scholarly journals ENTROPY OF SOME INNER AUTOMORPHISMS OF THE HYPERFINITE II1-FACTOR

1993 ◽  
Vol 04 (02) ◽  
pp. 319-322 ◽  
Author(s):  
ERLING STØRMER
Keyword(s):  
Cartan Subalgebra ◽  
Unitary Operator ◽  
Inner Automorphism ◽  
Inner Automorphisms

It is shown that the entropy of an inner automorphism Ad u of the hyperfinite II1-factor is zero if the unitary operator u belongs to a Cartan subalgebra.

scholarly journals Cartan subalgebras of regular extensions of von Neumann algebras

1988 ◽  
Vol 45 (2) ◽  
pp. 249-274
Author(s):  
Colin E. Sutherland
Keyword(s):  
Cartan Subalgebra ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Regular Representation ◽  
Projective Representation ◽  
Von Neumann ◽  
Left Regular ◽  
Cartan Subalgebras ◽  
Inner Automorphisms ◽  
The Given

AbstractWe analyse the structure of a regular extension ℳ ⋊ γ, υQ of a von Neumann algebra ℳ by an action (modulo inner automorphisms) γ of a discrete group Q, and a nonabelian 2-cycle υ for γ, under the assumption that the “action” γ of Q is cocycle conjugate to an “action”, α which leaves globally invariant a cartan subalgebra of ℳ. we show that ℳ ⋊ γ, υQ is isomorphic with the algebra of the left regular projective representation of a certain discrete, non-principal groupoid ℜ V Q determined by the action of Q on the given cartan subalgebrs, where ℜ is the Takesaki relation associated to the pair (ℳ, ) we apply this description to give a decomposition of the regular representation of a group G into irreducibles, where G is a split extension of a type I group K by an abelian group Q, and work out the details of the author's earlier abstract plancherel theorem in the case when K is abelian.

The finite inner automorphism groups of division rings

1995 ◽  
Vol 118 (2) ◽  
pp. 207-213 ◽  
Author(s):  
M. Shirvani
Keyword(s):  
Finite Groups ◽  
Galois Theory ◽  
Automorphism Groups ◽  
Outer Automorphism ◽  
Commutative Field ◽  
Division Rings ◽  
Inner Automorphism ◽  
Projective Representations ◽  
Inner Automorphisms ◽  
A Finite Group

Let G be a finite group of automorphisms of an associative ring R. Then the inner automorphisms (x↦ u−1xu = xu, for some unit u of R) contained in G form a normal subgroup G0 of G. In general, the Galois theory associated with the outer automorphism group G/G0 is quit well behaved (e.g. [7], 2·3–2·7, 2·10), while little group-theoretic restriction on the structure of G/G0 may be expected (even when R is a commutative field). The structure of the inner automorphism groups G0 does not seem to have received much attention so far. Here we classify the finite groups of inner automorphisms of division rings, i.e. the finite subgroups of PGL (1, D), where D is a division ring. Such groups also arise in the study of finite collineation groups of projective spaces (via the fundamental theorem of projective geometry, cf. [1], 2·26), and provide examples of finite groups having faithful irreducible projective representations over fields.

ON AUTOMORPHISMS OF THE ENDOMORPHISM SEMIGROUP OF A FREE UNIVERSAL ALGEBRA

2007 ◽  
Vol 17 (05n06) ◽  
pp. 1085-1106 ◽  
Author(s):  
G. MASHEVITZKY ◽  
B. I. PLOTKIN
Keyword(s):  
Lie Algebras ◽  
Universal Algebra ◽  
Automorphism Groups ◽  
Endomorphism Semigroup ◽  
Universal Algebras ◽  
Free Lie Algebras ◽  
Inner Automorphism ◽  
Groups Of Automorphisms ◽  
Inner Automorphisms

Let U be a universal algebra. An automorphism α of the endomorphism semigroup of U defined by α(φ) = sφs-1 for a bijection s : U → U is called a quasi-inner automorphism. We characterize bijections on U defining such automorphisms. For this purpose, we introduce the notion of a pre-automorphism of U. In the case when U is a free universal algebra, the pre-automorphisms are precisely the well-known weak automorphisms of U. We also provide different characterizations of quasi-inner automorphisms of endomorphism semigroups of free universal algebras and reveal their structure. We apply obtained results for describing the structure of groups of automorphisms of categories of free universal algebras, isomorphisms between semigroups of endomorphisms of free universal algebras, automorphism groups of endomorphism semigroups of free Lie algebras etc.

scholarly journals Ext and OrderExt Classes of Certain Automorphisms of C*-Algebras Arising from Cantor Minimal Systems

2001 ◽  
Vol 53 (2) ◽  
pp. 325-354 ◽  
Author(s):  
Hiroki Matui
Keyword(s):  
Transformation Group ◽  
Minimal System ◽  
Full Group ◽  
C Algebras ◽  
Inner Automorphism ◽  
Inner Automorphisms ◽  
Cantor Minimal Systems

AbstractGiordano, Putnam and Skau showed that the transformation group C*-algebra arising from a Cantor minimal system is an AT-algebra, and classified it by its K-theory. For approximately inner automorphisms that preserve C(X), we will determine their classes in the Ext and OrderExt groups, and introduce a new invariant for the closure of the topological full group. We will also prove that every automorphism in the kernel of the homomorphism into the Ext group is homotopic to an inner automorphism, which extends Kishimoto’s result.

Automorphisms of G-Azumaya Algebras

1985 ◽  
Vol 37 (6) ◽  
pp. 1047-1058 ◽  
Author(s):  
Margaret Beattie
Keyword(s):  
Finite Abelian Group ◽  
Equivalence Classes ◽  
Usual Sense ◽  
Smash Product ◽  
Azumaya Algebras ◽  
Azumaya Algebra ◽  
Linear Automorphism ◽  
Inner Automorphism ◽  
Image Position ◽  
Inner Automorphisms

Let R be a commutative ring, G a finite abelian group of order n and exponent m, and assume n is a unit in R. In [10], F. W. Long defined a generalized Brauer group, BD(R, G), of algebras with a G-action and G-grading, whose elements are equivalence classes of G-Azumaya algebras. In this paper we investigate the automorphisms of a G-Azumaya algebra A and prove that if Picm(R) is trivial, then these automorphisms are all, in some sense, inner.In fact, each of these “inner” automorphisms can be written as the composition of an inner automorphism in the usual sense and a “linear“ automorphism, i.e., an automorphism of the typewith r(σ) a unit in R. We then use these results to show that the group of gradings of the centre of a G-Azumaya algebra A is a direct summand of G, and thus if G is cyclic of order pr, A is the (smash) product of a commutative and a central G-Azumaya algebra.

Nielsen's commutator test for two-generator groups

1993 ◽  
Vol 114 (2) ◽  
pp. 295-301 ◽  
Author(s):  
Narain Gupta ◽  
Vladimir Shpilrain
Keyword(s):  
Free Group ◽  
Final Section ◽  
Commutator Subgroup ◽  
Metabelian Group ◽  
Free Metabelian Group ◽  
Sufficiency Condition ◽  
Inner Automorphism ◽  
Tame Automorphism ◽  
Polynilpotent Group ◽  
Inner Automorphisms

Nielsen [14] gave the following commutator test for an endomorphism of the free group F = F2 = 〈x, y; Ø〉 to be an automorphism: an endomorphism ø: F → F is an automorphism if and only if the commutator [ø(x), ø(y)] is conjugate in F to [x, y]±1. He obtained this test as a corollary to his well-known result that every IA-automorphism of F (i.e. one which fixes F modulo its commutator subgroup) is an inner automorphism. Bachmuth et al. [4] have proved that IA-automorphisms of most two-generator groups of the type F/R′ are inner, and it becomes natural to ask if Nielsen's commutator test remains valid for those groups as well. Durnev[7] considered this question for the free metabelian group F/F″ and confirmed the validity of the commutator test in this case. Here we prove that Nielsen's test does not hold for a large class of F/R′ groups (Theorem 3·1) and, as a corollary, deduce that it does not hold for any non-metabelian solvable group of the form F/R″ (Corollary 3·2). In view of our Theorem 3·1, Nielsen's commutator test in these situations seems to have less appeal than his result that the IA-automorphisms of F are precisely the inner automorphisms of F. We explore some applications of this important result with respect to non-tameness of automorphisms of certain two- generator groups F/R (i.e. automorphisms of F/R which are not induced by those of the free group F). For instance, we show that a two-generator free polynilpotent group F/V, , has non-tame automorphisms except when V = γ2(F) or V = γ3(F), or when V is of the form [yn(U), γ(U)], n ≥ 2 (Theorem 4·2). This complements the results of [9] and [16] rather nicely, and is shown to follow from a more general result (Proposition 4·1). We also include an example of an endomorphism θ: x → xu, y→y of F which induces a non-tame automorphism of F/γ6(F) while the partial derivative ∂(u)/∂(x) is ‘balanced’in the sense of Bryant et al. [5] (Example 4·4). This gives an alternative solution of a problem in [5] which has already been resolved by Papistas [15] in the negative. In our final section, we consider groups of the type F/[R′,F] and, in contrast to groups of the type F/R′, we show that the Nielsen's commutator test does hold in most of these groups (Theorem 5·1). We conclude with a sufficiency condition under which Nielsen's commutator test is valid for a given pair of generating elements ofF modulo [R′,F] (Proposition 5·2).

scholarly journals Rigid automorphisms of linking systems

2021 ◽  
Vol 9 ◽  
Author(s):  
George Glauberman ◽  
Justin Lynd
Keyword(s):  
Finite Group ◽  
Sylow Subgroup ◽  
Fusion Systems ◽  
Outer Automorphisms ◽  
Inner Automorphism ◽  
Inner Automorphisms ◽  
A Finite Group

Abstract A rigid automorphism of a linking system is an automorphism that restricts to the identity on the Sylow subgroup. A rigid inner automorphism is conjugation by an element in the center of the Sylow subgroup. At odd primes, it is known that each rigid automorphism of a centric linking system is inner. We prove that the group of rigid outer automorphisms of a linking system at the prime $2$ is elementary abelian and that it splits over the subgroup of rigid inner automorphisms. In a second result, we show that if an automorphism of a finite group G restricts to the identity on the centric linking system for G, then it is of $p'$ -order modulo the group of inner automorphisms, provided G has no nontrivial normal $p'$ -subgroups. We present two applications of this last result, one to tame fusion systems.

A note on extending partial automorphisms of abelian groups

1970 ◽  
Vol 11 (1) ◽  
pp. 37-41 ◽  
Author(s):  
C. G. Chehata ◽  
A. Shawky
Keyword(s):  
Abelian Groups ◽  
Index Set ◽  
Inner Automorphism ◽  
Group 3 ◽  
Inner Automorphisms

Given a group G and a partial automorphism μ of G, i.e. an isomorphism mapping a subgroup A of G onto another subgroup B of G, then it is known [3] that μ can always be extended to a total automorphism, in fact an inner one, of a supergroup of G; that is there exists a group G* ⊇ G with an inner automorphism μ* whose effect on the elements of A is the same as that of μ. Also any number of partial automorphisms μσ, where a ranges over some index set Σ can be simultaneously extended to inner automorphisms of one and the same group [3, Theorem II].

The Clifford Algebra and the Group of Similitudes

1962 ◽  
Vol 14 ◽  
pp. 45-59 ◽  
Author(s):  
Maria J. Wonenburger
Keyword(s):  
Quadratic Form ◽  
Clifford Algebra ◽  
Orthogonal Group ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Clifford Group ◽  
Inner Automorphism ◽  
Invertible Elements ◽  
Inner Automorphisms ◽  
Natural Way

LetC(M, Q)be the Clifford algebra of an even dimensional vector spaceMrelative to a quadratic formQ. WhenQis non-degenerate, it is well known that there exists an isomorphism of the orthogonal groupO(Q)onto the group of those automorphisms ofC(M, Q)which leave invariant the spaceM⊂C(M, Q). These automorphisms are inner and the group of invertible elements ofC(M, Q)which define such inner automorphisms is called the Clifford group.If instead of the groupO(Q)we take the group of similitudesγ(Q)or even the group of semi-similitudesΓγ(Q), it is possible to associate in a natural way with any element of these groups an automorphism or semi-automorphism, respectively, of the subalgebra of even elementsC+(M, Q)⊂C(M, Q). Each one of the automorphisms ofC+(M, Q)so defined can be extended, as it is shown here (Theorem 2), to an inner automorphism ofC(M, Q), although the extension is not unique.

scholarly journals On Maximal Abelian Subrings of Factors of Type II

1960 ◽  
Vol 12 ◽  
pp. 289-296 ◽  
Author(s):  
Lajos Pukánszky
Keyword(s):  
Hilbert Space ◽  
Unitary Operator ◽  
Type I ◽  
Type Ii ◽  
Bounded Operators ◽  
Unitary Space ◽  
Infinite Dimensional ◽  
Complete Knowledge ◽  
Inner Automorphism

Although we possess a fairly complete knowledge of the abelian subrings of rings of operators in a Hilbert space which are algebraically isomorphic to the ring of all bounded operators of a finite or infinite dimensional unitary space, that is of factors of Type I, very little is known of abelian subrings of factors of Type II1. In (1), Dixmier investigated several properties of maximal abelian subrings of factors of Type II. It turned out that their structure differs essentially from that of maximal abelian subrings of factors of Type I. He showed the existence of maximal abelian subrings in approximately finite factors, possessing the property that every inner*-automorphism carrying this subring into itself is necessarily implemented by a unitary operator of this subring. These maximal abelian subrings are called singular. In addition, he constructed a IIi factor containing two singular abelian subrings which cannot be connected by an inner *automorphism of this ring.

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