When is a Markovian Semigroup Induced by a Semiflow?

t , for t ≥ 0, be a strongly continuous Markovian semigroup acting on C(X), where X is a compact Hausdorf space, and let D denote the domain of its infinitesimal generator Z. Suppose D contains a (perhaps finite) family of functions f separating the points of X and satisfying Zf² = 2fZf. If either (1) there exists δ > 0 such that (T^t f)²∈ D if 0 ≤ t ≤δ for each f in this family; or (1′) for some core D′ of Z, g ∈ D′ implies g²∈ D, then the underlying Markoff process on X is deterministic. That is, there exists a semiflow — a semigroup (under composition) of continuous functions φ_t from X into X — such that T^tf(x) = f(φ_t (x)). If the domain D should be an algebra then conditions (1) and (1′) hold trivially. Conversely, if we have a separating family satisfying Zf² = 2fZf then each of these conditions implies that D is an algebra. It is an open question as to whether these conditions are redundant. If the functions φ_t are homeomorphisms from X onto X, then of course we have a Markovian group induced by a flow. This result is obtained by first providing general results about the null-space N of the (function-valued) positive semidefinite quadratic form defined by < f, g > = Z(fg) - fZg - gZf. The set N can be defined for any generator Z of a strongly continuous Markovian semigroup and is equivalently given by N = {f ∈ D| f²∈ D and Zf² = 2fZf} = {f ∈ D| T^t(f²)-(T^tf)² is o(t²) in C(X)}. In the general case N is an algebra closed under composition with any C¹-function φ from the reals to the reals, and Z(φ[f]) = (Zf)φ′[f] if f ∈ N. This "chain rule" on N (on which Z must act as a derivation) is a special case of a theorem for C²-functions φ which holds more generally for all f in d, viz., Z(φ[f] = (Zf) φ′[f] + ? <f, f> φ″[f], Provided Z is a local operator and D is an algebra. In this case the form < f, g > itself enjoys the relation < φ[f], ψ[g] > = φ′ [f] ψ′[g] < f, g >, for C²functions φ and ψ. Some of the results and their proofs continue to hold when the setting is switched from the commutative C*-algebra C(X) to a general (noncommutative) C*-algebra A. In the norm continuous case we obtain a sharp characterization of Markovian semigroups that are groups: Let T^t = e^tz , defined for t ≥ 0, be a Markovian semigroup acting on a C*-algebra A that is norm continuous, i.e., ||T^t - I|| ⇒ 0 as t ⇒ 0 +. Assume Z(a²) = a(Za) + (Za) a for some (perhaps finite) set of self-adjoint elements a that generate a Jordan algebra dense among the self-adjoint elements of A. The e^tz , -∞ < t < ∞, is a group of Markovian operators.     

On λ[S] Semigroup of Semigroup

Wehaveintroducedσconstructionofrings[1],[2]withhomomorphismofmodulesofringsinringtheoryanddiscussedsomerelationsbetweenringRand[R]σandRσ.Thecongruencesandidealsareequevalent,sohomomorphismsandidealsarecorrespondingonebyoneinringtheory;buthomomorphismsandidealsarenotequevalentinsemigrouptheory,homomor-phismsandcongreuncesarecorrespondingonebyone,thekernerofhomomorphismsisacon-greunces,hencethediscussionofsimilarproblemsinringtheoryandsemigrouptheoryhasgreatdifference.Firstweintroducethenoti…     

Semigroup Algebras and Noetherian Maximal Orders: a Survey

A survey is given on recent results describing when a semigroup algebra K[S] of a submonoid S of a polycyclic-by-finite group is a prime Noetherian maximal order. As an application one constructs concrete classes of finitely presented algebras that have the listed properties. Also some open problems are stated.     

The Maximal Operator Associated to a Nonsymmetric Ornstein–Uhlenbeck Semigroup

Let (ℋ _t ) _t≥0 be the Ornstein–Uhlenbeck semigroup on ℝ ^d with covariance matrix I and drift matrix λ(R−I), where λ>0 and R is a skew-adjoint matrix, and denote by γ _∞ the invariant measure for (ℋ _t ) _t≥0. Semigroups of this form are the basic building blocks of Ornstein–Uhlenbeck semigroups which are normal on L ²(γ _∞). We prove that if the matrix R generates a one-parameter group of periodic rotations, then the maximal operator ℋ_* f(x)=sup  _t≥o |ℋ _t f(x)| is of weak type 1 with respect to the invariant measure γ _∞. We also prove that the maximal operator associated to an arbitrary normal Ornstein–Uhlenbeck semigroup is bounded on L ^p (γ _∞) if and only if 1<p≤∞.      

Free Subordination and Belinschi–Nica Semigroup

We realize the Belinschi–Nica semigroup of homomorphisms as a free multiplicative subordination. This realization allows to define more general semigroups of homomorphisms with respect to free multiplicative convolution. For these semigroups we show that a differential equation holds, generalizing the complex Burgers equation. We give examples of free multiplicative subordination and find a relation to the Markov–Krein transform, Boolean stable laws and monotone stable laws. A similar idea works for additive subordination, and in particular we study the free additive subordination associated to the Cauchy distribution and show that it is a homomorphism with respect to monotone, Boolean and free additive convolutions.     

Serre's Condition Rℓ for Affine Semigroup Rings

In this note we characterize the affine semigroup rings K[S] over an arbitrary field K that satisfy condition R_? of Serre. Our characterization is in terms of the face lattice of the positive cone pos(S) of S. We start by reviewing some basic facts about the faces of pos(S) and consequences for the monomial primes of K[S]. After proving our characterization we turn our attention to the Rees algebras of a special class of monomial ideals in a polynomial ring over a field. In this special case, some of the characterizing criteria are always satisfied. We give examples of non-normal affine semigroup rings that satisfy R₂.     

The K°-relation on the Congruence Lattice of a Regular Semigroup with an Inverse Transversal

Let S be a regular semigroup with an inverse transversal S° and C(S) the congruence lattice of S. A relation K° on C(S) is introduced as follows: if ρ, θ∈ C(S), then we say that ρ and θ are K°-related if Ker ρ° = Ker θ° , where ρ° = ρ|S°. Expressions for the least and the greatest congruences in the same K°-class as ρ are provided. A number of equivalent conditions for K° being a congruence are given.     

Higher-order Riesz operators for the Ornstein–Uhlenbeck Semigroup

We prove that the second-order Riesz transforms associated to the Ornstein–Uhlenbeck semigroup are weak type (1,1) with respect to the Gaussian measure in finite dimension. We also show that they are given by a principal value integral plus a constant multiple of the identity. For the Riesz transforms of order three or higher, we present a counterexample showing that the weak type (1,1) estimate fails.     

The Semigroup Characterizations of Positive Implicative BCK—algebras

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＢｙａＢＣＩ－ａｌｇｅｂｒａｗｅｍｅａｎａｎａｌｇｅｂｒａ（Ｘ，，０）ｏｆｔｙｐｅ（２，０）ｗｉｔｈｔｈｅｆｏｌｌｏｗｉｎｇｃｏｎｄｉ－ｔｉｏｎｓ：（１）（（ｘｙ）（ｘｚ））（ｚｙ）＝０；（２）（ｘ（ｘｙ））ｙ＝０；（３）ｘｘ＝０；（４）ｘｙ＝ｙｘ＝０ｉｍｐｌｉｅｓｘ＝ｙ．ＩｆａＢＣＩ－ａｌｇｅｂｒａ（Ｘ，，０）ｓａｔｉｓｆｉｅｓ（５）０ｘ＝０．ｔｈｅｎｉｔｉｓｃａｌｌｅｄａＢＣＫ－ａｌｇｅｂｒａ．ＩｎａＢＣＩ－ａｌｇｅｂｒａ，ｔｈｅｆ…     

Approximate Solutions of Variational Inequalities on Sets of Common Fixed Points of a One-Parameter Semigroup of Nonexpansive Mappings

Let \(\mathcal{T}\) be a one-parameter semigroup of nonexpansive mappings on a nonempty closed convex subset C of a strictly convex and reflexive Banach space X. Suppose additionally that X has a uniformly Gâteaux differentiable norm, C has normal structure, and \(\mathcal{T}\) has a common fixed point. Then it is proved that, under appropriate conditions on nonexpansive semigroups and iterative parameters, the approximate solutions obtained by the implicit and explicit viscosity iterative processes converge strongly to the same common fixed point of \(\mathcal{T}\), which is a solution of a certain variational inequality.     

Theory of Grobner Basesin Semigroup Algebra k[A]

TheoryofGr6bnerbasesinpolynomialalgebrak[X]isthemostfundamentalcomputationaltheoryincompilationalalgebraicgeometyll--5].Nevertheless,efficienciesofthealgorithmsofGrobnerbasesareverylowlybecauseofspecialpropertyofsymboliccomputation.Algorithmsinsemigr...     

Numerical Semigroup Rings and Almost Prüfer v-Multiplication Domains

Let D be an integral domain with quotient field K, X be an indeterminate over D, Γ be a numerical semigroup with Γ ? ?₀, D[Γ] be the semigroup ring of Γ over D (and hence D ? D[Γ] ? D[X]), and D + X ⁿ K[X] = {a + X ⁿ g∣a ∈ D and g ∈ K[X]}. We show that there exists an order-preserving bijection between Spec(D[X]) and Spec(D[Γ]), which also preserves t-ideals. We also prove that D[Γ] is an APvMD (resp., AGCD-domain) if and only if D[X] is an APvMD (resp., AGCD-domain) and char(D) ≠ 0. We show that if n ≥ 2, then D is an APvMD (resp., AGCD-domain, AGGCD-domain, AP-domain, AB-domain) and char(D) ≠ 0 if and only if D + X ⁿ K[X] is an APvMD (resp., AGCD-domain, AGGCD-domain, AP-domain, AB-domain). Finally, we give some examples of APvMDs which are not AGCD-domains by using the constructions D[Γ] and D + X ⁿ K[X].     

Twisted Semigroup Crossed Products and Twisted Toeplitz Algebras of Ordered Groups

Let г^＋ be the positive cone of a totally ordered abelian group г, and σa cocycle in г. We study the twisted crossed products by actions of г＋ as endomorphisms of C^＊-algebras, and use this to generalize the theorem of Ji.     

On the Structure of E-unitary Covers for an Orthodox Semigroup

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A Semigroup Treatment to Viscosity Solutions of the Hanilton-Jacobi Fquation

芬1 .Int一oduction R以泥nUyC,二ndall,Lio愁and Eva哪[4,5]co川ide,记a new kind of genO21)五过solution,wlliCll认以U记viscosity solution,for the椒milton一为cobi阅uations. In tll此Papel‘忱从u勿the Cauclly ProblOll for the Haff己ton一Jaco以equation.(1 .1):‘,+F恤V:‘)二0,沂R”,‘>0,andV“=gladxl‘,from the point of view of noul毗咖igbuP tl初卿.讹get the follo诩ng心ult: 肠句魄川1之Suppo黯产CJ(R xR”火)satisfy此lbllo期。g山ndition:(F)F(o,0)二o,and几(:‘,z,))0 for any‘(“,l))峨“R:卫祀n,for…     

On the Structure of the Semigroup of Entire Étale Mappings

The motivation for this paper comes from new ideas for solving the two-dimensional Jacobian Conjecture. The Jacobian Conjecture is one of the most famous open problems in algebraic geometry. This long-standing conjecture is no doubt one of the central problems in this well developed field of mathematics and hence the importance of investigating it. We can consider a semigroup of local diffeomorphisms on the affine space with a composition of mappings as its binary operation. We put a geometric fractal-like structure on this semigroup after equipping it with a natural metric (this is heavily dependent on the fact that our mappings are local diffeomorphisms). This structure is much more general than the structure of the ind-variety suggested by Kambayashi for étale polynomial mappings in the algebraic context. Hence, it applies to other semigroups such as the semigroup of all the entire functions in one complex variable with a nonvanishing first order derivative. This last semigroup is the theme of the current paper. We hope that the corresponding Hausdorff measure and Hausdorff dimension will enable us to relate the structure of the semigroup with arithmetic machinery such as certain Zeta functions.     

Semigroup and Riesz transform for the Dunkl–Schrödinger operators

Semigroup Forum - Let $$L_k=-\Delta _k+V$$ be the Dunkl–Schrödinger operators, where $$\Delta _k=\sum _{j=1}^dT_j^2$$ is the Dunkl Laplace operator associated to the Dunkl operators...     

Heat Semigroup Bounds on BV(Ω) and Related Estimates

Let \(\overline{\Omega}\) be a compact Riemannian manifold with nonempty boundary. We note that if \(f\in C^\infty(\overline{\Omega})\) does not vanish identically on the boundary, then the heat semigroup \(e^{t\Delta_D}\) (with the Dirichlet boundary condition) acting on f produces a family bounded in \(H^{1,p}(\overline{\Omega})\) if and only if p?=?1. This observation motivates the main result of this paper, which is that the heat semigroup is uniformly bounded on BV(Ω), the space of functions on Ω with bounded variation.     

The Bochner–Schoenberg–Eberlein Property for Totally Ordered Semigroup Algebras

The concepts of BSE property and BSE algebras were introduced and studied by Takahasi and Hatori in 1990 and later by Kaniuth and Ülger. This abbreviation refers to a famous theorem proved by Bochner and Schoenberg for \(L^1({\mathbb {R}})\), where \({\mathbb {R}}\) is the additive group of real numbers, and by Eberlein for \(L^1(G)\) of a locally compact abelian group G. In this paper we investigate this property for the Banach algebra \(L^p(S,\mu )\;(1\le p<\infty )\) where S is a compact totally ordered semigroup with multiplication \(xy=\max \{x,y\}\) and \(\mu \) is a regular bounded continuous measure on S. As an application, we have shown that \(L^1(S,\mu )\) is not an ideal in its second dual.