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A complete sup-lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q1300101.png" /> together with an associative product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q1300102.png" /> satisfying the distributive laws
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{{MSC|06F07}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q1300103.png" /></td> </tr></table>
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A complete sup-lattice $Q$ together with an associative product $\otimes$ satisfying the distributive laws
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$$
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a \otimes \left({ \bigvee_i b_i }\right) = \bigvee_i a \otimes b_i
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$$
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$$
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\left({ \bigvee_i b_i }\right) \otimes a= \bigvee_i b_i \otimes a
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$$
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for all $a, b_i \in Q$ (cf. also [[Lattice]]; [[Distributivity]]; [[Associativity]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q1300104.png" /></td> </tr></table>
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The name  "quantale" was introduced by C.J. Mulvey [[#References|[a1]]] to provide a non-commutative extension of the concept of [[locale]]. The intention was to develop the concept of non-commutative topology introduced by R. Giles and H. Kummer [[#References|[a2]]], while providing a constructive, and non-commutative, context for the foundations of quantum mechanics and, more generally, non-commutative logic. The observation that the closed right ideals of a [[C*-algebra|$C^*$-algebra]] form a quantale satisfying the conditions that each element is right-sided ($a \otimes 1_Q \le a$) and idempotent ($a \otimes a = a$)) led certain authors to restrict the term  "quantale" to mean only quantales of this kind [[#References|[a3]]], but the term is now applied only in its original sense.
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q1300105.png" /> (cf. also [[Lattice|Lattice]]; [[Distributivity|Distributivity]]; [[Associativity|Associativity]]).
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The realization by J. Rosický [[#References|[a4]]] that the development of topological concepts such as regularity required additional structure led [[#References|[a5]]] to the consideration of involutive quantales, and of the spectrum $\text{Max} A$ of a $C^*$-algebra $A$ (cf. also [[Spectrum of a C*-algebra|Spectrum of a $C^*$-algebra]]) as the quantale of closed linear subspaces of $A$, together with the operations of join given by closed linear sum, product given by closed linear product of subspaces, and involution by involution within the $C^*$-algebra. The right-sided elements of the spectrum $\text{Max} A$ are the closed right ideals of the $C^*$-algebra $A$ (cf. [[#References|[a2]]], [[#References|[a6]]]). By the existence of approximate units, each element $a \in R(\text{Max} A)$ of the sup-lattice of right-sided elements satisfies the condition that $a \otimes a^* \otimes a = a$. By a ''Gel'fand quantale'' $Q$ is meant an involutive unital quantale in which the right-sided (equivalently, left-sided) elements satisfy this condition.
  
The name  "quantale"  was introduced by C.J. Mulvey [[#References|[a1]]] to provide a non-commutative extension of the concept of [[Locale|locale]]. The intention was to develop the concept of non-commutative topology introduced by R. Giles and H. Kummer [[#References|[a2]]], while providing a constructive, and non-commutative, context for the foundations of quantum mechanics and, more generally, non-commutative logic. The observation that the closed right ideals of a [[C*-algebra|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q1300106.png" />-algebra]] form a quantale satisfying the conditions that each element is right-sided (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q1300107.png" />) and idempotent (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q1300108.png" />) led certain authors to restrict the term  "quantale"  to mean only quantales of this kind [[#References|[a3]]], but the term is now applied only in its original sense.
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Generalizing an observation in [[#References|[a4]]], the right-sided elements of any involutive quantale $Q$ may be shown to admit a pseudo-orthocomplement, defined by $a^\perp = \bigvee_{a^* \otimes b = 0_Q} b$. In any Gel'fand quantale $Q$, the right-sided elements are idempotent, and the two-sided elements form a locale.
  
The realization by J. Rosický [[#References|[a4]]] that the development of topological concepts such as regularity required additional structure led [[#References|[a5]]] to the consideration of involutive quantales, and of the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001010.png" /> of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001011.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001012.png" /> (cf. also [[Spectrum of a C*-algebra|Spectrum of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001013.png" />-algebra]]) as the quantale of closed linear subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001014.png" />, together with the operations of join given by closed linear sum, product given by closed linear product of subspaces, and involution by involution within the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001015.png" />-algebra. The right-sided elements of the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001016.png" /> are the closed right ideals of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001017.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001018.png" /> (cf. [[#References|[a2]]], [[#References|[a6]]]). By the existence of approximate units, each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001019.png" /> of the sup-lattice of right-sided elements satisfies the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001020.png" />. By a Gel'fand quantale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001021.png" /> is meant an involutive unital quantale in which the right-sided (equivalently, left-sided) elements satisfy this condition.
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Observing that relations on a set $X$ forming a quantale with respect to arbitrary union and composition is applied implicitly by C.A. Hoare and He Jifeng when considering the weakest pre-specification of a program [[#References|[a7]]], and noting that the quantale $\mathcal{Q}(X)$ in question is exactly that of endomorphisms of the sup-lattice $\mathcal{P}(X)$ of subsets of $X \times X$, led to the consideration [[#References|[a8]]] of the quantale $\mathcal{Q}(S)$ of endomorphisms of any orthocomplemented sup-lattice $S$, in which the involute $\alpha^*$ of a sup-preserving mapping $\alpha$ is defined by $s \alpha^* = \left({ \bigvee_{t \alpha \le s^\perp} t} \right)^\perp$ for each $s \in S$. In the quantale $\mathcal{Q}(X)$ of relations on a set $X$, this describes the [[Transposed relation|reverse]] of a relation in terms of complementation of subsets. Observing that the quantale $\mathcal{Q}(H)$ of endomorphisms of the orthocomplemented sup-lattice of closed linear subspaces of a [[Hilbert space]] $H$ provides a motivating example for this quantization of the calculus of relations, the term ''Hilbert quantale'' was introduced for any quantale isomorphic to the quantale $Q9S)$ of an orthocomplemented sup-lattice $S$.
  
Generalizing an observation in [[#References|[a4]]], the right-sided elements of any involutive quantale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001022.png" /> may be shown to admit a pseudo-orthocomplement, defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001023.png" />. In any Gel'fand quantale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001024.png" />, the right-sided elements are idempotent, and the two-sided elements form a locale.
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Noting that the weak spectrum $\text{Max}_{\text{W}}(B)$ of a [[von Neumann algebra]] $B$ is a Gel'fand quantale of which the right-sided elements correspond to the projections of $B$ and on which the right pseudo-orthocomplement corresponds to orthocomplementation of projections, a Gel'fand quantale $Q$ is said to be a ''von Neumann quantale'' if $(a^\perp)^\perp = a$ for any right-sided element $a \in Q$. For any von Neumann quantale $Q$, the locale $I(Q)$ of two-sided elements is a complete [[Boolean algebra]]. Any Hilbert quantale $Q$ is a von Neumann quantale, and a von Neumann quantale $Q$ is a Hilbert quantale exactly if the canonical homomorphism $\mu_Q : Q \rightarrow \mathcal{Q}(R(Q))$, assigning to each $a \in Q$ the sup-preserving mapping $b \in R(Q) \mapsto a^* \otimes b \in R(Q)$ on the orthocomplemented sup-lattice $R(Q)$ of right-sided elements of $Q$, is an isomorphism [[#References|[a8]]]. Any Hilbert quantale $Q$ is a von Neumann factor quantale in the sense that $I(Q)$ is exactly $\mathbf2]$. The weak spectrum $\text{Max}_{\text{W}}(B)$ of a von Neumann algebra $B$ is a factor exactly if $B$ is a factor [[#References|[a9]]] (cf. also [[von Neumann algebra]]).
  
Observing that relations on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001025.png" /> forming a quantale with respect to arbitrary union and composition is applied implicitly by C.A. Hoare and He Jifeng when considering the weakest pre-specification of a program [[#References|[a7]]], and noting that the quantale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001026.png" /> in question is exactly that of endomorphisms of the sup-lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001027.png" /> of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001028.png" />, led to the consideration [[#References|[a8]]] of the quantale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001029.png" /> of endomorphisms of any orthocomplemented sup-lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001030.png" />, in which the involute <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001031.png" /> of a sup-preserving mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001032.png" /> is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001033.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001034.png" />. In the quantale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001035.png" /> of relations on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001036.png" />, this describes the reverse of a relation in terms of complementation of subsets. Observing that the quantale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001037.png" /> of endomorphisms of the orthocomplemented sup-lattice of closed linear subspaces of a [[Hilbert space|Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001038.png" /> provides a motivating example for this quantization of the calculus of relations, the term Hilbert quantale was introduced for any quantale isomorphic to the quantale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001039.png" /> of an orthocomplemented sup-lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001040.png" />.
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A homomorphism $\phi : Q \rightarrow \mathcal{Q}(S)$ from a Gel'fand quantale $Q$ to the Hilbert quantale $\mathcal{Q}(S)$ of an orthocomplemented sup-lattice $S$ is said to be a representation of $Q$ on $S$ [[#References|[a10]]]. A representation is said to be ''irreducible'' provided that $s \in S$ invariant (in the sense that $s \phi_a \le s$ for all $a \in Q$) implies $s = 0_Q$ or $s = 1_Q$. The irreducibility of a representation $\phi : Q \rightarrow \mathcal{Q}(S)$ is equivalent to the homomorphism being ''strong'', in the sense that $\phi(1_Q) = 1_{\mathcal{Q}(S)}$. A homomorphism $Q' \rightarrow Q$ of Gel'fand quantales is strong exactly if $Q' \rightarrow Q \rightarrow \mathcal{Q}(S)$ is irreducible whenever $Q \rightarrow \mathcal{Q}(S)$ is irreducible. A representation $\phi : Q \rightarrow \mathcal{Q}(S)$ of $Q$ on an atomic orthocomplemented sup-lattice $S$ is said to be ''algebraically irreducible'' provided that for any atoms $x,y \in S$ there exists an $a \in Q$ such that $x\phi_a = y$ (cf. also [[atomic lattice]]). Any algebraically irreducible representation is irreducible: the algebraically irreducible representations are those for which every atom is a cyclic generator. An algebraically irreducible representation $\phi : Q \rightarrow \mathcal{Q}(S)$ on an atomic orthocomplemented sup-lattice $S$ is said to be a ''point'' of the Gel'fand quantale $Q$. The points of the spectrum $\text{Max} A$ of a $C^*$-algebra $A$ correspond bijectively to the equivalence classes of irreducible representations of $A$ on a Hilbert space [[#References|[a10]]]. (Presently (2000), this is subject to the conjecture that every point of $\text{Max} A$ is non-trivial in the sense that there exists a pure state that maps properly. For a discussion of the role of pure states in this context, see [[#References|[a10]]].) In particular, the spectrum $\text{Max} A$ is an invariant of the $C^*$-algebra $A$. It may be noted that the Hilbert quantale $\mathcal{Q}(S)$ of an atomic orthocomplemented sup-lattice has, to within equivalence, a unique point; moreover, the reflection of such a Gel'fand quantale into the category of locales is exactly $\mathbf{2}$. In particular, the points of any locale are exactly its points in the sense of the theory of locales.
  
Noting that the weak spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001041.png" /> of a [[Von Neumann algebra|von Neumann algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001042.png" /> is a Gel'fand quantale of which the right-sided elements correspond to the projections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001043.png" /> and on which the right pseudo-orthocomplement corresponds to orthocomplementation of projections, a Gel'fand quantale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001044.png" /> is said to be a von Neumann quantale if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001045.png" /> for any right-sided element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001046.png" />. For any von Neumann quantale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001047.png" />, the locale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001048.png" /> of two-sided elements is a complete [[Boolean algebra|Boolean algebra]]. Any Hilbert quantale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001049.png" /> is a von Neumann quantale, and a von Neumann quantale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001050.png" /> is a Hilbert quantale exactly if the canonical homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001051.png" />, assigning to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001052.png" /> the sup-preserving mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001053.png" /> on the orthocomplemented sup-lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001054.png" /> of right-sided elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001055.png" />, is an isomorphism [[#References|[a8]]]. Any Hilbert quantale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001056.png" /> is a von Neumann factor quantale in the sense that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001057.png" /> is exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001058.png" />. The weak spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001059.png" /> of a von Neumann algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001060.png" /> is a factor exactly if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001061.png" /> is a factor [[#References|[a9]]] (cf. also [[Von Neumann algebra|von Neumann algebra]]).
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A von Neumann quantale $Q$ is said to be ''atomic'' provided that the orthocomplemented sup-lattice $R(Q)$ of its right-sided elements is atomic. For any atomic von Neumann quantale $Q$ the complete Boolean algebra of two-sided elements $I(Q)$ is atomic. Moreover, the canonical homomorphism $\mu_Q : Q \rightarrow \mathcal{Q}(R(Q))$ is algebraically irreducible exactly if $Q$ is a von Neumann factor quantale. A Gel'fand quantale $Q$ is said to be ''discrete'' provided that it is an atomic von Neumann quantale that admits a central decomposition of the unit $e_Q \in Q$, in the sense that the atoms of the complete Boolean algebra $I(Q)$ majorize a family of central projections with join $e_Q \in Q$. For any atomic von Neumann algebra $B$, the weak spectrum $\text{Max}_{\text{W}} B$ is a discrete von Neumann quantale. A locale $L$ is a discrete von Neumann quantale exactly if it is a complete atomic Boolean algebra, hence the power set of its set of points. A homomorphism $X \rightarrow Q$ of Gel'fand quantales is said to be:
  
A homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001062.png" /> from a Gel'fand quantale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001063.png" /> to the Hilbert quantale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001064.png" /> of an orthocomplemented sup-lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001065.png" /> is said to be a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001066.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001067.png" /> [[#References|[a10]]]. A representation is said to be irreducible provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001068.png" /> invariant (in the sense that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001069.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001070.png" />) implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001071.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001072.png" />. The irreducibility of a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001073.png" /> is equivalent to the homomorphism being strong, in the sense that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001074.png" />. A homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001075.png" /> of Gel'fand quantales is strong exactly if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001076.png" /> is irreducible whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001077.png" /> is irreducible. A representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001078.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001079.png" /> on an atomic orthocomplemented sup-lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001080.png" /> is said to be algebraically irreducible provided that for any atoms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001081.png" /> there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001082.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001083.png" /> (cf. also [[Atomic lattice|Atomic lattice]]). Any algebraically irreducible representation is irreducible: the algebraically irreducible representations are those for which every atom is a cyclic generator. An algebraically irreducible representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001084.png" /> on an atomic orthocomplemented sup-lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001085.png" /> is said to be a point of the Gel'fand quantale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001086.png" />. The points of the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001087.png" /> of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001088.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001089.png" /> correspond bijectively to the equivalence classes of irreducible representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001090.png" /> on a Hilbert space [[#References|[a10]]]. (Presently (2000), this is subject to the conjecture that every point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001091.png" /> is non-trivial in the sense that there exists a pure state that maps properly. For a discussion of the role of pure states in this context, see [[#References|[a10]]].) In particular, the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001092.png" /> is an invariant of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001093.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001094.png" />. It may be noted that the Hilbert quantale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001095.png" /> of an atomic orthocomplemented sup-lattice has, to within equivalence, a unique point; moreover, the reflection of such a Gel'fand quantale into the category of locales is exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001096.png" />. In particular, the points of any locale are exactly its points in the sense of the theory of locales.
+
''algebraically strong'' if $X \rightarrow Q \rightarrow \mathcal{Q}(S)$ is algebraically irreducible whenever $Q \rightarrow \mathcal{Q}(S)$ is an algebraically irreducible representation of $Q$ on an atomic orthocomplemented sup-lattice $S$;
  
A von Neumann quantale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001097.png" /> is said to be atomic provided that the orthocomplemented sup-lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001098.png" /> of its right-sided elements is atomic. For any atomic von Neumann quantale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q13001099.png" /> the complete Boolean algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010100.png" /> of two-sided elements is atomic. Moreover, the canonical homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010101.png" /> is algebraically irreducible exactly if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010102.png" /> is a von Neumann factor quantale. A Gel'fand quantale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010103.png" /> is said to be discrete provided that it is an atomic von Neumann quantale that admits a central decomposition of the unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010104.png" />, in the sense that the atoms of the complete Boolean algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010105.png" /> majorize a family of central projections with join <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010106.png" />. For any atomic von Neumann algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010107.png" />, the weak spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010108.png" /> is a discrete von Neumann quantale. A locale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010109.png" /> is a discrete von Neumann quantale exactly if it is a complete atomic Boolean algebra, hence the power set of its set of points. A homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010110.png" /> of Gel'fand quantales is said to be:
+
a ''right embedding'' if it restricts to an embedding $R(X) \rightarrow R(Q)$ of the lattices of right-sided elements;
  
algebraically strong if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010111.png" /> is algebraically irreducible whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010112.png" /> is an algebraically irreducible representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010113.png" /> on an atomic orthocomplemented sup-lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010114.png" />;
+
''discrete'' if it is an algebraically strong right embedding.  
  
a right embedding if it restricts to an embedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010115.png" /> of the lattices of right-sided elements;
+
A Gel'fand quantale $X$ is said to be ''spatial'' if it admits a discrete homomorphism $X \rightarrow Q$ into a discrete von Neumann quantale $Q$ [[#References|[a11]]]. For any $C^*$-algebra $A$, the canonical homomorphism
 
+
$$
discrete if it is an algebraically strong right embedding. A Gel'fand quantale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010116.png" /> is said to be spatial if it admits a discrete homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010117.png" /> into a discrete von Neumann quantale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010118.png" /> [[#References|[a11]]]. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010119.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010120.png" />, the canonical homomorphism
+
\text{Max} A \rightarrow \text{Max}_{\text{W}} B
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010121.png" /></td> </tr></table>
+
of its spectrum $\text{Max} A$ into the weak spectrum of its enveloping atomic von Neumann algebra $B$ is discrete, hence $\text{Max} A$ is spatial. Similarly, a locale $L$ is spatial as a Gel'fand quantale exactly if its canonical homomorphism into the power set of its set of points is discrete. More generally, a Gel'fand quantale $Q$ is spatial exactly if it has enough points, in the sense that if $a,b \in R(Q)$ are distinct, then there is an algebraically irreducible representation $\phi : Q \rightarrow \mathcal{Q}(S)$ on an atomic orthocomplemented sup-lattice $S$ such that $\phi_a, \phi_b \in R(\mathcal{Q}(S))$ are distinct [[#References|[a11]]].
 
 
of its spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010122.png" /> into the weak spectrum of its enveloping atomic von Neumann algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010123.png" /> is discrete, hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010124.png" /> is spatial. Similarly, a locale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010125.png" /> is spatial as a Gel'fand quantale exactly if its canonical homomorphism into the power set of its set of points is discrete. More generally, a Gel'fand quantale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010126.png" /> is spatial exactly if it has enough points, in the sense that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010127.png" /> are distinct, then there is an algebraically irreducible representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010128.png" /> on an atomic orthocomplemented sup-lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010129.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010130.png" /> are distinct [[#References|[a11]]].
 
  
 
In other important directions, Girard quantales have been shown [[#References|[a12]]] to provide a semantics for non-commutative linear logic, and Foulis quantales to generalize the Foulis semi-groups of complete orthomodular lattices [[#References|[a13]]]. The concepts of quantal set and of sheaf have been introduced [[#References|[a14]]] for the case of idempotent right-sided quantales, generalizing those for any locale. These concepts may be localized [[#References|[a15]]] to allow the construction of a fibration from which the quantale may be recovered directly. The representation of quantales by quantales of relations has also been examined [[#References|[a16]]].
 
In other important directions, Girard quantales have been shown [[#References|[a12]]] to provide a semantics for non-commutative linear logic, and Foulis quantales to generalize the Foulis semi-groups of complete orthomodular lattices [[#References|[a13]]]. The concepts of quantal set and of sheaf have been introduced [[#References|[a14]]] for the case of idempotent right-sided quantales, generalizing those for any locale. These concepts may be localized [[#References|[a15]]] to allow the construction of a fibration from which the quantale may be recovered directly. The representation of quantales by quantales of relations has also been examined [[#References|[a16]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.J. Mulvey,  "&amp;"  ''Rend. Circ. Mat. Palermo'' , '''12'''  (1986)  pp. 99–104</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Giles,  H. Kummer,  "A non-commutative generalization of topology"  ''Indiana Univ. Math. J.'' , '''21'''  (1971)  pp. 91–102</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K.I. Rosenthal,  "Quantales and their applications" , ''Pitman Research Notes in Math.'' , '''234''' , Longman  (1990)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Rosický,  "Multiplicative lattices and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010131.png" />-algebras"  ''Cah. Topol. Géom. Diff. Cat.'' , '''30'''  (1989)  pp. 95–110</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  C.J. Mulvey,  "Quantales" , ''Invited Lecture, Summer Conf. Locales and Topological Groups, Curaçao''  (1989)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  C.A. Akemann,  "Left ideal structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130010/q130010132.png" />-algebras"  ''J. Funct. Anal.'' , '''6'''  (1970)  pp. 305–317</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  C.A.R. Hoare,  He Jifeng,  "The weakest prespecification"  ''Inform. Proc. Lett.'' , '''24'''  (1987)  pp. 127–132</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  C.J. Mulvey,  J.W. Pelletier,  "A quantisation of the calculus of relations" , ''Category Theory 1991, CMS Conf. Proc.'' , '''13''' , Amer. Math. Soc.  (1992)  pp. 345–360</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  J.W. Pelletier,  "Von Neumann algebras and Hilbert quantales"  ''Appl. Cat. Struct.'' , '''5'''  (1997)  pp. 249–264</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  C.J. Mulvey,  J.W. Pelletier,  "On the quantisation of points"  ''J. Pure Appl. Algebra'' , '''159'''  (2001)  pp. 231–295</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  C.J. Mulvey,  J.W. Pelletier,  "On the quantisation of spaces"  ''J. Pure Appl. Math.''  (to appear)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  D. Yetter,  "Quantales and (non-commutative) linear logic"  ''J. Symbolic Logic'' , '''55'''  (1990)  pp. 41–64</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  C.J. Mulvey,  "Foulis quantales"  ''to appear''</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  C.J. Mulvey,  M. Nawaz,  "Quantales: Quantal sets" , ''Non-Classical Logics and Their Application to Fuzzy Subsets: A Handbook of the Mathematical Foundations of Fuzzy Set Theory'' , Kluwer Acad. Publ.  (1995)  pp. 159–217</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  U. Berni-Canani,  F. Borceux,  R. Succi-Cruciani,  "A theory of quantale sets"  ''J. Pure Appl. Algebra'' , '''62'''  (1989)  pp. 123–136</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  C. Brown,  D. Gurr,  "A representation theorem for quantales"  ''J. Pure Appl. Algebra'' , '''85'''  (1993)  pp. 27–42</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  C.J. Mulvey,  "&amp;"  ''Rend. Circ. Mat. Palermo'' , '''12'''  (1986)  pp. 99–104 {{ZBL|0633.46065}}</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Giles,  H. Kummer,  "A non-commutative generalization of topology"  ''Indiana Univ. Math. J.'' , '''21'''  (1971)  pp. 91–102</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  K.I. Rosenthal,  "Quantales and their applications" , ''Pitman Research Notes in Math.'' , '''234''' , Longman  (1990)</TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Rosický,  "Multiplicative lattices and $C^*$-algebras"  ''Cah. Topol. Géom. Diff. Cat.'' , '''30'''  (1989)  pp. 95–110</TD></TR>
 +
<TR><TD valign="top">[a5]</TD> <TD valign="top">  C.J. Mulvey,  "Quantales" , ''Invited Lecture, Summer Conf. Locales and Topological Groups, Curaçao''  (1989)</TD></TR>
 +
<TR><TD valign="top">[a6]</TD> <TD valign="top">  C.A. Akemann,  "Left ideal structure of $C^*$-algebras"  ''J. Funct. Anal.'' , '''6'''  (1970)  pp. 305–317</TD></TR>
 +
<TR><TD valign="top">[a7]</TD> <TD valign="top">  C.A.R. Hoare,  He Jifeng,  "The weakest prespecification"  ''Inform. Proc. Lett.'' , '''24'''  (1987)  pp. 127–132 {{ZBL|0622.68025}}</TD></TR>
 +
<TR><TD valign="top">[a8]</TD> <TD valign="top">  C.J. Mulvey,  J.W. Pelletier,  "A quantisation of the calculus of relations" , ''Category Theory 1991, CMS Conf. Proc.'' , '''13''' , Amer. Math. Soc.  (1992)  pp. 345–360 {{ZBL|0793.06008}}</TD></TR>
 +
<TR><TD valign="top">[a9]</TD> <TD valign="top">  J.W. Pelletier,  "Von Neumann algebras and Hilbert quantales"  ''Appl. Cat. Struct.'' , '''5'''  (1997)  pp. 249–264</TD></TR>
 +
<TR><TD valign="top">[a10]</TD> <TD valign="top">  C.J. Mulvey,  J.W. Pelletier,  "On the quantisation of points"  ''J. Pure Appl. Algebra'' , '''159'''  (2001)  pp. 231–295</TD></TR>
 +
<TR><TD valign="top">[a11]</TD> <TD valign="top">  C.J. Mulvey,  J.W. Pelletier,  "On the quantisation of spaces"  ''J. Pure Appl. Math.''  '''175''' (2002) pp.289-325 {{ZBL|1026.06018}}</TD></TR>
 +
<TR><TD valign="top">[a12]</TD> <TD valign="top">  D. Yetter,  "Quantales and (non-commutative) linear logic"  ''J. Symbolic Logic'' , '''55'''  (1990)  pp. 41–64</TD></TR>
 +
<TR><TD valign="top">[a13]</TD> <TD valign="top">  C.J. Mulvey,  "Foulis quantales"  ''to appear''</TD></TR>
 +
<TR><TD valign="top">[a14]</TD> <TD valign="top">  C.J. Mulvey,  M. Nawaz,  "Quantales: Quantal sets" , ''Non-Classical Logics and Their Application to Fuzzy Subsets: A Handbook of the Mathematical Foundations of Fuzzy Set Theory'' , Kluwer Acad. Publ.  (1995)  pp. 159–217</TD></TR>
 +
<TR><TD valign="top">[a15]</TD> <TD valign="top">  U. Berni-Canani,  F. Borceux,  R. Succi-Cruciani,  "A theory of quantale sets"  ''J. Pure Appl. Algebra'' , '''62'''  (1989)  pp. 123–136</TD></TR>
 +
<TR><TD valign="top">[a16]</TD> <TD valign="top">  C. Brown,  D. Gurr,  "A representation theorem for quantales"  ''J. Pure Appl. Algebra'' , '''85'''  (1993)  pp. 27–42</TD></TR>
 +
</table>
 +
 
 +
{{TEX|done}}

Revision as of 19:06, 15 August 2016

2020 Mathematics Subject Classification: Primary: 06F07 [MSN][ZBL]

A complete sup-lattice $Q$ together with an associative product $\otimes$ satisfying the distributive laws $$ a \otimes \left({ \bigvee_i b_i }\right) = \bigvee_i a \otimes b_i $$ $$ \left({ \bigvee_i b_i }\right) \otimes a= \bigvee_i b_i \otimes a $$ for all $a, b_i \in Q$ (cf. also Lattice; Distributivity; Associativity).

The name "quantale" was introduced by C.J. Mulvey [a1] to provide a non-commutative extension of the concept of locale. The intention was to develop the concept of non-commutative topology introduced by R. Giles and H. Kummer [a2], while providing a constructive, and non-commutative, context for the foundations of quantum mechanics and, more generally, non-commutative logic. The observation that the closed right ideals of a $C^*$-algebra form a quantale satisfying the conditions that each element is right-sided ($a \otimes 1_Q \le a$) and idempotent ($a \otimes a = a$)) led certain authors to restrict the term "quantale" to mean only quantales of this kind [a3], but the term is now applied only in its original sense.

The realization by J. Rosický [a4] that the development of topological concepts such as regularity required additional structure led [a5] to the consideration of involutive quantales, and of the spectrum $\text{Max} A$ of a $C^*$-algebra $A$ (cf. also Spectrum of a $C^*$-algebra) as the quantale of closed linear subspaces of $A$, together with the operations of join given by closed linear sum, product given by closed linear product of subspaces, and involution by involution within the $C^*$-algebra. The right-sided elements of the spectrum $\text{Max} A$ are the closed right ideals of the $C^*$-algebra $A$ (cf. [a2], [a6]). By the existence of approximate units, each element $a \in R(\text{Max} A)$ of the sup-lattice of right-sided elements satisfies the condition that $a \otimes a^* \otimes a = a$. By a Gel'fand quantale $Q$ is meant an involutive unital quantale in which the right-sided (equivalently, left-sided) elements satisfy this condition.

Generalizing an observation in [a4], the right-sided elements of any involutive quantale $Q$ may be shown to admit a pseudo-orthocomplement, defined by $a^\perp = \bigvee_{a^* \otimes b = 0_Q} b$. In any Gel'fand quantale $Q$, the right-sided elements are idempotent, and the two-sided elements form a locale.

Observing that relations on a set $X$ forming a quantale with respect to arbitrary union and composition is applied implicitly by C.A. Hoare and He Jifeng when considering the weakest pre-specification of a program [a7], and noting that the quantale $\mathcal{Q}(X)$ in question is exactly that of endomorphisms of the sup-lattice $\mathcal{P}(X)$ of subsets of $X \times X$, led to the consideration [a8] of the quantale $\mathcal{Q}(S)$ of endomorphisms of any orthocomplemented sup-lattice $S$, in which the involute $\alpha^*$ of a sup-preserving mapping $\alpha$ is defined by $s \alpha^* = \left({ \bigvee_{t \alpha \le s^\perp} t} \right)^\perp$ for each $s \in S$. In the quantale $\mathcal{Q}(X)$ of relations on a set $X$, this describes the reverse of a relation in terms of complementation of subsets. Observing that the quantale $\mathcal{Q}(H)$ of endomorphisms of the orthocomplemented sup-lattice of closed linear subspaces of a Hilbert space $H$ provides a motivating example for this quantization of the calculus of relations, the term Hilbert quantale was introduced for any quantale isomorphic to the quantale $Q9S)$ of an orthocomplemented sup-lattice $S$.

Noting that the weak spectrum $\text{Max}_{\text{W}}(B)$ of a von Neumann algebra $B$ is a Gel'fand quantale of which the right-sided elements correspond to the projections of $B$ and on which the right pseudo-orthocomplement corresponds to orthocomplementation of projections, a Gel'fand quantale $Q$ is said to be a von Neumann quantale if $(a^\perp)^\perp = a$ for any right-sided element $a \in Q$. For any von Neumann quantale $Q$, the locale $I(Q)$ of two-sided elements is a complete Boolean algebra. Any Hilbert quantale $Q$ is a von Neumann quantale, and a von Neumann quantale $Q$ is a Hilbert quantale exactly if the canonical homomorphism $\mu_Q : Q \rightarrow \mathcal{Q}(R(Q))$, assigning to each $a \in Q$ the sup-preserving mapping $b \in R(Q) \mapsto a^* \otimes b \in R(Q)$ on the orthocomplemented sup-lattice $R(Q)$ of right-sided elements of $Q$, is an isomorphism [a8]. Any Hilbert quantale $Q$ is a von Neumann factor quantale in the sense that $I(Q)$ is exactly $\mathbf2]$. The weak spectrum $\text{Max}_{\text{W}}(B)$ of a von Neumann algebra $B$ is a factor exactly if $B$ is a factor [a9] (cf. also von Neumann algebra).

A homomorphism $\phi : Q \rightarrow \mathcal{Q}(S)$ from a Gel'fand quantale $Q$ to the Hilbert quantale $\mathcal{Q}(S)$ of an orthocomplemented sup-lattice $S$ is said to be a representation of $Q$ on $S$ [a10]. A representation is said to be irreducible provided that $s \in S$ invariant (in the sense that $s \phi_a \le s$ for all $a \in Q$) implies $s = 0_Q$ or $s = 1_Q$. The irreducibility of a representation $\phi : Q \rightarrow \mathcal{Q}(S)$ is equivalent to the homomorphism being strong, in the sense that $\phi(1_Q) = 1_{\mathcal{Q}(S)}$. A homomorphism $Q' \rightarrow Q$ of Gel'fand quantales is strong exactly if $Q' \rightarrow Q \rightarrow \mathcal{Q}(S)$ is irreducible whenever $Q \rightarrow \mathcal{Q}(S)$ is irreducible. A representation $\phi : Q \rightarrow \mathcal{Q}(S)$ of $Q$ on an atomic orthocomplemented sup-lattice $S$ is said to be algebraically irreducible provided that for any atoms $x,y \in S$ there exists an $a \in Q$ such that $x\phi_a = y$ (cf. also atomic lattice). Any algebraically irreducible representation is irreducible: the algebraically irreducible representations are those for which every atom is a cyclic generator. An algebraically irreducible representation $\phi : Q \rightarrow \mathcal{Q}(S)$ on an atomic orthocomplemented sup-lattice $S$ is said to be a point of the Gel'fand quantale $Q$. The points of the spectrum $\text{Max} A$ of a $C^*$-algebra $A$ correspond bijectively to the equivalence classes of irreducible representations of $A$ on a Hilbert space [a10]. (Presently (2000), this is subject to the conjecture that every point of $\text{Max} A$ is non-trivial in the sense that there exists a pure state that maps properly. For a discussion of the role of pure states in this context, see [a10].) In particular, the spectrum $\text{Max} A$ is an invariant of the $C^*$-algebra $A$. It may be noted that the Hilbert quantale $\mathcal{Q}(S)$ of an atomic orthocomplemented sup-lattice has, to within equivalence, a unique point; moreover, the reflection of such a Gel'fand quantale into the category of locales is exactly $\mathbf{2}$. In particular, the points of any locale are exactly its points in the sense of the theory of locales.

A von Neumann quantale $Q$ is said to be atomic provided that the orthocomplemented sup-lattice $R(Q)$ of its right-sided elements is atomic. For any atomic von Neumann quantale $Q$ the complete Boolean algebra of two-sided elements $I(Q)$ is atomic. Moreover, the canonical homomorphism $\mu_Q : Q \rightarrow \mathcal{Q}(R(Q))$ is algebraically irreducible exactly if $Q$ is a von Neumann factor quantale. A Gel'fand quantale $Q$ is said to be discrete provided that it is an atomic von Neumann quantale that admits a central decomposition of the unit $e_Q \in Q$, in the sense that the atoms of the complete Boolean algebra $I(Q)$ majorize a family of central projections with join $e_Q \in Q$. For any atomic von Neumann algebra $B$, the weak spectrum $\text{Max}_{\text{W}} B$ is a discrete von Neumann quantale. A locale $L$ is a discrete von Neumann quantale exactly if it is a complete atomic Boolean algebra, hence the power set of its set of points. A homomorphism $X \rightarrow Q$ of Gel'fand quantales is said to be:

algebraically strong if $X \rightarrow Q \rightarrow \mathcal{Q}(S)$ is algebraically irreducible whenever $Q \rightarrow \mathcal{Q}(S)$ is an algebraically irreducible representation of $Q$ on an atomic orthocomplemented sup-lattice $S$;

a right embedding if it restricts to an embedding $R(X) \rightarrow R(Q)$ of the lattices of right-sided elements;

discrete if it is an algebraically strong right embedding.

A Gel'fand quantale $X$ is said to be spatial if it admits a discrete homomorphism $X \rightarrow Q$ into a discrete von Neumann quantale $Q$ [a11]. For any $C^*$-algebra $A$, the canonical homomorphism $$ \text{Max} A \rightarrow \text{Max}_{\text{W}} B $$ of its spectrum $\text{Max} A$ into the weak spectrum of its enveloping atomic von Neumann algebra $B$ is discrete, hence $\text{Max} A$ is spatial. Similarly, a locale $L$ is spatial as a Gel'fand quantale exactly if its canonical homomorphism into the power set of its set of points is discrete. More generally, a Gel'fand quantale $Q$ is spatial exactly if it has enough points, in the sense that if $a,b \in R(Q)$ are distinct, then there is an algebraically irreducible representation $\phi : Q \rightarrow \mathcal{Q}(S)$ on an atomic orthocomplemented sup-lattice $S$ such that $\phi_a, \phi_b \in R(\mathcal{Q}(S))$ are distinct [a11].

In other important directions, Girard quantales have been shown [a12] to provide a semantics for non-commutative linear logic, and Foulis quantales to generalize the Foulis semi-groups of complete orthomodular lattices [a13]. The concepts of quantal set and of sheaf have been introduced [a14] for the case of idempotent right-sided quantales, generalizing those for any locale. These concepts may be localized [a15] to allow the construction of a fibration from which the quantale may be recovered directly. The representation of quantales by quantales of relations has also been examined [a16].

References

[a1] C.J. Mulvey, "&" Rend. Circ. Mat. Palermo , 12 (1986) pp. 99–104 Zbl 0633.46065
[a2] R. Giles, H. Kummer, "A non-commutative generalization of topology" Indiana Univ. Math. J. , 21 (1971) pp. 91–102
[a3] K.I. Rosenthal, "Quantales and their applications" , Pitman Research Notes in Math. , 234 , Longman (1990)
[a4] J. Rosický, "Multiplicative lattices and $C^*$-algebras" Cah. Topol. Géom. Diff. Cat. , 30 (1989) pp. 95–110
[a5] C.J. Mulvey, "Quantales" , Invited Lecture, Summer Conf. Locales and Topological Groups, Curaçao (1989)
[a6] C.A. Akemann, "Left ideal structure of $C^*$-algebras" J. Funct. Anal. , 6 (1970) pp. 305–317
[a7] C.A.R. Hoare, He Jifeng, "The weakest prespecification" Inform. Proc. Lett. , 24 (1987) pp. 127–132 Zbl 0622.68025
[a8] C.J. Mulvey, J.W. Pelletier, "A quantisation of the calculus of relations" , Category Theory 1991, CMS Conf. Proc. , 13 , Amer. Math. Soc. (1992) pp. 345–360 Zbl 0793.06008
[a9] J.W. Pelletier, "Von Neumann algebras and Hilbert quantales" Appl. Cat. Struct. , 5 (1997) pp. 249–264
[a10] C.J. Mulvey, J.W. Pelletier, "On the quantisation of points" J. Pure Appl. Algebra , 159 (2001) pp. 231–295
[a11] C.J. Mulvey, J.W. Pelletier, "On the quantisation of spaces" J. Pure Appl. Math. 175 (2002) pp.289-325 Zbl 1026.06018
[a12] D. Yetter, "Quantales and (non-commutative) linear logic" J. Symbolic Logic , 55 (1990) pp. 41–64
[a13] C.J. Mulvey, "Foulis quantales" to appear
[a14] C.J. Mulvey, M. Nawaz, "Quantales: Quantal sets" , Non-Classical Logics and Their Application to Fuzzy Subsets: A Handbook of the Mathematical Foundations of Fuzzy Set Theory , Kluwer Acad. Publ. (1995) pp. 159–217
[a15] U. Berni-Canani, F. Borceux, R. Succi-Cruciani, "A theory of quantale sets" J. Pure Appl. Algebra , 62 (1989) pp. 123–136
[a16] C. Brown, D. Gurr, "A representation theorem for quantales" J. Pure Appl. Algebra , 85 (1993) pp. 27–42
How to Cite This Entry:
Quantale. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quantale&oldid=39041
This article was adapted from an original article by C.J. Mulvey (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article