Involutive Banach Algebras Research Articles

The Relative (Co)homology Theory through Operator Algebras

This paper introduces a new idea in the unital involutive Banach algebras and its closed subset. This paper aims to study the cohomology theory of operator algebra. We will study the Banach algebra as an applied example of operator algebra, and the Banach algebra will be denoted by <img src=image/13426521_01.gif>. The definitions of cyclic, simplicial, and dihedral cohomology group of <img src=image/13426521_01.gif> will be introduced. We presented the definition of <img src=image/13426521_14.gif>-relative dihedral cohomology group that is given by: <img src=image/13426521_02.gif>, and we will show that the relation between dihedral and <img src=image/13426521_14.gif>-relative dihedral cohomology group <img src=image/13426521_11.gif> <img src=image/13426521_12.gif> <img src=image/13426521_13.gif> can be obtained from the sequence <img src=image/13426521_03.gif>. Among the principal results that we will explain is the study of some theorems in the relative dihedral cohomology of Banach algebra as a Connes-Tsygan exact sequence, since the relation between the relative Banach dihedral and cyclic cohomology group (<img src=image/13426521_04.gif> and <img src=image/13426521_05.gif>) of <img src=image/13426521_01.gif> will be proved as the sequence <img src=image/13426521_06.gif>. Also, we studied and proved some basic notations in the relative cohomology of Banach algebra with unity and defined its properties. So, we showed the Morita invariance theorem in a relative case with maps <img src=image/13426521_07.gif> and <img src=image/13426521_08.gif>, and proved the Connes-Tsygan exact sequence that relates the relative cyclic and dihedral (co)homology of <img src=image/13426521_01.gif>. We proved the Mayer-Vietoris sequence of <img src=image/13426521_04.gif> in a new form in the Banach B-relative dihedral cohomology: <img src=image/13426521_09.gif> <img src=image/13426521_10.gif>. It should be borne in mind that the study of the cohomology theory of operator algebra is concerned with studying the spread of Covid 19.

Strongly amenable involutive representations of involutive Banach algebras

Let $$\mathfrak{A }$$ be a Banach $$*$$ -algebra and let $$\varphi $$ be a nonzero self-adjoint character on $$\mathfrak{A }$$ . For a $$*$$ -representation $$\pi $$ of $$\mathfrak{A }$$ on a Hilbert space $$\mathcal{H }$$ , we introduce and study strong $$\varphi $$ -amenability of $$\pi $$ in terms of certain states on the von Neumann algebra of bounded operators on $$\mathcal{H }$$ . We then give some characterizations of this notion in terms of certain positive functionals on $$\mathfrak{A }$$ . We finally investigate some hereditary properties of strong $$\varphi $$ -amenability of Banach algebras.

Noncommutative spectral synthesis for the involutive Banach algebra associated with a topological dynamical system

If $\Sigma=(X,\sigma)$ is a topological dynamical system, where $X$ is a compact Hausdorff space and $\sigma$ is a homeomorphism of $X$, then a crossed product involutive Banach algebra $\ell^1$ is naturally associated with these data. If $X$ consists of one point, then $\ell^1$ is the group algebra of the integers, hence the general$\ell^1$could be regarded as a noncommutative $\ell^1$-algebra. In this paper, we study spectral synthesis for the closed ideals of $\ell^1$ in two versions, one modeled after $C(X)$and one modeled after $\ell^1(\mathbb{Z})$. We identify the closed ideals which are equal to (what is the analogue of) the kernel of their hull, and determine when this holds for all closed ideals, i.e., when spectral synthesis holds. In both models, this is the case precisely when $\Sigma$ is free.

A characterisation of $C^*$-algebras through positivity of functionals

‎We show that a unital involutive Banach algebra‎, ‎with identity of‎ ‎norm one and continuous involution‎, ‎is a $C^*$-algebra‎, ‎with the‎ ‎given involution and norm‎, ‎if every continuous linear functional‎ ‎attaining its norm at the identity is positive‎.

Riccati Equations on Noncommutative Banach Algebras

Conditions for the existence of a solution of a Riccati equation to be in some prescribed noncommutative involutive Banach algebras are given. The Banach algebras are inverse-closed subalgebras of the space of bounded linear operators on some Hilbert space, and the Riccati equation has an exponentially stabilizing solution on this larger space. Applications to spatially distributed systems are given.

$\mathrm{C}^*$-Algebras

A \mathrm{C}^* -algebra is an involutive Banach algebra A which satisfies the \mathrm{C}^* -condition \|a^*a\|=\|a\|^2 for all a\in A . The theory of \mathrm{C}^* -algebras goes back to work of Murray and von Neumann, who first studied a special variant now known as von Neumann algebras. The theory developed rapidly after some ground breaking work of Gelfand and Naimark in 1943 in which they showed that The aim of the workshop \mathrm{C}^* -algebras , organized by Claire Anantharaman–Delaroche, Siegfried Echterhoff, Uffe Haagerup, and Dan Voiculescu, is to bring together leading researchers from basically all areas related to the field. This gives a unique opportunity to maintain a broad view on the subject and to create new cooperations between researchers with different background. Among the 42 participants was a good number of young researchers, some of them already on the top of the field. There have been 27 lectures presented at the workshop with topics ranging from classification of \mathrm{C}^* -algebras, group actions on \mathrm{C}^* -algebras, orbit equivalence of dynamical systems, Jones' theory of subfactors, fundamental groups of II _1 -factors, L^2 -invariants, duality for quantum groups and quantum groupoids, Index theorems, \mathrm{C}^* -algebras related to number theory, free probability, the relation between \mathrm{C}^* -algebras and Harmonic Analysis, and others. Among the most exciting recent developments in the field we mention the breathtaking advances of Popa and Vaes on the fundamental groups of II _1 -factors in the theory of von Neumann algebras, with beautiful applications to the classification of equivalence relations and dynamical systems. Other milestones are the progress of Kirchberg and coworkers on the classification of actions on \mathcal O_2 and the beautiful results of Dadarlat on continuous bundles of \mathrm{C}^* -algebras. Important progress has been achieved also in the classification theory of simple and non-simple \mathrm{C}^* -algebras by Elliot, Rørdam, Toms, and Winter–with some further progress being made during the workshop. There were many other exciting contributions to this very successful workshop, which can be seen from the reports presented in this issue. It is a pleasure for the organizers to thank all participants of the workshop for their beautiful lectures and fruitful discussions. We also want to use this opportunity to thank the Mathematisches Forschungs-institut Oberwolfach for providing a very stimulating environment and strong support for organizing this conference. Special thanks also go to the very competent and helpful staff of the institute.

Moore-Penrose Inverse in Kreĭn Spaces

We discuss the notion of Moore-Penrose inverse in Kreĭn spaces for both bounded and unbounded operators. Conditions for the existence of a Moore-Penrose inverse are given. We then investigate its relation with adjoint operators, and study the involutive Banach algebra \({\mathfrak{B}}({\mathcal{H}})\). Finally applications to the Schur complement are given.

INVOLUTION AND THE HAAGERUP TENSOR PRODUCT

AbstractWe show that the involution $\theta(a\otimes b)=a^*\otimes b^*$ on the Haagerup tensor product $A\otimes_{\mrm{H}}B$ of $C^*$-algebras $A$ and $B$ is an isometry if and only if $A$ and $B$ are commutative. The involutive Banach algebra $A\otimes_{\mrm{H}}A$ arising from the involution $a\otimes b\to b^*\otimes a^*$ is also studied.AMS 2000 Mathematics subject classification: Primary 46L05; 46M05

QUATERNION HOMOLOGY OF BANACH SPACE

In this article we are concerned with the quaternion Banach spaces and their homology. We obtain the relation between the quaternion and the dihedral homology for a unital involutive Banach algebra.

Numerical ranges of elements of involutive Banach algebras and commutativity

Let $ {\cal A} $ be a unital involutive Banach algebra. Bonsall and Duncan defined the numerical range for each element x in $ {\cal A} $ by $ V(x) = \{f(x):f \in {\cal A'}, f(e) = 1 =|| f || \} $ , where e is the unit. We introduce another numerical range by $ W (x) = \{ f(x) : f \in {\cal A'}, f \geq 0, f(e) = 1 \} $ , and we call $ w(x) = {\rm sup} \{|z|: z \in W (x) \} $ the numerical radius of x. We give a few conditions for $ {\cal A} $ to be a C *-algebra, and we see that some mapping theorems for numerical ranges of elements of $ {\cal A} $ hold. We show that if w (x) ≤ 1 implies $ w (\varphi (x) ) \leq 1 $ for every Mobius transform $ \varphi $ of the unit disk, then $ {\cal A} $ is commutative.

Applications du calcul fonctionnel harmonique dans une algèbre de Banach involutive quotient

We dene the spectral radius and Ptak’s function on a quotient involutive Banach algebra and we extend certains results of V.Ptak to thes quotients. As an application of functional harmonic calculus we obtain theorem similar to those of K.Fan and Von Neumann.

Regular Fréchet–Lie Groups of Invertible Elements in Some Inverse Limits of Unital Involutive Banach Algebras

Abstract We consider a wide class of unital involutive topological algebras provided with a C*-norm and which are inverse limits of sequences of unital involutive Banach algebras; these algebras are taking a prominent position in noncommutative differential geometry, where they are often called unital smooth algebras. In this paper we prove that the group of invertible elements of such a unital solution smooth algebra and the subgroup of its unitary elements are regular analytic Fréchet–Lie groups of Campbell–Baker–Hausdorff type and fulfill a nice infinite-dimensional version of Lie's second fundamental theorem.

Regular Fréchet-Lie groups of invertible elements in some inverse limits of unital involutive Banach algebras

We consider a wide class of unital involutive topological algebras provided with aC*-norm and which are inverse limits of sequences of unital involutive Banach algebras; these algebra sare taking a prominent position in noncommutative differential geometry, where they are often called unital smooth algebras. In this paper we prove that the group of invertible elements of such a unital solution smooth algebra and the subgroup of its unitary elements are regular analytic Frechet-Lie groups of Campbell-Baker-Hausdorff type and fulfill a nice infinite-dimensional version of Lie's second fundamental theorem.

The logic of quantum systems with diagonal singularities

The work of the Brussels-Austin groups on irreversibility over the last years has shown that Quantum Large Poincare systems with diagonal singularity lead to an extension of the conventional formulation of dynamics at the level of mixtures which is manifestly time asymmetric. States with diagonal singularity acquire meaning as linear fractionals over the involutive Banach algebra of operators with diagonal singularity. We show in this paper that the logic of quantum systems with diagonal singularity is not the conventional logic of Hilbert space, because only finite combinations of prepositions are allowed.

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.

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,

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.

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