# Tagged Questions

A Banach algebra is an algebra over the real or complex numbers which is equipped with a complete norm such that |xy| ≤ |x||y|. The study of Banach algebras is a major topic in functional analysis. If you are about to ask a question on C*-algebras or von Neumann algebras please use (c-star-algebras) ...

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### example of the arens multiplication; I want to understand the construction

I want to understand the double dual as a $C^*$-algebra of a given $C^*-$algebra $A$ but my first problem is to understand the Arens multiplication on the double dual $A^{**}$ (considered as Banach ...
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### Completely positive, orthogonaly preserving maps

First of all, here is the setting for my problem: Definition: Let $A$ be a $C^*$-Algebra, $a,b\in A$. We say, a and b are orthogonal, if $ab=ba=a^*b=ab^*=0$. Proposition: Let $A$ be a $C^*$-Algebra, ...
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### Maximal ideal space of $L^{\infty}(m)$ separable?

Let $m$ denote the Lebesgue-measure on the unit interval and let $L^{\infty}(m)$ be the set of measurable essentially bounded functions on that interval (i.e. $f \in L^{\infty}(m)$ if and only if ...
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### Elements near the identity of a linear subspace

I am currently trying to understand a proof and ran into the following problem. The proof states (everything takes place in a commutative, unital Banach-Algebra): A linear subspace $X$ with ...
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### On existence of square root of positive elements of a unital $C^*$-algebra

Given a unital $\mathcal{C}^*$-algebra $A$ and a positive element $a \in A$, I am trying to prove the existence of a square root $a^{\frac{1}{2}}$ i.e. a positive element $b \in A$ such that $b^2 = a$....
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### The Gelfand and norm topologies are equal on the character space of $L^1(\mathbb Z)$

We know that the character space of the Banach algebra $L^{1}(\mathbb Z)$ is homeomorphic to the unit circle $\mathbb T$, but I can't show that the Gelfand and norm topologies are equal on that.
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### Algebra with element having empty spectrum?

The definition of the spectrum makes sense for any algebra. I guess we can go to the unitization to make sense of it even non-unital algebras. Recalling the well-known fact that for normed algebras, ...
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Let $D = \{ z \in \mathbb{C} : |z| < 1 \}$ be the open unit disk and $\mathbb{T} = \{ z \in \mathbb{C} : |z| = 1 \}$ its boundary. We will naturally write $\bar{D}$ for its closure $\{ z \in \... 1answer 83 views ### Countable set in a Banach space which spans densely? Let$\mathcal{C}(\mathbf{T})$be the algebra of continuous complex functions on the unit circle$\mathbf{T}$. Consider the following two statements: The$*$-subalgebra generated by$1$and$z$spans ... 2answers 93 views ### Prove Operator is a Projector Let$\mathscr{H}$be a complex Hilbert space. A projector is a linear map$P:\mathscr{H}\to\mathscr{H}$such that$P\circ P = P$. I'm trying to prove the following claim, from the information given ... 1answer 46 views ###$||a||\leq \sup_{||b||\leq 1} ||ab||$in a C*-algebra I would like to prove that, if$a$is an element in a C*-algebra then $$\|a\|\leq \sup_{\|b\|\leq 1} \|ab\|$$ It is obvious if the algebra is unital. What if it is not? 0answers 38 views ### Proof of theorems in the field of banach-and$c^*$-algebras in a categorial language At the moment I'm studying the basics in the theory of banach- and$c^*$-algebras. There are many results in the theory of$c^*$-algebra which you first prove in the unital case and then in the ... 0answers 40 views ### positive elements in$L(H)$At the moment I'm studying positive elements in$C^*$-algebras. Let$H$be a complex Hilbert space,$L(H)$the linear bounded operators$H\to H$and$T^*=T. Then: $$\sigma(T)\subseteq [0,\|T\| ]\; \... 0answers 56 views ### Choice of a dense subset of a separable Banach space I recently came across the following statement, and still can't prove it: Statement: Suppose X is a separable,closed subspace of L^1(G), where G is a locally compact group. Since X is ... 1answer 25 views ### positive linear maps which are involutive Let S be an operator system, B a c^*-algebra and \phi:S\to B a positive linear map. Then \phi is involutive, i.e. \phi(x^*)=\phi(x)^* for all x\in S. I want to prove this claim but I'm ... 1answer 40 views ### positive linear maps of c^*-algebras are bounded Let A, B be c^*-algebras and \phi:A\to B a positive, linear map. Then \phi is bounded. Proof: It is sufficient to proof boundedness of \phi on the unitarization (I missunderstood that, see ... 0answers 28 views ### Applications of C^\ast algebras in differential topology I was wondering if there were any useful ways C^\ast algebras come into play within differential topology. I know this is a fairly broad question,so any type of input is applicable. 2answers 86 views ### Can {L^{1}}(G) be a C^{*} -algebra? Let G be a locally compact abelian group. Then {L^{1}}(G) is a commutative algebra when equipped with convolution. Is there an involution ^{*} on {L^{1}}(G) so that it becomes a C^{*}... 1answer 30 views ### A property of a ideal of Banach algebras Let B be a Banach algebra and A be a bi-ideal of it. Suppose that for any b\in B, Ab=\{0\} implies b=0. Now could we say that for some c\in B if cA=\{0\} then c=0? 1answer 45 views ### Banach algebra norms on M_n(A) Let A be a Banach algebra (not sure whether I need A to be unital). I saw the claim that all Banach algebra norms on M_n(A) with continuous projections on entries are equivalent. How does one ... 0answers 30 views ### When do injective and projective tensor norms agree? For C^*-algebra tensor products, one talks about the min and max tensor norms, and they agree when one of the C^*-algebras is nuclear. For general Banach algebras, what is the analog of nuclearity?... 1answer 53 views ### if f is in Banach space, then \nabla f is in the dual space? I am not very deep in advanced real analysis. Could you help me decipher the following two phrases hold? 1) if f is in Banach space \mathcal{B}, then \nabla f is in the dual space \mathcal{B}^... 1answer 37 views ### Multiplicative linear functionals on subalgebras If A is a commutative C^\ast algebra and C is a C^\ast sub algebra of A is it true that the characters on C are just restrictions of characters on A. The reason I am asking this because ... 2answers 100 views ### Algebra is Generated by nilpotent Lie Algebra X Banach space. In B(X), we can define a Lie product [ , ]:[T_1,T_2]=T_1T_2-T_2T_1 for any T_1,T_2 \in B(X). Let \mathcal{L} Lie Algebra. \mathcal{L}^1=\mathcal{L} , \mathcal{L}^2=[\... 1answer 55 views ### realizing/ understanding C^*(\phi_g(C([0,1]))) and "support projection of an element of a C^*-algebra someone asked me the following question but I don't know the answer, but and I'm interested in it too. Let C([0,1])=\{f:[0,1]\to\mathbb{C}; \text{f is continuous}\}, g\in C([0,1]) be a positive ... 0answers 41 views ### Banach algebras for which the Gelfand transform is 1-1 but not onto Are there examples of Banach Algebra's for which the Gelfand transform is 1-1, (ie: the intersection of all maximal ideals of the algebra is \{0\}) but not onto? Context Page 96 of Kaniuth's "... 1answer 46 views ### almost unital Banach algebra's Let A be an "almost unital Banach algebra", in the sense that it satisfies all the usual axioms but not necessarily that \|1\|=1. From the product inequality \forall x,y \in A \|... 1answer 61 views ### preserving problem Define:$$\begin{align*} \varphi &:L^2[0,1] \to L^2[0,1], \\ &(\varphi f)(x)=‎\int_0^x f(t)dt. \end{align*}‎$$Is it true that, if ‎‎‎‎B is a dense ‎subset ‎of ‎‎‎‎L^2[0,1], then ‎‎‎\... 0answers 56 views ### Ideal in Matrix algebra Let A be a Banach algebra and suppose that \Lambda be a nonempty set. With the following product and norm the matrix space$$M_\Lambda(A)=\{(a_{ij})_{i,j\in\Lambda}:\sum_{i,j\in\Lambda}\|a_{ij}\|<... 1answer 53 views ### Quotient Rings of Algebras So let us take the commutative Banach algebraB=\mathscr{l}^1(\mathbb{Z}_n)$over$\mathbb{R}$with convolution as multiplication$(f\ast g)(x)=\underset{y\in \mathbb{Z}_n}{\sum} f(y)g(x-y)$. I know ... 1answer 76 views ### Topological modules and relative homological algebra. This question might be a bit dumb but I'm tired right now and this is just going over my head at the moment, in "The homology of Banach and topological algebras" Helemskii said that relative ... 1answer 40 views ### If$X$is compact Hausdorff, is every isomorphicm$\mathcal{C}(X) \to \mathcal{C}(X)$continuous? If$X$is a compact Hausdorff space, is then every isomorphism from${\mathcal C}(X)$onto${\mathcal C}(X)$is continuous? 1answer 71 views ### GNS Construction on non-unital algebra STATEMENT: If A has a multiplicative identity 1, then it is immediate that the equivalence class$ξ$in the GNS Hilbert space H containing 1 is a cyclic vector for the above representation. If$A$is ... 1answer 92 views ### Quaternions as a counterexample to the Gelfand–Mazur theorem It seems that by the Gelfand–Mazur theorem quaternions are isomorphic to complex numbers. That is clearly wrong. So where is the catch? I think that I found the problem but it seams so subtle that ... 0answers 36 views ### Equivalent statements involving 'little o' Let$A$be a Banach algebra and$a\in A$.$\|z(z-a)^{-1}\|=1+o(\frac{1}{z})$as$z\rightarrow +\infty$iff$\|(z-a)^{-1}\|^{-1}=z+o(1)$as$z\rightarrow +\infty$. i.e.,$lim_{z\rightarrow +\infty}\|(...
Reading about Gelfand-Naimark theorem I've seen that the Fourier transform is a special case of Gelfand transform for the space $L^1(\mathbb{R})$ with the convolution product. In a related question on ...