between two left-aligned blocks by inserting \quad or \qquad between the consecutive & symbols.

• The alignat environments requires an argument, an integer. To calculate this integer, take the maximal number of & symbols in any row, add 1, and divide by 2. In the example below, the maximal number & symbols is 5; adding 1 and dividing by 2 gives 3. (If the maximal number is an even number, add 1 before proceeding.)

Some additional comments: I would replace all instances of ... with \dots in order to create typographical ellipses; and I'd encase the 3 : symbols in curly braces in order to avoid the extra spacing that otherwise gets inserted.

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{alignat}{3} % max. # of '&' symbols in any row: 5. (5+1)/2 = 3 
& a[i{:}j] &&= \langle\,\rangle            
      &\qquad&\text{if $i > j$} \\ % or '\quad', if you prefer
& a[i{:}j] &&= \langle a_i, \dots, a_j \rangle 
      &&\text{if $a = \langle a_0, \dots, a_i, \dots a_j, \dots \rangle$} \\
& a[i{:}]  &&= \langle a_i, \dots \rangle  
      &&\text{if $a = \langle a_o, \dots, a_i, \dots \rangle$}
\end{alignat}
\end{document}

Use alignat, and read the documentation carefully. It has quite funny ideas of how to align column contents, you have to pick the ones to fill with some care. A bit of experimentation (and some dummy columns, perhaps for spacing only) will get you the results you desire.