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Ross Stokke

Publications and source records attributed to Ross Stokke.

10 recordsLinked to original sources
arXiv2026

(Generalized) Spine Subalgebras of Fourier-Stieltjes algebras and their Homomorphisms

For any upper semilattice ${\cal D}$ of locally precompact topologies on a locally compact group $G$, we define an associated generalized spine subalgebra $A^*_{\cal D}(G)$ of the Fourier-Stieltjes algebra $B(G)$. We show that $A^*_{\cal D}(G)$ is a semilattice-graded $\ell^1$-direct sum of maximal copies of Fourier algebras and we identify its spectrum as a semilattice of groups. We build a collection of examples of generalized spine algebras over whose spectra we exhibit fine control. We define notions of compatible fusions of homomorphisms and affine maps, and use these definitions to characterize all completely positive, completely contractive and, when $G$ is amenable, all completely bounded homomorphisms from a generalized spine algebra $A^*_{\cal D}(G)$ to a Fourier-Stieltjes algebra $B(H)$. These results are new, even when $A^*_{\cal D}(G)$ is the full spine algebra $A^*(G)$ and even when $G$ and $H$ are abelian. We provide examples illustrating the scope of our theorems.

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arXiv2026

Totally Disconnected Semigroup Compactifications: Non-Introversion of the Full Boolean Algebra of Clopen Sets

In terms of the existence of a single clopen set and two related nets, we characterize when the full Boolean algebra, ${\mathfrak B}(G)$, of clopen subsets of a topological group $G$ is left introverted. We employ this characterization to show that when $G$ is a first countable, $\sigma$-compact, totally disconnected locally compact group, ${\mathfrak B}(G)$ is left introverted if and only if $G$ is compact or discrete, thus providing a strong positive answer to a question posed in Stephens and Stokke (Q J Math 2023). Examples of clopen sets and nets witnessing our non-introversion theorem are presented. Some hereditary properties of left introversion of ${\mathfrak B}(G)$ are proved and then employed to extend our main result to other classes of topological groups.

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arXiv2022

Totally disconnected semigroup compactifications of topological groups

We introduce the notion of an introverted Boolean algebra $\cal B$ of closed-and-open subsets of a topological group $G$, show that the associated Stone space $(ν_{\cal B} G, ν_{\cal B})$ is a totally disconnected semigroup compactification of $G$, and show that every totally disconnected semigroup compactification of $G$ takes this form. We identify and study the universal totally disconnected semigroup compactification, the universal totally disconnected semitopological semigroup compactification and the universal totally disconnected group compactification of $G$. Our main results are obtained independently of Gelfand theory and well-known properties of the (typically non-totally disconnected) universal compactifications $G^{LUC}$, $G^{WAP}$ and $G^{AP}$, though we do employ Gelfand theory to clarify the relationship between these familiar universal compactifications and their totally disconnected counterparts.

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arXiv2021

Norm-multiplicative homomorphisms of Beurling algebras

We introduce and study "norm-multiplicative" homomorphisms $φ: {\cal L}^1(F) \rightarrow {\cal M}_r(G)$ between group and measure algebras, and $φ: {\cal L}^1(ω_F) \rightarrow {\cal M}(ω_G)$ between Beurling group and measure algebras, where $F$ and $G$ are locally compact groups with continuous weights $ω_F$ and $ω_G$. Through a unified approach we recover, and sometimes strengthen, many of the main known results concerning homomorphisms and isomorphisms between these (Beurling) group and measure algebras. We provide a first description of all positive homomorphisms $φ: {\cal L}^1(F) \rightarrow {\cal M}_r(G)$. We state versions of our results that describe a variety of (possibly unbounded) homomorphisms $φ: \mathbb{C} F \rightarrow \mathbb{C} G$ for (discrete) groups $F$ and $G$.

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arXiv2021

On Beurling measure algebras

We show how the measure theory of regular compacted-Borel measures defined on the $δ$-ring of compacted-Borel subsets of a weighted locally compact group $(G,ω)$ provides a compatible framework for defining the corresponding Beurling measure algebra ${\cal M}(G,ω)$, thus filling a gap in the literature.

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arXiv2020

Homomorphisms of Fourier-Stieltjes algebras

Every homomorphism $φ: B(G) \rightarrow B(H)$ between Fourier-Stieltjes algebras on locally compact groups $G$ and $H$ is determined by a continuous mapping $α: Y \rightarrow Δ(B(G))$, where $Y$ is a set in the open coset ring of $H$ and $Δ(B(G))$ is the Gelfand spectrum of $B(G)$ (a $*$-semigroup). We exhibit a large collection of maps $α$ for which $φ=j_α: B(G) \rightarrow B(H)$ is a completely positive/completely contractive/completely bounded homomorphism and establish converse statements in several instances. For example, we fully characterize all completely positive/completely contractive/completely bounded homomorphisms $φ: B(G) \rightarrow B(H)$ when $G$ is a Euclidean- or $p$-adic-motion group. In these cases, our description of the completely positive/completely contractive homomorphisms employs the notion of a "fusion map of a compatible system of homomorphisms/affine maps" and is quite different from the Fourier algebra situation.

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arXiv2018

Fourier spaces and completely isometric representations of Arens product algebras

Motivated by the definition of a semigroup compactification of a locally compact group and a large collection of examples, we introduce the notion of an (operator) "homogeneous left dual Banach algebra" (HLDBA) over a (completely contractive) Banach algebra $A$. We prove a Gelfand-type representation theorem showing that every HLDBA over $A$ has a concrete realization as an (operator) homogeneous left Arens product algebra: the dual of a subspace of $A^*$ with a compatible (matrix) norm and a type of left Arens product ${\scriptscriptstyle \square}$. Examples include all left Arens product algebras over $A$, but also -- when $A$ is the group algebra of a locally compact group -- the dual of its Fourier algebra. Beginning with any (completely) contractive (operator) $A$-module action $Q$ on a space $X$, we introduce the (operator) Fourier space $({\cal F}_Q(A^*), \| \cdot \|_Q)$ and prove that $({\cal F}_Q(A^*)^*, {\scriptscriptstyle \square})$ is the unique (operator) HLDBA over $A$ for which there is a weak$^*$-continuous completely isometric representation as completely bounded operators on $X^*$ extending the dual module representation. Applying our theory to several examples of (completely contractive) Banach algebras $A$ and module operations, we provide new characterizations of familiar HLDBAs over $A$ and we recover -- and often extend -- some (completely) isometric representation theorems concerning these HLDBAs.

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arXiv2014

Extension of derivations, and Connes-amenability of the enveloping dual Banach algebra

If $D:A \to X$ is a derivation from a Banach algebra to a contractive, Banach $A$-bimodule, then one can equip $X^{**}$ with an $A^{**}$-bimodule structure, such that the second transpose $D^{**}: A^{**} \to X^{**}$ is again a derivation. We prove an analogous extension result, where $A^{**}$ is replaced by $\F(A)$, the \emph{enveloping dual Banach algebra} of $A$, and $X^{**}$ by an appropriate kind of universal, enveloping, normal dual bimodule of $X$. Using this, we obtain some new characterizations of Connes-amenability of $\F(A)$. In particular we show that $\F(A)$ is Connes-amenable if and only if $A$ admits a so-called WAP-virtual diagonal. We show that when $A=L^1(G)$, existence of a WAP-virtual diagonal is equivalent to the existence of a virtual diagonal in the usual sense. Our approach does not involve invariant means for $G$.

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arXiv2012

Matrix coefficients of unitary representations and associated compactifications

We study, for a locally compact group $G$, the compactifications $(π,G^π)$ associated with unitary representations $π$, which we call {\it $π$-Eberlein compactifications}. We also study the Gelfand spectra $Φ_{\mathcal{A}}(π)}$ of the uniformly closed algebras $\mathcal{A}(π)$ generated by matrix coefficients of such $π$. We note that $Φ_{\mathcal{A}(π)}\cup\{0\}$ is itself a semigroup and show that the Šilov boundary of $\mathcal{A}(π)$ is $G^π$. We study containment relations of various uniformly closed algebras generated by matrix coefficients, and give a new characterisation of amenability: the constant function 1 can be uniformly approximated by matrix coefficients of representations weakly contained in the left regular representation if and only if $G$ is amenable. We show that for the universal representation $ω$, the compactification $(ω,G^ω)$ has a certain universality property: it is universal amongst all compactifications of $G$ which may be embedded as contractions on a Hilbert space, a fact which was also recently proved by Megrelishvili. We illustrate our results with examples including various abelian and compact groups, and the $ax+b$-group. In particular, we witness algebras $\fA(π)$, for certain non-self-conjugate $π$, as being generalised algebras of analytic functions.

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