I am reading a book about C*-algebra. But I meet with some problems. In the book, the author says:
If $I$ is an ideal in a C*-algebra $A$, then $B=I\cap I^{\ast}$ is a C*-subalgebra. However, I suppose that $B$ may not be a C*-subalgebra if $I$ is not closed. Is this conclusion correct?
Implicit in the quoted statement, it is assumed that $I$ is a closed ideal. Otherwise, the statement would be false in general. For example, if $A=B(H)$ and $I$ is the set of finite rank operators on $H$, then $I$ is a non-closed two-sided ideal in $A$, and $I\cap I^*=I$ is not a C*-subalgebra because it is not norm-closed.
Some books on the subject say at some point early on that the word 'ideal' will generally be used for closed ideals unless otherwise stated. (In that case, the authors have to be more careful when referring to not-necessarily-closed ideals.) Whether that was said or not, it is hard to keep all language in a book entirely consistent, so there may be occasional slips of precision requiring good attentiveness of the reader, such as your question.