Let $G$ be a quasi-complete p-group and let $A$ be a subgroup of $G$ such that there exists a direct summand $L$ of $G$ containing $A$ which is minimal among the direct summands of $G$ that contain $A$. Such a direct summand $L$ is said to be a minimal direct summand of $G$ containing $A$. We prove that all minimal direct summands of G containing $A$ are isomorphic.

In this paper we consider closed subalgebras of $C(M)$ and study the structure of algebras $\mathscr{A}$ satisfying $\mathscr{A}_{b}=C(M)$. We show that the Bourgain algebra of $A$ is contained in $H^{\infty}(D)+C(\overline{D})$ if $A$ is between the disk algebra $A(D)$ and $H^{\infty}(D)$ or between $H^{\infty}(D)$ and $H^{\infty}(D)+C(\overline{D})$, and the Bourgain algebra of $H^{\infty}\circ L_{m}$ is contained in $H^{\infty}(D)+C(\overline{D})$ if $m$ is a nontrivial point.

If $f X\arrow X$ is a continuous function, $x\in X$ is called a periodic point if some iterate $f^{k}$ of $f$ leaves $x$ fixed. In 1953 F. B. Fuller proved a basic result for the existence of periodic points. He showed that if $X$ is a compact absolute neighborhood retract with non-zero Euler characteristic and if $N(X)$ is the larger of the sums of the odd and even real Betti numbers of $X$, then for any homeomorphism $f$ on $X$, $f^{k}$ has a fixed point for some $k, $1\leq k\leq N(X)$. Actually Fuller’s result holds for any space$X$ for which the Lefschetz Fixed Point Theorem is valid.In this paper the authors first establish a number of extensions of the Fuller Theorem. They next consider G-spaces and extend results of Mann on periodic points and invariant orbits.In the last section they turn to the Riemannian category using the volume function on the orbit space to obtain additional results and examples on periodic points and invariant orbits for groups of isometries. They conclude the paper by using the Fuller Theorem, together with a result of Kobayashi relating the Lefschetz number of an isometry and the Euler characteristic of its fixed point set, to establish a new result on the existence of ${\it common}$ periodic points for ${\it commuting}$ isometries.

It is shown that a class of Dirac operators acting in the abstract Boson-Fermion Fock space, which were introduced in a previous paper (A. Arai, J. Funct. Anal. 105 (1992), 342-408), is essentially self-adjoint. The result is applied to the Wess-Zumino models of supersymmetric quantum field theory to prove the essential self-adjointness of their supercharges and Hamiltonians.