We present Atacama Large Millimeter/submillimeter Array (ALMA) observations of multiple protostar systems in the Perseus molecular cloud previously detected by the Karl G. Jansky Very Large Array (VLA). We observed 17 close ($<$600~AU separation) multiple systems at 1.3~mm in continuum and five molecular lines (i.e., twco, cateo, thco, H$_2$CO, SO) to characterize the circum-multiple environments in which these systems are forming. We detect at least one component in the continuum for the 17 multiple systems. In three systems, one companion is not detected, and for two systems the companions are unresolved at our observed resolution. We also detect circum-multiple dust emission toward 8 out of 9 Class 0 multiples. Circum-multiple dust emission is not detected toward any of the 8 Class I multiples. Twelve systems are detected in the dense gas tracers toward their disks/inner envelopes. For these 12 systems, we use the dense gas observations to characterize their formation mechanism. The velocity gradients in the circum-multiple gas are clearly orthogonal to the outflow directions in 8 out of the 12 systems, consistent with disk fragmentation. Moreover, only two systems with separations $<$200~AU are textit{inconsistent} with disk fragmentation, in addition to the two widest systems ($>$500~AU). Our results suggest that disk fragmentation via gravitational instability is an important formation mechanism for close multiple systems, but further statistics are needed to better determine the relative fraction formed via this method.
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