1,150 research outputs found
A note on drastic product logic
The drastic product is known to be the smallest -norm, since whenever . This -norm is not left-continuous, and hence it
does not admit a residuum. So, there are no drastic product -norm based
many-valued logics, in the sense of [EG01]. However, if we renounce standard
completeness, we can study the logic whose semantics is provided by those MTL
chains whose monoidal operation is the drastic product. This logic is called
in [NOG06]. In this note we justify the study of this
logic, which we rechristen DP (for drastic product), by means of some
interesting properties relating DP and its algebraic semantics to a weakened
law of excluded middle, to the projection operator and to
discriminator varieties. We shall show that the category of finite DP-algebras
is dually equivalent to a category whose objects are multisets of finite
chains. This duality allows us to classify all axiomatic extensions of DP, and
to compute the free finitely generated DP-algebras.Comment: 11 pages, 3 figure
On the universal module of -adic spherical Hecke algebras
Let be a split connected reductive group with connected
center over a local non-Archimedean field of residue characteristic
, let be a hyperspecial maximal compact open subgroup in
. Let be a commutative ring, let be a finitely generated
-free -module. For an -algebra and a character
of the spherical Hecke
algebra we
consider the specialization of the universal -module . For large classes of (including
and ), , and , arguing
geometrically on the Bruhat Tits building we give a sufficient criterion for
to be -free and to admit a -equivariant
resolution by a Koszul complex built from finitely many copies of . This criterion is the exactness of
certain fairly small and explicit -equivariant -module
complexes, where is the group of -valued points
of the unipotent radical of a Borel subgroup in . We verify it
if and if is an irreducible -representation with highest weight in the (closed) bottom
-alcove, or a lift of it to . We use this to construct
-adic integral structures in certain locally algebraic representations of
Bimodules over Cartan MASAs in von Neumann Algebras, Norming Algebras, and Mercer's Theorem
In a 1991 paper, R. Mercer asserted that a Cartan bimodule isomorphism
between Cartan bimodule algebras A_1 and A_2 extends uniquely to a normal
*-isomorphism of the von Neumann algebras generated by A_1 and A_2 [13,
Corollary 4.3]. Mercer's argument relied upon the Spectral Theorem for
Bimodules of Muhly, Saito and Solel [15, Theorem 2.5]. Unfortunately, the
arguments in the literature supporting [15, Theorem 2.5] contain gaps, and
hence Mercer's proof is incomplete.
In this paper, we use the outline in [16, Remark 2.17] to give a proof of
Mercer's Theorem under the additional hypothesis that the given Cartan bimodule
isomorphism is weak-* continuous. Unlike the arguments contained in [13, 15],
we avoid the use of the Feldman-Moore machinery from [8]; as a consequence, our
proof does not require the von Neumann algebras generated by the algebras A_i
to have separable preduals. This point of view also yields some insights on the
von Neumann subalgebras of a Cartan pair (M,D), for instance, a strengthening
of a result of Aoi [1].
We also examine the relationship between various topologies on a von Neumann
algebra M with a Cartan MASA D. This provides the necessary tools to
parametrize the family of Bures-closed bimodules over a Cartan MASA in terms of
projections in a certain abelian von Neumann algebra; this result may be viewed
as a weaker form of the Spectral Theorem for Bimodules, and is a key ingredient
in the proof of our version of Mercer's theorem. Our results lead to a notion
of spectral synthesis for weak-* closed bimodules appropriate to our context,
and we show that any von Neumann subalgebra of M which contains D is synthetic.
We observe that a result of Sinclair and Smith shows that any Cartan MASA in
a von Neumann algebra is norming in the sense of Pop, Sinclair and Smith.Comment: 21 pages, paper is a completely reworked and expanded version of an
earlier preprint with a similar titl
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