There is a specific explanation for each of the examples.
an algebra with unit e;
Every algebra had a unique unit which is a property of it; for this algebra the unit is e. It cannot be "a unit" because there is only one unit. This is perhaps a contraction of "with the unit being e".
an operator with domain H2;
Similar. Every operator has a domain, and only one domain.
a solution with vanishing Cauchy data;
"Data" is plural, so no article is needed.
a cube with sides parallel to the axes;
Plural, and all sides are parallel to the axes, so no article. If only one side is parallel, it would have to have an aritcle; "a side parallel" or "one side parallel".
a domain with smooth boundary;
As before, the boundary is referred to as a property of the domain, and not as just a curve; there is only one boundary, and it is smooth. Can be "with the boundary being smooth".
an equation with constant coefficients;
a function with compact support;
Property of a function
random variables with zero expectation
The expectation is a property of the distribution, which is a property of the RV.
Let В be a Banach space with a weak symplectic form w.
W is some weak symplectic form, it is not determined, so use an indeterminate article.
Two random variables with a common distribution.
Some distribution, which is common to both, but otherwise not specified. If it was specified, the article would be omitted: "two random variables with common distribution N(0, 1)".