Abstract
It is shown that the Banach algebra \(C_{ub}(X,d)\) of bounded uniformly \({{\mathbb {K}}}\)-valued continuous functions on a metric space (X, d) is coherent if and only if d is a uniformly discrete metric, or equivalently, if X does not contain twin sequences. The proof is based on Neville’s result that \(C(X, {\mathbb {R}})\) for a Tychonov space is coherent if and only if X is basically disconnected. Since \(C_{ub}(X,d)\) is self-adjoint, we also include in the survey part some general results on the spectrum M(A) of general self-adjoint Banach function algebras which we need for the special case here.
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Notes
Writing \(C(X,{{\mathbb {K}}})\) here would be too restrictive here, as we need to consider real algebras of complex-valued functions. Note that in our notation, the \({{\mathbb {C}}}\) in \(C(X,{{\mathbb {C}}})\) just denotes the target space of the functions and not the underlying field of scalars on which \(C(X,{{\mathbb {C}}})\) operates.
The first line in the displayed formula is only a formal expression, since complex scalars are not necessarily elements of \({\mathcal {R}}\)
We do not assume that singletons are closed.
For instance, we may take \({\mathcal {F}}=C(X, {\mathbb {R}})\).
Note that the elements in \(X/_\sim \) are subsets of X.
Recall that \(x^*=x\) is possible.
Note that when defining \(f(E):=\{f(x):x\in E\}\), we have \(\pi (E)=\bigcup _{x\in E} \{\pi (x)\}\subseteq X/_\sim \), which is of course different from \(\underline{\pi }(E)=\bigcup _{x\in E} \underline{ \pi }(\{x\})= \bigcup _{x\in E} \pi (x)\subseteq X\).
This property is motivated by [23, Definition 32.120] introducing L-sets, or \({\mathcal {R}}\)-rationally convex sets in real Banach algebras.
Attention: in the second case \({\mathcal {H}}(x_0)\) is not necessarily a subset of L since L is assumed to contain only the real constants.
This is in [31] for complex algebras.
Without this condition, the algebra \(A(\varvec{D})_\mathrm{sym}|_{[-1,1]}\) considered above, yields a counter-example (here \(\tau (x)=x\) for \(x\in [-1,1]\)).
By the way, (2) \(\Longrightarrow \) (1) is much easier to prove than (3) \(\Longrightarrow \) (1).
This is a method presented in [23, Theorem 8.55]. For sequences we would simply have \(x_1,y_1,x_2,y_2,\ldots \)
In German: “maximale Rückspiegelung”; a terminology coined in [11] when studying closures of Gleason parts in \(H^\infty ({{\mathbb {D}}})\).
If \(B=A\) then we may also use (1.3) to conclude that \(\iota \) is a homeomorphism onto \(\Delta \).
See [23, Def. 1.258] for the definition used here.
Which we identify with X if X is compact.
The equivalence of (1) with (2) may be viewed as a very special case of [23, Theorem 8.30].
More precisely \(\overline{\iota _A(E)}^{M(A)}\cap \; \overline{\iota _A(F)}^{M(A)}=\emptyset \).
In particular, this holds if X is a compact metric space.
Use that \(L:=I\cap J=(da)\cap (db)\) with \(1=ua+vb\) for some u, v and that \(ax=by\) implies that b divides a ([23, Lemma 19.36]), to prove that \(L=(dab)\).
This is from [4], where it is claimed without proof.
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Acknowledgements
The authors thank Amol Sasane for some initial thoughts on Sect. 4 and Alexander Brudnyi for some information on \(C_{ub}({{\mathbb {D}}},\psi )\).
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Mortini , R., Rupp, R. A Characterization of Coherence of the Algebra of Bounded Uniformly Continuous Functions on a Metric Space and the Spectrum of General Self-Adjoint Banach Function Algebras. Results Math 76, 210 (2021). https://doi.org/10.1007/s00025-021-01523-1
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DOI: https://doi.org/10.1007/s00025-021-01523-1