C⁎-subproduct and product systems Floricel, Remus; Ketelboeter, Brian
Journal of mathematical analysis and applications,
12/2023, Volume:
528, Issue:
1
Journal Article
Peer reviewed
We introduce and study two-parameter subproduct and product systems of C⁎-algebras as the operator-algebraic analogues of, and in relation to, Tsirelson's two-parameter product systems of Hilbert ...spaces. Using several inductive limit techniques, we show that (i) any C⁎-subproduct system can be dilated to a C⁎-product system; and (ii) any C⁎-subproduct system that admits a unit, i.e., a co-multiplicative family of projections, can be assembled into a C⁎-algebra, which comes equipped with a one-parameter family of comultiplication-like homomorphisms. We also introduce and discuss co-units of C⁎-subproduct systems, consisting of co-multiplicative families of states, and show that they correspond to idempotent states of the associated C⁎-algebras. We then use the GNS construction to obtain Tsirelson subproduct systems of Hilbert spaces from co-units, and describe the relationship between the dilation of a C⁎-subproduct system and the dilation of the Tsirelson subproduct system of Hilbert spaces associated with a co-unit. All these results are illustrated concretely at the level of C⁎-subproduct systems of commutative C⁎-algebras.
Motivated by the search for idempotent cellular automata (CA), we study CA that act almost as the identity unless they read a fixed pattern p. We show that constant and symmetrical patterns always ...generate idempotent CA, and we characterize the quasi-constant patterns that generate idempotent CA. Our results are valid for CA over an arbitrary group G. Moreover, we study the semigroup theoretic natural partial order defined on idempotent CA. If G is infinite, we prove that there is an infinite independent set of idempotent CA, and if G has an element of infinite order, we prove that there is an infinite increasing chain of idempotent CA.
The concept of idempotent extending and summand idempotent is a proper generalizations of extending and uniform modules. An R-module M is called idempotent extending if every idempotent submodule of ...M is essential in a direct summand of M. An R-module M is called summand idempotent if every direct summand of M is an idempotent. We supply example showing that idempotent extending need not be extending. we have obtained conditions under which the converse holds.
Dynamics near an idempotent Shaikh, Md. Moid; Patra, Sourav Kanti; Ram, Mahesh Kumar
Topology and its applications,
08/2020, Volume:
282
Journal Article
Peer reviewed
Hindman and Leader first introduced the notion of the semigroup of ultrafilters converging to zero for a dense subsemigroup of ((0,∞),+). Using the algebraic structure of the Stone-Čech ...compactification, Tootkaboni and Vahed generalized and extended this notion to an idempotent instead of zero, that is a semigroup of ultrafilters converging to an idempotent e for a dense subsemigroup of a semitopological semigroup (R,+) and they gave the combinatorial proof of the Central Sets Theorem near e. Algebraically one can define quasi-central sets near e for dense subsemigroups of (R,+). In a dense subsemigroup of (R,+), C-sets near e are the sets, which satisfy the conclusions of the Central Sets Theorem near e. Patra gave dynamical characterizations of these combinatorially rich sets near zero. In this paper, we shall establish these dynamical characterizations for these combinatorially rich sets near e. We also study minimal systems near e in the last section of this paper.
Suppose
(
C
,
E
,
s
)
is an
n
-exangulated category. We show that the idempotent completion and the weak idempotent completion of
C
are again
n
-exangulated categories. Furthermore, we also show that ...the canonical inclusion functor of
C
into its (resp. weak) idempotent completion is
n
-exangulated and 2-universal among
n
-exangulated functors from
(
C
,
E
,
s
)
to (resp. weakly) idempotent complete
n
-exangulated categories. Furthermore, we prove that if
(
C
,
E
,
s
)
is
n
-exact, then so too is its (resp. weak) idempotent completion. We note that our methods of proof differ substantially from the extriangulated and
(
n
+
2
)
-angulated cases. However, our constructions recover the known structures in the established cases up to
n
-exangulated isomorphism of
n
-exangulated categories.
Recently, some authors studied the distributive laws of continuous t-norms and some families of the common classes of uninorms over overlap and grouping functions in 33, 41, but until now a complete ...characterization of the distributivity on idempotent uninorms over overlap or grouping functions widely used in image processing is still unresolved. Moreover, authors in 55 characterized the distributivity equations of uni-nullnorms with continuous Archimedean underlying operators over the above two functions. In this paper, we proceed with the distributivity characterization of idempotent uni-nullnorms over them which obviously generalizes the corresponding results of idempotent uninorms over these two functions. During the process, we introduce a class of weak overlap and grouping functions with weak coefficients, and obtain the full characterizations of the above functions by considering the different values of the underlying uninorms' associated functions of idempotent uni-nullnorms on the interval endpoints and comparing the values with the weak coefficients. These obtained results totally answer the question on the distributive solutions of idempotent uninorms over overlap functions, which have been mentioned as their future works in 33. In additions, we also obtain that overlap and grouping functions have particular structures with a constant domain where its value equals to the neutral element of the idempotent uninorm when they are distributive over idempotent uninorms.
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