Hopfian $\ell$-groups, MV-algebras and AF~{C}*-algebras

Daniele Mundici

doi:10.1515/forum-2015-0177

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Hopfian $\ell$-groups, MV-algebras and AF~{C}^*-algebras

Daniele Mundici

Published/Copyright: May 1, 2016

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From the journal Forum Mathematicum Volume 28 Issue 6

Abstract

An algebra is said to be hopfian if it is not isomorphic to a proper quotient of itself. We describe several classes of hopfian and of non-hopfian unital lattice-ordered abelian groups and MV-algebras. Using Elliott classification and K0-theory, we apply our results to other related structures, notably the Farey–Stern–Brocot AF C*-algebra and all its primitive quotients, including the Behnke–Leptin C*-algebras 𝒜k,q.

Keywords: Hopfian algebra; residually finite; MV-algebra; abelian lattice-ordered group; piecewise linear function; hull kernel topology; Yosida duality; Baker–Beynon duality; germ of a function; Bratteli diagram; Elliott classification; residually finite dimensional

MSC 2010: 06F20; 06D35; 08A10; 08A35; 08B20; 08B30; 08C05; 18A20; 18C05; 19A49; 20M05; 46L05; 46L80; 46M40; 52B55; 57Q05; 57Q25

Communicated by Manfred Droste

A Appendix: Background on MV-algebras

Here we collect a number of basic MV-algebraic results which have been used repeatedly in the previous sections. Each result comes with a reference where the interested reader can find a proof.

A.1 Representation

Lemma A.1

Lemma A.1 ([15, Section 1.2.8])

Let A and B be MV-algebras. If σ is a homomorphism of A onto B, then there is an isomorphism τ of A/ker⁡(σ) onto B such that τ⁢(x/ker⁡(σ))=σ⁢(x) for all x∈A.

Theorem A.2

The following statements hold.

[15, Theorem 3.5.1] An MV-algebra A is simple if and only if it is isomorphic to a subalgebra of the standard MV-algebra [0,1].
[15, Theorem 9.1.5] For each cardinal κ, the free MV-algebra Freeκ on κ free generators is given by the McNaughton functions over [0,1]κ, with pointwise operations, i.e., those functions g:[0,1]κ→[0,1] such that there are ordinals α⁢(0)<…<α⁢(m-1)<κ and a McNaughton function f over [0,1]m having the following property: for each x∈[0,1]κ, g⁢(x)=f⁢(xα⁢(0),…,xα⁢(m-1)).
[15, Theorem 3.6.7] An MV-algebra A with κ generators is semisimple if and only if for some nonempty closed subset X⊆[0,1]κ, A is isomorphic to the MV-algebra ℳ⁢(X) of restrictions to X of all functions in Freeκ.

A.2 Yosida duality

For any nonempty compact Hausdorff space X≠∅ we let C⁢(X) denote the MV-algebra of all continuous [0,1]-valued functions on X, with the pointwise operations of the MV-algebra [0,1].

An MV-subalgebra A of C⁢(X) is said to be separating if for any two distinct points x,y∈X, there is f∈A such that f⁢(x)=0 and f⁢(y)>0. Following [33], for each ideal 𝔦 of A we let

𝒵𝔦=⋂{f-1⁢(0)∣f∈𝔦}

(𝒵𝔦 is denoted V𝔦 in [15]).

Proposition A.3

Proposition A.3 ([15, Proposition 3.4.5])

Let X be a compact Hausdorff space and let A be a separating subalgebra of Cont⁢(X). Then the map f/i↦f↾Zi is an isomorphism of A/i onto A↾Zi if and only if i is an intersection of maximal ideals of A.

For every MV-algebra A, we let hom⁡(A) denote the set of homomorphisms of A into the standard MV-algebra [0,1].

Theorem A.4

Theorem A.4 (Yosida duality, [33, Theorem 4.16])

Let A be an MV-algebra.

For any maximal ideal 𝔪 of A there is a unique pair (𝔪¯,I𝔪) with I𝔪 an MV-subalgebra of [0,1] and 𝔪¯ an isomorphism of A/𝔪 onto I𝔪.
The map ker:η↦ker⁡η is a one-to-one correspondence between hom⁡(A) and 𝝁⁢(A). The inverse map sends each 𝔪∈𝝁⁢(A) to the homomorphism η𝔪:A→[0,1] given by a↦𝔪¯⁢(a/𝔪). For each θ∈hom⁡(A) and a∈A, θ⁢(a)=ker⁡θ¯⁢(a/ker⁡θ).
The map :*a∈A↦a*∈[0,1]𝝁⁢(A) defined by a*⁢(𝔪)=𝔪¯⁢(a/𝔪), is a homomorphism of A onto a separating MV-subalgebra A* of C⁢(𝝁⁢(A)). The map a↦a* is an isomorphism of A onto A* if and only if A is semisimple.
Suppose X≠∅ is a compact Hausdorff space and B is a separating subalgebra of C⁢(X). Then the map ι:x∈X↦𝔪x={f∈B∣f⁢(x)=0} is a homeomorphism of X onto 𝝁⁢(B). The inverse map ι-1 sends each 𝔪∈𝝁⁢(B) to the only element of the set 𝒵⁢𝔪.
From the hypotheses of (iv) it follows that f*∘ι=f for each f∈B. Thus the map f*∈B*↦f*∘ι∈C⁢(X) is the inverse of the isomorphism :*B≅B* defined in (iii). In particular, f⁢(x)=f*⁢(𝔪x) for each x∈X.
[33, Corollary 4.18] For every nonempty closed subset Y of [0,1]κ, the map
ι:x∈Y↦𝔪x={f∈ℳ⁢(Y)∣f⁢(x)=0}
of (iv) is a homeomorphism of Y onto 𝝁⁢(ℳ⁢(Y)). The inverse map 𝔪↦x𝔪 sends every maximal ideal 𝔪 of ℳ⁢(Y)to the only element of 𝒵⁢𝔪.

Notational stipulations.

In the light of Theorem A.4 (i)–(iii), for every MV-algebra A, a∈A and 𝔪∈𝝁⁢(A) we will tacitly identify a/𝔪 with the real number 𝔪¯⁢(a/𝔪), and write a/𝔪=𝔪¯⁢(a/𝔪)=a*⁢(𝔪). Further, if B is a separating MV-subalgebra of C⁢(X) as in Theorem A.4 (iv)–(v), identifying B with B* and X with 𝝁⁢(B), we will write without fear of ambiguity, f⁢(x)=f⁢(𝔪x)=f/𝔪x for each x∈X and f∈B.

A.3 The Γ functor

Theorem A.5

The following statements hold.

[30, Theorem 3.9] For each unital ℓ-group (G,u) let Γ⁢(G,u) be the MV-algebra ([0,1],0,¬,⊕), where ¬⁢x=u-x and x⊕y=min⁡(u,x+y). Further, for every homomorphism ξ:(G,u)→(H,v) let Γ⁢(ξ) be the restriction of ξ to [0,u]. Then Γ is a categorical equivalence between unital ℓ-groups and MV-algebras.
[15, Theorem 7.2.2] Let A=Γ⁢(G,u). Then the correspondence ϕ:𝔦↦ϕ⁢(𝔦)={x∈G∣|x|∧u∈𝔦} is an order-isomorphism from the set of ideals of A onto the set of ideals of G, both sets being ordered by inclusion. The inverse isomorphism ψ is given by ψ:𝔧↦ψ⁢(𝔧)=𝔧∩[0,u].
[15, Theorem 7.2.4] For every ideal 𝔧 of G, the MV-algebra Γ⁢(G/𝔧,u/𝔧) is isomorphic to the quotient MV-algebra Γ⁢(G,u)/(𝔧∩[0,u]).

Acknowledgements

The author is grateful to the referee for his/her careful reading and valuable suggestions for improvement. He is also indebted to L. M. Cabrer for Remark 2.3.

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Received: 2015-9-10

Revised: 2015-12-7

Published Online: 2016-5-1

Published in Print: 2016-11-1

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Keywords for this article

Hopfian algebra; residually finite; MV-algebra; abelian lattice-ordered group; piecewise linear function; hull kernel topology; Yosida duality; Baker–Beynon duality; germ of a function; Bratteli diagram; Elliott classification; residually finite dimensional