  In this paper, we study simplicial complexes whose Stanley-Reisner rings are
almost Gorenstein and have $a$-invariant zero. We call such a simplicial
complex an almost Gorenstein* simplicial complex. To study the almost
Gorenstein* property, we introduce a new class of simplicial complexes which we
call uniformly Cohen-Macaulay simplicial complexes. A $d$-dimensional
simplicial complex $\Delta$ is said to be uniformly Cohen-Macaulay if it is
Cohen-Macaulay and, for any facet $F$ of $\Delta$, the simplicial complex
$\Delta \setminus\{F\}$ is Cohen-Macaulay of dimension $d$. We investigate
fundamental algebraic, combinatorial and topological properties of these
simplicial complexes, and show that almost Gorenstein* simplicial complexes
must be uniformly Cohen-Macaulay. By using this fact, we show that every almost
Gorenstein* simplicial complex can be decomposed into those of having one
dimensional top homology. Also, we give a combinatorial criterion of the almost
Gorenstein* property for simplicial complexes of dimension $\leq 2$.
