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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positive elements in c*algebras and states

I have problems to prove that an element $a $ is a $C^*$-algebra is positive if and only if $f(a) \geq 0$ for all states $f$. The definitions I use: -$f:A\to\mathbb{C}$ linear functional on a ...
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18 views

limit of state is zero

Let A be a C$^*$-algebra, $a\in A$ strictly positive (this means: for every state $\varphi$ of A is $\varphi(a)>0$). Let $u_n=(\frac{1}{n}+a)^{-1}$. Then for all $b\in A$ and all states $\varphi$ ...
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18 views

Proof that the spectrum of an element of a Banach Algebra is non-empty

I don't see why the line indicated with ***** in the following proof is true in the proof that spectrum of an element of a Banach Algebra is non-empty (Arveson, p.27) : For every $\lambda_0 \not\in ...
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20 views

strictly positive element iff A contains a countable approximative unit

I search a proof of: Let A be a c$^*$-algebra and let $(u_n)_{n\in\mathbb{N}}$ an approximative unit in A. Then $a=\sum\limits_{n=1}^{\infty}\frac{u_n}{2^n}$ is strictly positive. Could anybody tell ...
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42 views

Positive Elements in a C*algebra

Let A be a C$^*$-Algebra, $a\in A$. Why is $a\ge 0$ (a is called "positive") iff $\forall \varphi\in S(A): \varphi\ge0$? S(A) is the set of linear positive functional $\eta:A\to\mathbb{C}$ with ...
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27 views

Polar decomposition

Every $x\in B(H)$ has a representation such as $x= u|x|$ (by polar decomposition) where $u$ is a partial isometry. In a paper, the author claims $$u = strong - \lim_{\epsilon\to 0} ...
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C*-Algebras: Contractive Morphism

Problem Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$ with $\mathbb{1}_\mathcal{A}\in\mathcal{A}$. Consider an algebraic morphism $\pi:\mathcal{D}\subseteq\mathcal{A}\to\mathcal{B}$ with ...
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20 views

Show that $\pi(M)'' = \pi(M'')$

Let $M$ is a $*-$ subalgebra of $B(H)$. Let $\bar H$ denote the direct sum $\sum H_i$ where $\{H_i\}$ is a family of replicas of $H$. Define $$\pi :x\in B(H) \to \bar x \in B(\bar ...
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17 views

Literature: Derivations in C*-Algebras

Do you have some nice reference for dynamical systems in C*-algebras (including discussion of their derivations!) like notes, papers, books, etc.?
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$(\lim x_n)y = \lim (x_n y)$ when $x_n, y \in B(H)$

Please give me a hint to prove $(\lim x_n)y = \lim (x_n y)$ when $x_n, y \in B(H)$ for every n. Thanks in advance.
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36 views

If $A$ is a $*-$ Banach algebra then $\bar A^{wot} = \bar A^{weak^*}$?

If $A$ is a $*-$ subalgebra of $B(H)$, then clearly $\bar A^{weak^*}\subset \bar A^{wot}$ (wot means weak operator topology). Also on every bounded subset of $A$, two topologies equal. Now my question ...
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20 views

$A y= b$ in $C(X)$

Let $X$ be a compact Hausdorff topological space, and $C(X)$ denote the ring of all complex valued continuous functions on $X$. If $A\in C(X)^{m\times n}$, $b\in C(X)^{m\times 1}$, and for all $x\in ...
2
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1answer
52 views

Are the invertible elements of a Banach algebra closed in the set of left-invertible elements?

Let $A$ be a unital Banach algebra. Denote by $\mathrm{Inv}(A)$ the invertible elements in $A$, and $\mathrm{Inv}_\ell(A)$ the left-invertible elements. That is, $a \in \mathrm{Inv}_\ell(A)$ if and ...
2
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1answer
74 views

Locally Compact Stone-Weierstrass Theorem

STATEMENT: Let $X$ be a locally compact Hausdorff space, and let $A = C_∞(X)$ be the algebra of continuous real-valued functions on $X$ that vanish at infinity, as above, equipped with the supremum ...
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21 views

Derivations: Analytic Vectors

Given a C*-algebra with unit $1\in\mathcal{A}$. Consider a dynamics $\tau$. Define its derivative: $$\delta(A):=\lim_{t\to0}\frac{1}{t}\left(\tau^t(A)-A\right)\quad A\in\mathcal{D}(\delta)$$ (The ...
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1answer
44 views

Why is $\widehat{a}$ bijective?

I am trying to understand the proof of the following theorem: Let $A$ be a unital Banach algebra generated by $1$ and $a$. Then $A$ is abelian and the map $\widehat{a}:\Omega(A) \to \sigma (a), \tau ...
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1answer
40 views

Comparing weak and weak operator topology

We can compare topologies on $B(H)$. For instance, Sot topology is stronger than wot topology or $\sigma-$ weak topology is equivalent to weak* topology. I would like to compare wot topology and weak ...
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125 views

Is the left regular representation of an algebra, always faithful?

Let $\mathcal{A}$ be a unital associative algebra with a countable basis $\mathcal{b}$ over $\mathbb{C}$. Let $H=l^2(b)$ be the Hilbert space generated by $\mathcal{b}$. Let $H_0 = \{v \in H \ \vert \ ...
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1answer
25 views

Where is a mistake in my proof concerning the spectrum of elements of a unital Banach algebra?

I was going to prove: Let $A$ be a unital Banach algebra. Then $$\sigma(a) = \{\tau(a) \mid \tau \in \Omega (A)\}$$ and I started the following argument: Let $\lambda \in \sigma (a)$ and let ...
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30 views

The spectrum of an element in a non-unital Banach algebra

Let $A$ be a commutative non-unital Banach algebra. Let $\widetilde{A}$ denote its unitisation. I am trying to understand the proof of $$ \sigma (a) = \{\tau (a) \mid \tau \in \Omega (A)\}\cup ...
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1answer
34 views

Restriction of a spectral measure

Let $x$ be a self-adjoint operator on $H$. By spectral theorem, there is a spectral measure $\mu$ correspondence to $*-$ homomorphism $\pi:C(\sigma(x)) \to B(H)$ such that $x=\int_{-||x||}^{||x||} ...
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1answer
111 views

An abelian Banach algebra without characters

Can one give an example of an abelian Banach algebra with empty character space? Such algebra must be necessarily non-unital. I couldn't find any examples of such algebras. Thanks!
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23 views

Banach fixed-point theorem and Newton

I have to combine the Newton method and the Banach fixed-point theorem: Let $I \subset \mathbb{R}$ a closed interval and $\phi: I \rightarrow I$ Lipschitz continous. Let $f: I \rightarrow ...
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75 views

A lemma by Foguel and Weiss [1973]

so I am reading Krengel's text on Ergodic theorems. And the next lemma bugs me as for the proof of it. It's by Foguel and Weiss. Statement: If $P_1, P_2$ are commuting elements of a Banach algebra ...
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1answer
23 views

Natural *-isomorphism

Show that if $X,Y$ are locally compact spaces, then there is a natural *-isomorphism from $C_0(X,C_0(Y))$ onto $C_0(X\times Y)$. My attempt: I define $\phi:C_0(X,C_0(Y))\to C_0(X\times Y)$ such that ...
2
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1answer
53 views

The set of analytic functions on unit circle is not a C*-algebra

Let $\mathbb{D}$ be the open unit disc on the complex plane and consider the set $$A=\{f\in C({\rm cl}\, {\Bbb D})\colon f \text{ is an analytic function on } {\Bbb D}\}.$$ It is certainly closed ...
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1answer
22 views

Is $C(\Omega)$ a C*-algebra if $\Omega$ is not locally compact, nor compact?

We always say if $\Omega$ is compact or locally compact, then C(\Omega) is a C*-algebra. Now is $C(\Omega)$ a C*-algebra if $\Omega$ is not compact nor locally compact? If not, I want to know which ...
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1answer
30 views

spectral projection of an element in a C*-algebra

I'm studying Takesaki's Operator theory and I preferred "spectral projection "in the page 43 of this book while he didn't speak about it before. I searched it, but I could not find it. Please explain ...
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1answer
38 views

Positive linear functional on an involutive Banach algebra

Why is every positive linear functional on an involutive Banach algebra with a bounded approximate continuous?
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32 views

Approximate unit of an involutive Banach algebra

I know that every C*-algebra has an approximate unit. I have two questions: why we cannot show that every involutive Banach algebra has an approximate unit? I need an example of an involutive Banach ...
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1answer
30 views

Positive elements in a Banach algebra

Let $A$ be a unital Banach algebra. If $a$ is an element of $A$ with $||1-a||_{sp}<1$, then there exists $b\in A$ such that $b^2=a$. Furthermore, if $A$ is an involutive Banach algebra and if $a$ ...
2
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1answer
35 views

Cyclic representation on $L^2(\mu)$

Show that if $(X,\Omega,\mu)$ is a $\sigma-$ finite measure space and $H=L^2(\mu)$, then $\pi:L^\infty(\mu)\to B(H)$ defined by $\pi(\phi)=M_\phi$ is a cyclic representation and find all the cyclic ...
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Approximate unit of a separable C*-algebra

The following is a corollary of Takesaki's Operator Theory: My question: I do not know why the author says"there exists an n such that $||x(1-v_n)^\frac{1}{2}||<\epsilon$" . Please help me to ...
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34 views

Norm of an element in a C*-algebra

The following is a part of a proof in Takasaki's Operator theory: Let $\epsilon>0$ and $A$ is a C*-algebra. For an $x\in A$, put $h=x^*x$ and $u_\epsilon=(h+\epsilon)^{-1}h$. We have then ...
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1answer
30 views

Spectrum of Rank 1 Operators

Given $\psi$ and $\phi$ in a Hilbert space $H$, we let $T$ be the rank-1 operator such that $$T\varphi=<\psi,\varphi>\phi.$$ It is easy to find the eigenvalues of $T$, they are $0$ and ...
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36 views

The positive element in a C*-algebra

The following is a theorem of Conway's Functional Analysis: for the proof ($c\to a$), I think we can say: for $\lambda\in \sigma(a)\subset \Bbb R$, there is a character $h:C(\sigma(a))\to\Bbb C$ ...
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1answer
19 views

A question about Invariant subspaces of an algebra.

I feel that this is a very simple problem, but somehow I don't see the solution. I want to show that if $A$ is a subalgebra of $B(H)$ containing $1$ then if $B\in SOTcl(A)$, for every n, ...
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1answer
33 views

norm of a matrix that its entries are operators in B(H).

Let S is a subset of B(H). Define $M_2(S)=\{T= \left( \begin{array}{ccc} A & B \\ C & D \\ \end{array} \right) : A,B,C,D \in S\}$. what is the relationship between $||T||$ and ...
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2answers
43 views

Self-adjoint elements in a C*-algebra

I have a simple question which confused me. Suppose $A$ is a C*-algebra. every $x\in A$ has a representation such as $x=a+ib$ where $a,b$ are self-adjoint elements of $A$. Also we claim that $x^*x$ ...
3
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0answers
36 views

*-isomorphism of a C*-algebra into an involutive Banach algebra is norm increasing

The following is a proposition of Takesaki's Operator theory: My question: How does he assume, considering the C*-subalgebra generated by k instead of $*$-Banach algebra B? Are we sure that the ...
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15 views

An element a of a complex normed algebra is Hermitian iff $\|\text{exp}(i\alpha a)\|=1 \, \forall \alpha \in \mathbb{R}$

The (algebraic) numerical range $V(a)$ of an element $a$ of a complex unital normed algebra $A$ is defined as: $V(a)=\{f(a):f \in A^{'}, \|f\|\le1, f(1)=1\}$. $a$ is said to be Hermitian iff ...
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Limit of an element in a unital C*-algebra

Let $A$ be a unital C*-algebra. Show that an element $x$ of $A$ is self-adjoint if and only if $\lim_{t\to 0}\frac{1}{t}(||1+itx||-1)$=0. My attempt: Suppose $x=x^*$. By functional calculus of x, ...
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28 views

There is no continuous function h on unit circle such that u=exp ih when spectrum u is entire unit circle

Let $\Gamma$ be the unit circle. Let u be the unitary element in $C(\Gamma)$ defined by $u(\lambda)=\lambda$. Show that there is no continuous function h on unite circle such that u=exp ih. ...
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functional calculus on a set of normal elements is continuous

Let $K$ be a compact subset of $\Bbb C$. Let $A_K$ denote the set of all normal elements $x$ with $\sigma_A(x)\subset K$. If $f$ is a continuous function on $K$, then the functional calculus :$x\in ...
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19 views

Why alg(v) is not normclosed in the Banach algebra V in general?

Let V be an unital Banach algebra, v $\in V$. The smallest subalgebra of V, with contains v and e, is defined as follows: $\overline{alg(v)}:=\overline{\{\sum\limits_{k=0}^n\lambda_kv^k: ...
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Homeomorphism between locally compact space $\Omega$ and maximal ideals space of $C_0(\Omega)$

the following is a proposition: If $\Omega$ is locally compact and $\Sigma$ is the maximal ideal space of $C_0(\Omega)$, then the map $x\to \delta_x$ is a homeomorphism. To prove it, the author ...
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2answers
67 views

Is there any multiplicative linear functional on B(H)?

If A is a Banach algebra, we say that $\Phi: A \longrightarrow \mathbb{C}$ is a multiplicative linear functional if $\Phi$ is nontrivial, linear and $\Phi(xy)=\Phi(x)\Phi(y)$. It is easy to see that ...
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59 views

Closed unit ball of B(H) is not compact in strong operator topology of B(H).

In operator theory we prove that closed unit ball of B(H) is compact in weak operator topology and is closed in strong operator topology. But a book of operator theory states that closed unit ball of ...
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1answer
43 views

The map $T\longmapsto \|T\|$ is not continuous in the strong operator topology of $\mathscr B(H)$

In the context of Strong and Weak operator topologies on $\mathscr B(H)$ there is an statement that says: the map on $\mathscr B(H)$ that $T\longmapsto \|T\|$ is not continuous in the strong operator ...
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1answer
29 views

an exercise about the spectrum of an element in Banach algebra.

An exercise of Banach algebra, section of spectrum has wanted the proof of this statement: Let $A$ be a Banach algebra and $x\in A$. Show that for every open set $U$ in $\mathbb{C}$ that contains ...