Involutive Algebras Research Articles

Semisimple radical classes of involution algebras

J. Wichman has asked about semisimple radical classes of involution algebras. In the present paper we describe the semisimple radical classes of involution algebras over a field K* with involution *. If K* is infinite, then there are only trivial semisimple radical classes. If K* is finite then these classes are subdirect closures of strongly hereditary finite sets of finite idempotent algebras. In proving this result we determine the structure of certain simple involution algebras. We prove that the variety of symmetric involution algebras over Z(2) does not have attainable identities, answering a problem posed by Gardner [2]. Most of the results are valid also for involution rings (over the integers).

An order-theoretic approach to involutive algebras

An order-theoretic approach to involutive algebras

A characterization of homomorphisms in certain Banach involution algebras

A characterization of homomorphisms in certain Banach involution algebras

Partially ordered baer involutive algebras

A survey is given of the basic results of the theory of AW*-algebras, the properties of the rings of measurable and locally measurable operators associated with von Neumann algebras and AW*-algebras are described, Baer*-algebras admitting partial orderings compatible with their algebraic structures are investigated.

Korovkin approximation of homomorphisms in involution algebras

Korovkin approximation of homomorphisms in involution algebras

Hermitian geometry and involutive algebras

Hermitian geometry and involutive algebras

On the Definition of C*-Algebras II

The theory of noncommutative involutive Banach algebras (briefly Banach *-algebras) owes its origin to Gelfand and Naimark, who proved in 1943 the fundamental representation theorem that a Banach *-algebra with C*-condition(C*)is *-isomorphic and isometric to a norm-closed self-adjoint subalgebra of all bounded operators on a suitable Hilbert space.At the same time they conjectured that the C*-condition can be replaced by the B*-condition.(B*)In other words any B*-algebra is actually a C*-algebra. This was shown by Glimm and Kadison [5] in 1960.

Extending positive definite linear forms

In the classical literature two properties, called here symmetry and bounded variation, of a positive definite linear form a ′ a’ on an involutive Banach algebra A \mathcal {A} are given. Together they form a necessary and sufficient condition that a ′ a’ admit a positive definite linear extension to the involutive algebra obtained from A \mathcal {A} by adjoining an identity. In this note we show that bounded variation alone suffices, in that it already implies symmetry.

Automatic continuity of positive functionals on topological involution algebras

This paper surveys the known results on automatic continuity of positive functionals on topological *-algebras and then shows how two theorems on Banach *-algebras extend to complete metrizable topological *-algebras. The two theorems concerned are Loy's theorem on separable Banach *-algebras A with centre Z such that AZ is of countable codimension and Varopoulos' result on Banach *-algebras with bounded approximate identity. Both theorems have the conclusion that all positive functionals on such algebras are continuous. The extension of the second theorem requires the algebra to be locally convex and the approximate identity to be ‘uniformly bounded’. Neither extension requires the algebra to be LMC. This means that the proof of the first theorem is quite different from the corresponding Banach algebra result (which used spectral theory). The proof of the second is closer to the previously known LMC version, but actually neater by being more general. It is also shown that the well-known estimate of |f(a*ba)| for a positive functional f on a Banach *-algebra may be obtained without the usual use of spectral theory. The paper concludes with a list of open questions.

The Group C ∗ -Algebra of the Desitter Group

Let G denote the universal-covering of the DeSitter group and C*(G) the group C*-algebra of G. In this paper we use the extension theory of C. Delaroche to describe the structure of C*(G). Introduction. Let G denote the universal-covering of the DeSitter group and C*(G) the group C*-algebra of G, i.e., the enveloping C*-algebra of the involutive Banach algebra LX(G) (see (2)). The main goal of this paper is to give a complete description of the structure of C*(G). Briefly, the main result is that C*(G) is isomorphic to the restricted product of certain C*-algebras whose structures have concrete descriptions given by the extension theory of C. Delaroche (1). In §1 of this paper we summarize the classification of the irreducible unitary representations of G given by J. Dixmier (3) and the character formulas for these representations given by T. Hirai (6). We refer to (3) or (9) for all information concerning the structure of G. In §2 we investigate the behavior of the irreducible characters and then follow the method of J. M. G. Fell (4) to describe the topology on G. An important step in this program is that of proving a Riemann-Lebesgue lemma for G. This we also do in §2. In §3 we determine the structure of C*(G). Since there are an infinite number of points where G fails to be Hausdorff, the methods of (1) do not apply directly. However, we are able to express C*(G) as the restricted product of certain C*-algebras each of which is describable via Theorem VI.3.8 of (11 When G = SL(2, C), the structure of C*(G) was first described by Fell (5) and later by Delaroche (1). For G = SL(2, R), the structure of C*(G) was determined by Milicic (7) by using methods similar to those of Fell in the SL(2, C) case. For the remaining Lorentz groups, one should be able to use the parameterization of G given by Thieleker (10), the character formulas given by Hirai (6), and the Delaroche extension theory to obtain results similar to those in this paper. This problem reduces to knowing the topologi- cal behavior at the ends of the complementary series representations,

Root-theory of involutive Banach-Lie algebras

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Local completeness of operator algebras

A normed ∗ \ast -algebra A \mathcal {A} is called a local C ∗ {C^\ast } -algebra, if all its maximal commutative ∗ \ast -subalgebras are C ∗ {C^\ast } -algebras. It is shown that any local C ∗ {C^\ast } -algebra dense in K ( H ) \mathcal {K}(\mathcal {H}) , the algebra of compact operators on the Hilbert space H \mathcal {H} equals K ( H ) \mathcal {K}(\mathcal {H}) . The same result holds also for local C ∗ {C^\ast } -algebras dense in A W ∗ A{W^\ast } -algebras without a II 1 {\text {II}_1} summand.

Positive linear maps on operator algebras

Given a family of completely positive maps, indexed by a group, from aC*-algebra into itself, we are concerned with its dilation to a group of *-automorphisms on a larger algebra. A Schwarz-type inequality forn-positive *-linear mappings from an involutive algebra into the bounded linear operators on a hilbert space is obtained. Strongly continuous one-parameter semigroups and groups onC*-algebras, which have certain positivity properties, are studied.

On symmetry of some Banach algebras

In recent years there was a growing interest in the problem of symmetry of involutive Banach algebras. In particular very substantial progress has been made towards a solution of the problem of characterizations of such locally compact groups G for which the group algebra Lι(G) is symmetric. The most striking results in this direction are due to J. Jenkins, who first proved that the discrete a% + 6 -group has a nonsymmetric algebra [3] and that the same

On representations of tensor products of involutive Banach algebras

On representations of tensor products of involutive Banach algebras

A Characterization of Group Algebras as a Converse of Tannaka-Stinespring-Tatsuuma Duality Theorem

Let G be a locally compact group with a left invariant Haar mneasure p. Then one can construct two involutive Banach algebras L -(G, /A) anld Ll(G. a) associated with G, and the former is the conjugate space of the latter as Banach space. By the terminology of group algebra, one has been expressing the algebra L1(G p), and then one's main attention has been traditionally concentrated only on L'(G, u). It should be, however, pointed out that there is a remarkable duality between them not only as Banach spaces but also as Banach algebras. For example, taking an arbitrary pair f, g in the IHilbert space L2(G,u), which may be regarded as a neutral space in the duality between L1 (G, K) and LX (G, p), the product fg belonigs to L1 (G, ,) and the convolution f $ y belongs to LX (G, 4), where v and g' are given by y(s) g(s) and g' (s) = g (s-1). Moreover, in the study of Tatsuuma duality theorem in [29], the duality property between Ll (G,,u) and L(G, u) is implicitly used and plays important r3le. To emphasize the importanec of the duality system {L1 (G, ), LX (G, u) }, let us assunme G abelian for a little while. Then the spectrum of Ll (G, u) forms a locallv compact abelian group G with the normalized dual Haar measure ji, which is called the dual group of G. The Fourier transform 5 carries the algebra Ll (G, u) with the convolution into the algebra L(G, A) with the multiplication. Conversely, the Fourier inverse transfornm 5 carries

Double Centralizers and Extensions of C ∗ -Algebras

useful in analysis, and is related to the left centralizer concept which J. G. Wendel used to investigate group algebras. The commutative version of this idea was first introduced by S. Helgason under the name of the algebra of multipliers (see (4)), and was recently developed by J. Wang (see (9)) and others. The connection with the extension problem and the work of Hochschild has apparently not been made by those using the centralizer concept. The definitions of double centralizer given in this paper are Johnson's. The proof that M(A) is an involutive algebra is also due to him, as is most of the proof of Propositions 2.5, 3.1. We show that M(A) is a C*-algebra if A is, and the abstract proof presented is new, however the fact can be deduced by using a result of G. A. Reid (see (7, Proposition 3, p. 1021)). We then use what we call the strict topology to investigate M(A) for certain particular A. This topology is a generalization of a topology with the same name which was defined in the commutative case by R. C. Buck, and the resulting proof of 3.9 is new, although the result was proved by Johnson. The main result of the paper is that the classes of extensions of a C*-algebra A by a C*-algebra C are in one to one correspondence with the *-homomorphisms from C to the quotient algebra M(A)/A. The author would like to thank Professor Peter Freyd for the suggestion of using pullback arguments in this section (?3). This eliminated several pages of long calculations. The remaining sections of this paper are devoted to the description of extensions and their primitive ideal spaces by the use of the main result. This paper formed the basis for the author's doctoral

A representation theorem for positive functionals on involution algebras

A representation theorem for positive functionals on involution algebras

Associative algebras with involution and Jordan algebras

Associative algebras with involution and Jordan algebras

A complement to ``On the unitary equivalence among the components of decompositions of representations of involutive Banach algebras and the associated diagonal algebras''

A complement to ``On the unitary equivalence among the components of decompositions of representations of involutive Banach algebras and the associated diagonal algebras''