Will a Verizon note 4 will work with T-Mobile 4g LTE
It should, but it is only minimally compatible. The only LTE compatible band is 4 (no band 2 or band 12 compatibility). Similarly, you only have one comparable band (band 2) on WCDMA and 2G.
I suggest using a more compatible device.
band 4 / 1700 MHz AWS (Americas)
band 13 / 750 MHz Upper 700 (USA (Verizon))
WCDMA (3G / 4G)
band 1 (I) / 2100 MHz IMT (Europe, Asia, Africa, Oceania, Brazil, India, Israel)
band 2 (II) / 1900 MHz PCS (Americas)
band 5 (V) / 850 MHz Cellular (Americas, Oceania, Brazil, Israel)
CDMA (2G / 3G)
850 MHz / band 5 / BC0 Cellular (Americas, Oceania, Brazil, Israel)
1900 MHz / band 2 / BC1 PCS (Americas)
850 MHz / band 5 Cellular (Americas, Oceania, Brazil, Israel)
900 MHz / band 8 (Europe, Asia, Africa)
1800 MHz / band 3 DCS (Europe, Asia, Africa)
1900 MHz / band 2 PCS (Americas)