We prove completeness for the main examples of infinite-dimensional Lie groups and some related topological groups. Consider a sequence $G_{1}\subseteq G_{2}\subseteq \cdots \,$ of topological groups $G_{n}$ n such that $G_{n}$ is a subgroup of $G_{n+1}$ and the latter induces the given topology on $G_{n}$, for each $n\in \mathbb{N}$. Let $G$ be the direct limit of the sequence in the category of topological groups. We show that $G$ induces the given topology on each $G_{n}$ whenever $\cup _{n\in \mathbb{N}}V_{1}V_{2}\cdots V_{n}$ is an identity neighbourhood in $G$ for all identity neighbourhoods $V_{n}\subseteq G_{n}$. If, moreover, each $G_{n}$ is complete, then $G$ is complete. We also show that the weak direct product $\oplus _{j\in J}G_{j}$ is complete for each family $(G_{j})_{j\in J}$ of complete Lie groups $G_{j}$. As a consequence, every strict direct limit $G=\cup _{n\in \mathbb{N}}G_{n}$ of finite-dimensional Lie groups is complete, as well as the diffeomorphism group $\text{Diff}_{c}(M)$ of a paracompact finite-dimensional smooth manifold $M$ and the test function group $C_{c}^{k}(M,H)$, for each $k\in \mathbb{N}_{0}\cup \{\infty \}$ and complete Lie group $H$ modelled on a complete locally convex space.

We show that the direct sum $\mathop{({X}_{1} \oplus \cdots \oplus {X}_{r} )}\nolimits_{\psi } $ with a strictly monotone norm has the weak fixed point property for nonexpansive mappings whenever $M({X}_{i} )\gt 1$ for each $i= 1, \ldots , r$. In particular, $\mathop{({X}_{1} \oplus \cdots \oplus {X}_{r} )}\nolimits_{\psi } $ enjoys the fixed point property if Banach spaces ${X}_{i} $ are uniformly nonsquare. This combined with the earlier results gives a definitive answer for $r= 2$: a direct sum ${X}_{1} {\mathop{\oplus }\nolimits}_{\psi } {X}_{2} $ of uniformly nonsquare spaces with any monotone norm has the fixed point property. Our results are extended to asymptotically nonexpansive mappings in the intermediate sense.

The notion of BSE algebras was introduced and first studied by Takahasi and Hatori and later studied by Kaniuth and Ülger. This notion depends strongly on the multiplier algebra $M( \mathcal{A} )$ of a commutative Banach algebra $ \mathcal{A} $. In this paper we first present a characterisation of the multiplier algebra of the direct sum of two commutative semisimple Banach algebras. Then as an application we show that $ \mathcal{A} \oplus \mathcal{B} $ is a BSE algebra if and only if $ \mathcal{A} $ and $ \mathcal{B} $ are BSE. We also prove that if the algebra $ \mathcal{A} \hspace{0.167em} {\mathop{\times }\nolimits}_{\theta } \hspace{0.167em} \mathcal{B} $ with $\theta $-Lau product is a BSE algebra and $ \mathcal{B} $ is unital then $ \mathcal{B} $ is a BSE algebra. We present some examples which show that the BSE property of $ \mathcal{A} \hspace{0.167em} {\mathop{\times }\nolimits}_{\theta } \hspace{0.167em} \mathcal{B} $ does not imply the BSE property of $ \mathcal{A} $, even in the case where $ \mathcal{B} $ is unital.