Spectrally bounded traces on C*-algebras

A linear map $T$ from a Banach algebra $A$ into another $B$ preserves zero products if $T(a)T(b)=0$ whenever $a,b\in A$ are such that $ab=0$. This paper is mainly concerned with the question of whether every continuous linear surjective map $T\colon A\rig

The paper begins with an introduction to the cohomology theory for Banach algebras needed in the remainder of the paper. A perturbation of the multiplication π of a Banach algebra A is a continuous associative multiplication ρ which is near ρ in the usual norm for bilinear maps. We consider the question of whether such a multiplication is necessarily of the form ρ(a, b) = W−1(WaWb) where W is a bounded linear map from A into A near the identity. A cohomological condition is given for this to hold. A linear map T from one Banach algebra A into another B is “almost” a homomorphism if the bilinear map \(\mathop{T}\limits^{ \circ } (a,b) = T(a,b) - T(a)T(b)\) is small. This occurs if T is near a genuine homomorphism and we consider whether every map which is almost a homomorphism is near a homomorphism. The main result is that if A is an amenable algebra and B is the dual of a suitable Banach space then the pair (A, B) has this property.

K-POTENT PRESERVING LINEAR MAPS

Automatic continuity conditions for a linear mapping from a banach algebra onto a semi-simple Banach algebra

In order to summarize the contents of this paper it is necessary to use some of the definitions in [5]. These definitions are repeated below at the beginning of Section 1. Although this paper adds to the material in [5] we have attempted to reduce the number of references to [o] to a minimum. An approximate diagonal in a Banach algebra 31 is a bounded net {ma} in 31 a for all a in 31 in the norm topologies where 31? 31 is the usual projective tensor product [2] and -k is the continuous linear map 31?31->31 defined by ir(a?b) =ab. The net {m 31) ** with aM = Ma and ?r** (M) a = a for all a ? 31. In the algebraic cohomology of algebras the existence of an element m of A A with ma = am and ir(m) the identity of A is equivalent to the statement that H1(A, X) =0 for all 31 modules X almost never holds?in fact, in the commutative case for all A bimodules X. Among Banach algebras the statement jEP^S^X) =0 (see [3; Theorem 1] and [5; Proposition 8.1]) we can always find a Banach 31 module ? with S^1(3I,36)=t^O unless 31 zz?n. The existence of an approxi? mate diagonal is not unusual and is equivalent to the statement !M* (31,36*) = 0 for all Banach 31 modules X; thatis 31 is an emenable Banach algebra. The element M plays the same role in the theory of amenable tlgebras that the invariant mean does for amenable groups. The definitions used from [5] and the theorem connecting amenability with approximate diagonals are given in ? 1. In ? 2 we show that if 31 has a virtual diagonal with Ma? 31? St for all a in 3T then 913 (31, 3E) ? .0 for all Banach 31 modules 3?. Algebras satisfying this condition are grdup algebras of compact topological groups and the algebra c0 of convergent sequences of complex numbers. Although we do not prove it the result also applies to any annihilator B* algebra with only finite dimensional minimal ideals. In ? 3 we show that if 31 is a semi simple commutative amenable Banach algebra then there is a Banach 31 module X with 3/2(3I,3E) ^0, unless 31 is finite

Let A be a unital C*-algebra of real rank zero and B be a unital semisimple complex Banach algebra. We characterize linear maps from A onto B preserving different essential spectral sets and quantities such as the essential spectrum, the (left, right) essential spectrum, the Weyl spectrum, the index and the essential spectral radius.

In this paper, we introduce the concept of generalized $g$-quasi-contractions in the setting of cone $b$-metric spaces over Banach algebras. By omitting the assumption of normality we establish common fixed point theorems for the generalized $g$-quasi-contractions with the spectral radius $r(\lambda)$ of the $g$-quasi-contractive constant vector $\lambda$ satisfying $r(\lambda) \in [0,\frac{1}{s})$ in the setting of cone $b$-metric spaces over Banach algebras, where the coefficient $s$ satisfies $s\ge 1$. The main results generalize, extend and unify several well-known comparable results in the literature.

For the Fr'{e}chet algebras $(A, (p_k))$ and $(B, (q_k))$ and $n in mathbb{N}$, $ngeq 2$, a linear map $T:A rightarrow B$ is called textit{almost $n$-multiplicative}, with respect to $(p_k)$ and $(q_k)$, if there exists $varepsilongeq 0$ such that$$q_k(Ta_1a_2cdots a_n-Ta_1Ta_2cdots Ta_n)leq varepsilon p_k(a_1) p_k(a_2)cdots p_k(a_n),$$for each $kin mathbb{N}$ and $a_1, a_2, ldots, a_nin A$. The linear map $T$ is called textit{weakly almost $n$-multiplicative}, if there exists $varepsilongeq 0$ such that for every $kin mathbb{N}$ there exists $n(k)in mathbb{N}$ with$$q_k(Ta_1a_2cdots a_n-Ta_1Ta_2cdots Ta_n)leq varepsilon p_{n(k)}(a_1) p_{n(k)}(a_2)cdots p_{n(k)}(a_n),$$for each $k in mathbb{N}$ and $a_1, a_2, ldots, a_nin A$.The linear map $T$ is called $n$-multiplicative if$$Ta_{1}a_{2} cdots a_{n} = Ta_{1} Ta_{2} cdots Ta_{n},$$for every $a_{1}, a_{2},ldots, a_{n} in A$.In this paper, we investigate automatic continuity of (weakly) almost $n$-multiplicative maps between certain classes of Fr'{e}chet algebras, including Banach algebras. We show that if $(A, (p_k))$ is a Fr'{e}chet algebra and $T: A rightarrow mathbb{C}$ is a weakly almost $n$-multiplicative linear functional, then either $T$ is $n$-multiplicative, or it is continuous. Moreover, if $(A, (p_k))$ and $(B, (q_k))$ are Fr'{e}chet algebras and $T:A rightarrow B$ is a continuous linear map, then under certain conditions $T$ is weakly almost $n$-multiplicative for each $ngeq 2$. In particular, every continuous linear functional on $A$ is weakly almost $n$-multiplicative for each $ngeq 2$.

We consider the Calkin algebra C R (A) and the Fredholm theory in a Banach algebra A, relative to some fixed ideal F of A. Our aim is to study linear maps between unital Banach algebras A and B which are surjective up to the inessential elements relative to F, and preserve Fredholm or semi-Fredholm elements in both directions or equivalently different relatively essential spectral sets such as essential spectrum, left or right essential spectrum, the boundary of essential spectrum or the full essential spectrum.We characterize such mappings when one of C R (A) or C R (B) is commutative and also investigate similar problems when A is assumed to be a unital C * -algebra of real rank zero and B is an arbitrary Banach algebra. RESUMENConsideramos el álgebra de Calkin C R (A) y la teoría de Fredholm en un álgebra de Banach A relativa a algún ideal fijo F de A. Nuestra meta es estudiar aplicaciones lineales entre álgebras de Banach unitales A y B las cuales son sobrejectivas salvo los elementos no esenciales relativos a F y preservan los elementos de Fredholm o semi-Fredholm en ambas direcciones o equivalentemente conjuntos espectrales esenciales relativos diferentes tales como el espectro esencial izquierdo o derecho, la frontera del espectro esencial o el espectro esencial completo.Caracterizamos dichas aplicaciones cuando uno de los C R (A) o C R (B) es conmutativo e investigamos problemas similares cuando A se asume que es una C * -álgebra unital de rango real cero y B es una álgebra de Banach cualquiera.

This chapter studies, in the setting of vector spaces over a field, the basics concerning multilinear functions, tensor products, spaces of linear functions, and algebras related to tensor products. Sections 1–5 concern special properties of bilinear forms, all vector spaces being assumed to be finite-dimensional. Section 1 associates a matrix to each bilinear form in the presence of an ordered basis, and the section shows the effect on the matrix of changing the ordered basis. It then addresses the extent to which the notion of “orthogonal complement” in the theory of inner-product spaces applies to nondegenerate bilinear forms. Sections 2–3 treat symmetric and alternating bilinear forms, producing bases for which the matrix of such a form is particularly simple. Section 4 treats a related subject, Hermitian forms when the field is the complex numbers. Section 5 discusses the groups that leave some particular bilinear and Hermitian forms invariant. Section 6 introduces the tensor product of two vector spaces, working with it in a way that does not depend on a choice of basis. The tensor product has a universal mapping property—that bilinear functions on the product of the two vector spaces extend uniquely to linear functions on the tensor product. The tensor product turns out to be a vector space whose dual is the vector space of all bilinear forms. One particular application is that tensor products provide a basis-independent way of extending scalars for a vector space from a field to a larger field. The section includes a number of results about the vector space of linear mappings from one vector space to another that go hand in hand with results about tensor products. These have convenient formulations in the language of category theory as “natural isomorphisms.” Section 7 begins with the tensor product of three and then $n$ vector spaces, carefully considering the universal mapping property and the question of associativity. The section defines an algebra over a field as a vector space with a bilinear multiplication, not necessarily associative. If $E$ is a vector space, the tensor algebra $T(E)$ of $E$ is the direct sum over $n\geq0$ of the $n$-fold tensor product of $E$ with itself. This is an associative algebra with a universal mapping property relative to any linear mapping of $E$ into an associative algebra $A$ with identity: the linear map extends to an algebra homomorphism of $T(E)$ into $A$ carrying 1 into 1. Sections 8–9 define the symmetric and exterior algebras of a vector space $E$. The symmetric algebra $S(E)$ is a quotient of $T(E)$ with the following universal mapping property: any linear mapping of $E$ into a commutative associative algebra $A$ with identity extends to an algebra homomorphism of $S(E)$ into $A$ carrying 1 into 1. The symmetric algebra is commutative. Similarly the exterior algebra $\bigwedge(E)$ is a quotient of $T(E)$ with this universal mapping property: any linear mapping $l$ of $E$ into an associative algebra $A$ with identity such that $l(v)^2=0$ for all $v\in E$ extends to an algebra homomorphism of $\bigwedge(E)$ into $A$ carrying 1 into 1. The problems at the end of the chapter introduce some other algebras that are of importance in applications, and the problems relate some of these algebras to tensor, symmetric, and exterior algebras. Among the objects studied are Lie algebras, universal enveloping algebras, Clifford algebras, Weyl algebras, Jordan algebras, and the division algebra of octonions.

Let $A$ and $B$ be two unital $C^{*}$-algebras and $varphi:A rightarrow B$ be a linear map. In this paper, we investigate the structure of linear maps between two $C^{*}$-algebras that preserve a certain property or relation. In particular, we show that if $varphi$ is unital, $B$ is commutative and $V(varphi(a)^{*}varphi(b))subseteq V(a^{*}b)$ for all $a,bin A$, then $varphi$ is a $*$-homomorphism. It is also shown that if $varphi(|ab|)=|varphi(a)varphi(b)|$ for all $a,bin A$, then $varphi$ is a unital $*$-homomorphism.

In this article we obtain 2 generalizations of the well known Gleason-Kahane-Zelazko Theorem. We consider a unital Banach algebra A, and a continuous unital linear mapping φ of A into Mn(C) – the n × n matrices over C. The first generalization states that if φ sends invertible elements to invertible elements, then the kernel of φ is contained in a proper two sided closed ideal of finite codimension. The second result characterizes this property for φ in saying that φ(Ainv) is contained in GLn(C) if and only if for each a in A and each natural number k: trace(φ(a)) = trace(φ(a)) .

Let A be a unital semiprime, complex normed ∗-algebra and let f, g, h : A → A be linear mappings such that f and g+h are continuous. Under certain conditions, we prove that if f(p ◦ p) = g(p) ◦ p + p ◦ h(p) holds for any projection p of A, then f and g+h are two-sided generalized derivations, where a◦b = ab+ba. We present some consequences of this result. Moreover, we show that if A is a semiprime algebra with the unit element e and n > 1 is an integer such that the linear mappings f, g : A → A satisfy f(x n ) = Pn j=1 x n−j g(x)x j−1 for all x ∈ A and further g(e) ∈ Z(A), then f and g are two-sided generalized derivations associated with the same derivation. Also, we show that if A is a unital, semiprime Banach algebra and F, G: A → A are linear mappings satisfying F(b) = −bG(b −1 )b for all invertible elements b ∈ A, then F and G are two-sided generalized derivations. Some other related results are also discussed.

This thesis is divided into two parts. In the first part, we consider an endomorphism \\alpha: A \\to A, with A being a commutative Banach algebra with unity, and X the spectrum of A. We ensure the existence of a unique partial dynamical system \\varphi: \\Delta \\subseteq X \\to X associated with the endomorphism. By adding the condition that A is a uniform algebra, we obtain an expression for the spectral radius of the weighted endomorphisms a\\alpha, a \\in A, which depends on the invariant or ergodic probabilities measures of the partial dynamical system (X, \\varphi). Moreover, this spectral radius will coincide with the spectral radius of the weighted shift operators aT, where \\alpha(a) = TaS for some operator S. In the second part of the project, for the shift operator \\sigma: \\Delta \\to X_A, where X_A is a generalized Markov shift space, we identify a *-endomorphism \\alpha: \\mathcal_A \\to \\mathcal_A associated with (X_A, \\sigma), where \\mathcal_A is the algebra whose spectrum is X_A. We give examples under conditions that guarantee the continuous extension of the shift operator and other examples where we show the impossibility of extending the shift operator continuously over the entire space. For the particular case of the renewal shift, we characterize every element of the algebra and provide an explicit formula to calculate r(a\\alpha) for all a \\in \\mathcal_A.

We prove that the norm of a hermitian element in a Banach algebra is equal to the spectral radius of the element.