Simple involutive quantales.

*(English)*Zbl 0894.06005A quantale \(Q\) is most easily described as a semigroup object in the category \({\mathbf S}{\mathbf L}\) of suplattices. If it is a monoid object it is called unital. (For more on quantales, see the reviewer’s book, Quantales and their applications (1990; Zbl 0703.06007).)

A quantale \(Q\) with operation \(\&\) is called involutive iff it has an involution \((\;)^0\) satisfying \(a^{00}= a\), \((a \& b)^0= b^0 \& a^0\). and \((\text{sup }a_i)^0= \sup a^0_i\) for all \(a\), \(b\), \(a_i\) in \(Q\). Among the important examples of involutive quantales are the so-called Gelfand quantales, which are unital, involutive and satisfy \(a \& a^0 \& a= a\) for all right-sided elements of \(Q\) (right-sided means \(a \& T= a\), where \(T\) is the top element of \(Q\)). Gelfand quantales include the examples of the spectrum \(\text{Max }A\) of a non-commutative \(C^*\)-algebra \(A\) with identity, as well as the examples \(\text{Rel}(X)\) of relations on a set \(X\).

The main object of study in this paper are the simple quantales, where a non-trivial quantale \(Q\) is simple iff any surjective homomorphism of involutive quantales from it is either an isomorphism or a constant mapping. The authors proceed to show that various quantales such as the Hilbert quantales are simple and they obtain various characterization theorems. The last section of the paper investigates notions for involutive quantales related to “spatiality” and separation of points and characterizes when a Gelfand quantale satisfies that every right-sided element is the intersection of primes in these terms.

A quantale \(Q\) with operation \(\&\) is called involutive iff it has an involution \((\;)^0\) satisfying \(a^{00}= a\), \((a \& b)^0= b^0 \& a^0\). and \((\text{sup }a_i)^0= \sup a^0_i\) for all \(a\), \(b\), \(a_i\) in \(Q\). Among the important examples of involutive quantales are the so-called Gelfand quantales, which are unital, involutive and satisfy \(a \& a^0 \& a= a\) for all right-sided elements of \(Q\) (right-sided means \(a \& T= a\), where \(T\) is the top element of \(Q\)). Gelfand quantales include the examples of the spectrum \(\text{Max }A\) of a non-commutative \(C^*\)-algebra \(A\) with identity, as well as the examples \(\text{Rel}(X)\) of relations on a set \(X\).

The main object of study in this paper are the simple quantales, where a non-trivial quantale \(Q\) is simple iff any surjective homomorphism of involutive quantales from it is either an isomorphism or a constant mapping. The authors proceed to show that various quantales such as the Hilbert quantales are simple and they obtain various characterization theorems. The last section of the paper investigates notions for involutive quantales related to “spatiality” and separation of points and characterizes when a Gelfand quantale satisfies that every right-sided element is the intersection of primes in these terms.

Reviewer: K.I.Rosenthal (Schenectady)

##### Keywords:

spatiality; involution; involutive quantales; Gelfand quantales; simple quantales; Hilbert quantales; separation of points; primes##### Citations:

Zbl 0703.06007
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\textit{J. W. Pelletier} and \textit{J. Rosický}, J. Algebra 195, No. 2, 367--386 (1997; Zbl 0894.06005)

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