Let L¹(G)** be the second dual of the group algebra L¹(G) of a locally compact group G. We study the question of involutions on L¹(G)**. A new class of subamenable groups is introduced which is universal for all groups. There is no involution on L¹(G)** for a subamenable group G.

The cardinal functions of pseudocharacter, closed pseudocharacter, and character are used to examine H-closed spaces and to contrast the differences between H-closed and minimal Hausdorff spaces. An H-closed space $X$ is produced with the properties that $\left|X\right|>2^{2^{\psi \left(X\right)}}$ and $\overline{\psi }\left(X\right)>2^{\psi \left(X\right)}$.

For a Tychonoff space $X$, $C\left(X\right)$ is the lattice-ordered group ($l$-group) of real-valued continuous functions on $X$, and $C^{*}\left(X\right)$ is the sub-$l$-group of bounded functions. A property that $X$ might have is (AP) whenever $G$ is a divisible sub-$l$-group of $C^{*}\left(X\right)$, containing the constant function 1, and separating points from closed sets in $X$, then any function in $C\left(X\right)$ can be approximated uniformly over $X$ by functions which are locally in $G$. The vector lattice version of the Stone-Weierstrass Theorem is more-or-less equivalent...

A point x is a (bow) tie-point of a space X if X∖x can be partitioned into (relatively) clopen sets each with x in its closure. We denote this as $X=A{\bowtie }_{x}B$ where A, B are the closed sets which have a unique common accumulation point x. Tie-points have appeared in the construction of non-trivial autohomeomorphisms of βℕ = ℕ* (by Veličković and Shelah Steprans) and in the recent study (by Levy and Dow Techanie) of precisely 2-to-1 maps on ℕ*. In these cases the tie-points have been the unique fixed point...

We study closed subspaces of $\kappa$-Ohio complete spaces and, for $\kappa$ uncountable cardinal, we prove a characterization for them. We then investigate the behaviour of products of $\kappa$-Ohio complete spaces. We prove that, if the cardinal ${\kappa }^{+}$ is endowed with either the order or the discrete topology, the space ${\left({\kappa }^{+}\right)}^{{\kappa }^{+}}$ is not $\kappa$-Ohio complete. As a consequence, we show that, if $\kappa$ is less than the first weakly inaccessible cardinal, then neither the space ${\omega }^{{\kappa }^{+}}$, nor the space ${ℝ}^{{\kappa }^{+}}$ is $\kappa$-Ohio complete.

Let ${(e_{\beta }:𝐐\to Y_{\beta })}_{\beta \in \mathbf{\text{Ord}}}$ be the large source of epimorphisms in the category $\mathbf{\text{Ury}}$ of Urysohn spaces constructed in [2]. A sink ${(g_{\beta }:Y_{\beta }\to X)}_{\beta \in \mathbf{\text{Ord}}}$ is called natural, if $g_{\beta }\circ e_{\beta }=g_{{\beta }^{\text{'}}}\circ e_{{\beta }^{\text{'}}}$ for all $\beta ,{\beta }^{\text{'}}\in \mathbf{\text{Ord}}$. In this paper natural sinks are characterized. As a result it is shown that $\mathbf{\text{Ury}}$ permits no $(Epi,ℳ)$-factorization structure for arbitrary (large) sources.