Quantale

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