Let
![$A$](img13.png)
and
![$B$](img14.png)
be C*-algebras such that
![$A$](img13.png)
or
![$B$](img14.png)
is exact.
We describe the largest ideal in
![$A \otimes B$](img15.png)
which has the
weak ideal property. For many C*-algebras
![$A$](img13.png)
and
![$B$](img14.png)
as above
we characterize when the largest ideal in
![$A \otimes B$](img15.png)
which
has the weak ideal property is the tensor product of the largest
ideals in
![$A$](img13.png)
and
![$B$](img14.png)
which have the weak ideal property (this is
not always true if
![$A$](img13.png)
or
![$B$](img14.png)
is exact). Assume that the C*-algebras
![$A$](img13.png)
and
![$B$](img14.png)
have the weak ideal property (and one of them is
exact).
We characterize (in an interesting particular case and also in
general) when
![$A \otimes B$](img15.png)
has the weak ideal property (these
two characterizations are totally different in nature).