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Abstract

Suppose that $A$ and $B$ are unital Banach algebras with units $1_A$ and $1_B$, respectively, $M$ is a unital Banach $A,B$-module, ${\cal T}= \left [{A\quad M\atop 0\quad B}\right]$ is the triangular Banach algebra, $X$ is a unital ${\cal T}$-bimodule, $X_{AA}=1_AX1_A$, $X_{BB}=1_BX1_B$, $X_{AB}=1_AX1_B$ and $X_{BA}=1_BX1_A$. Applying two nice long exact sequences related to $A$, $B$, ${\cal T}$, $X$, $X_{AA}$, $X_{BB}$, $X_{AB}$ and $X_{BA}$ we establish some results on (co)homology of triangular Banach algebras.