In the present paper we first introduce the notion of
LM^p_I(A)of a Banach space A, where I is an index set, and 1 �� p �� ��.
In the case where A is a Banach algebra and 1 �� p �� 2 we find necessary
and sufficient conditions for which LM^p_I(A) has a bounded approximate
identity. We also find the second dual of LM^p_I(A) for an arbitrary index
set I. In the case where 1 �� p �� 2 and LM^p_I(A) has a bounded
approximate identity we prove that LM^p_I(A) is unital if and only if
A is unital.