A C*-algebra is a complex Banach algebra together with an isometric antilinear involution satisfying (a b)* = b* a* and the C*-identity ‖a* a‖ = ‖a‖². Related tags: (banach-algebras), (von-neumann-algebras), (operator-algebras), (spectral-theory).

learn more… | top users | synonyms

0
votes
1answer
14 views

Partial Isometries: Characterization

Note: This thread is not to gain reputation!!! Given a C*-algebra. Any partial isometry satisfies: $$WW^*W=W$$ From this, one derives projections: $$W^*W,WW^*$$ Conversely, given projections: ...
0
votes
0answers
8 views

Extending isomorphisms between $*$-algebras to $C^*$-algebras

I'm quite sure I am correct about this but at the moment I can't think for the life of me why. Suppose $A$ and $B$ are $*$-algebras and there are $*$-homomorphisms $\pi_1 \colon A \to ...
1
vote
1answer
46 views

Sums of projections in a C*-algebra

Let $A$ be a $C^*$-algebra, and let $p_1, \ldots, p_n \in A$ be projections, meaning $p_i = p_i^* = p_i^2$. Now assume that the sum $p = p_1 + \ldots + p_n$ is also a projection. How can one show that ...
0
votes
0answers
13 views

pure state on a C*-algebra A and unit vector

Let $τ$ pure state on a C*-algebra A, and $y$ a unit vector in $H_{τ}$ such that $τ(a)= <\varphi _{τ}(a)(y),y>$ for all $a \in A$.Show that there is a scalar $b$ of modulus one such that $y=b ...
1
vote
1answer
22 views

characterizing an operator with projection whose spectrum is contained in $\{-1,1\}$

Let $\mathcal{A}$ be a $C^{*}$-algebra and $\sigma$ denote the spectrum. I want to show that if $\sigma (A)\subseteq \{-1,+1\}$ for $A\in \mathcal{A}$ then there is a projection $P$ such that ...
0
votes
0answers
36 views

Hilbert- Schmidt class is an ideal

Definitions: 1 - An operator $y\in B(H)$ is said to be of trace class if $y$ is compact, and also $\sum|\alpha_n| <\infty$ where $\alpha_n \in \sigma(y)$ and $y$ has a representation $\sum ...
2
votes
2answers
41 views

Compact operator space is the greatest ideal of $B(H)$

Suppose $H$ is a separable infinite dimensional Hilbert space. Show that if $A\in B(H)$ is noncompact, then there exist two operators $B,C$ such that $BAC=1$. Clearly if $A$ is invertible it holds, ...
1
vote
1answer
22 views

States: KMS-Condition

Given a C*-algebra $\mathcal{A}$ with dynamics $\tau$. Consider a state $\omega$. Does it suffice to have on a dense set the KMS-condition: ...
0
votes
1answer
23 views

Irridicible C*-algebra $A$ implies that projection $p$ is rank one if $pAp=\Bbb C p$

Let $A$ be an irreducible C*-subalgebra of $B(H)$ and $p$ be a nonzero projection in $B(H)$. Suppose $pAp=\Bbb C p$, show that $p$ is rank one. I do not have any idea about it. Please give me a ...
1
vote
0answers
53 views

What is so special about C*-Algebras?

Could anyone give me a brief explanation about C*-algebras and why they are so important in modern mathematics? I am specially interested in the way C*-algebras are intertwined with the ...
-1
votes
0answers
32 views

a positive compact operator on H is strictly positive in K(H) if and only if it has dense range

Let A be a C∗-algebra and A+ denote the positive elements. An element a∈A+ is called strictly positive if aAa¯=A. If H is a Hilbert space, Show that a positive compact operator on H is strictly ...
1
vote
2answers
46 views

Spectral measure of a unitary operator

The following is an Exercise of Conway's operator theory: 1-Show that if $A$ is hermitian operator, then $U=\text{exp}({iA})$ is unitary. 2- Show that every unitary can be so written. 3-Find the ...
0
votes
0answers
27 views

$M_{n}(A)$ is an AF-algebra

If $A$ is a $C^*$-algebra that contains an increasing sequence $(A_{n})_{n=1}^{\infty}$ of finite-dimensional $C^*$-subalgebras such that $\cup‎_{n=1}^{\infty} A_{n}$ is dense in $A$, show that ...
0
votes
1answer
39 views

Tensor Products of C*-Algebras

If A, B are C*-algebras, show that there exists a unique $*-isomorphism $ $ ‎\theta‎‎: A ‎\otimes‎_{*}‎‎ B ‎\longrightarrow‎ B ‎\otimes‎_{*}‎‎ A $ such that $\theta( a \otimes‎_{*} b) = b ...
0
votes
0answers
17 views

unital simple c*algebra and minimal projection

If an unital simple C*-algebra contains a non zero minimal projection is it finite dimensional? thanks
0
votes
0answers
16 views

AF algebra and approximate identity

Please prove following theorem: If $A$ is an AF-algebra, it has an approximate identity consisting of an increasing sequence of projection. thanks...
2
votes
1answer
43 views

Von Neumann algebra generated by a subalgebra

Let A be a C*-algebra of operators on a Hilbert space H. Show that if $A\subset K(H)$, then $\{A'\cap K(H)\}'\cap K(H) = A$ I do not have any idea about it. Please give me a hint. Thanks.
0
votes
1answer
24 views

Show that a nondegenerate *-Banach algebra is a C*-algebra

Takesaki in his operator theory says A C*-algebra $M$ of operators on Hilbert space $H$ means a nondegenerate ( $\text {cl} (MH) = H$) $*-$ subalgebra of $B(H)$ which is closed under the uniform ...
2
votes
2answers
107 views

Every normal operator on a separable Hilbert space has a square root that commutes with it

Show that every normal operator on a separable Hilbert space has a square root that commutes with it. Uniqueness? My attempt: Let $T$ be a normal operator. By polar decomposition $T=U|T|$ where ...
1
vote
1answer
60 views

On the weak closure of a sequence of projections

Let $H$ be a Hilbert space with $\text{dim}=\infty$ , and $\{e_n\}$ be an orthogonal sequence of projections in $B(H)$. Show that $\{\sqrt{n}e_n ; n\geq 1\}$ does not admit a subsequence converging to ...
0
votes
0answers
30 views

Construction of direct sum of Hilbert spaces

The following is a theorem of Takesaki's operator theory: I do not know why he puts $\bar H$ instead of $H$ for $n=-1,-2,\ldots\,{}$. Please help me. Thanks.
0
votes
0answers
21 views

States: Liouvilleans

Given a C*-algebra $\mathcal{A}$ with dynamics $\tau$. Consider an invariant state: $\omega\circ\tau^t\equiv\omega$ Then the dynamics is unitarily implementable: ...
1
vote
0answers
31 views

Closed unit ball of $B(H)$ with wot topology is compact

The following is a Theorem of Conway's operator theory: I can not understand how he proves it. I think $\phi(\text{ ball B(H)})$ is compact if $\phi(\text{ ball B(H)})$ is closed subset of compact ...
3
votes
2answers
68 views

For a hermitian element $a$ in a $C^*$-algebra, show that $\|a^{2n}\| = \|a\|^{2n}$

Let $\mathcal{A}$ be a $C^*$-algebra. Suppose that $a \in \mathcal{A}$ with the property that $a^* = a$ (that is, suppose that $a$ is hermitian). I would like to show that $\|a^{2n}\| = ...
1
vote
0answers
34 views

States: Approximate Unit

Given a C*-algebra $\mathcal{A}$. Consider a state: $\omega\geq0$ Especially one has: $\sup\omega(E^2)=\|\omega\|$ Can it actually fail to be a proper limit? The problem is that the square is not ...
-3
votes
1answer
40 views

Positive Elements: Square Root

Given a C*-algebra $\mathcal{A}$. Consider positive elements: $A\geq0$ Why does the square root lie within: $\sqrt{A}\in\mathcal{A}_A$
-1
votes
1answer
73 views

Positive Elements: Norm (Decomposition)

Given a C*-algebra $\mathcal{A}$. Then every element decomposes into: $Z=X_+-X_-+iY_+-iY_-=\sum_{\alpha=0\ldots3}i^\alpha Z_\alpha$ Obviously, one has: $\|Z\|\leq\sum_{\alpha=0\ldots3}\|Z_\alpha\|$ ...
1
vote
1answer
49 views

States: Positivity vs. Continuity

Definition Define positivity in terms of: $$\omega\geq0:\iff\omega(X^*X)\geq0$$ (This way it serves best for the GNS-construction.) Problem Given a C*-algebra $1\notin\mathcal{A}$. Consider a ...
1
vote
1answer
35 views

States: Extension

Definition Define positivity in terms of: $$\omega\geq0:\iff\omega(X^*X)\geq0$$ (This way it serves best for the GNS-construction.) Problem Given a C*-algebra $1\notin\mathcal{A}$ and adjoin a unit ...
2
votes
1answer
46 views

An elementary perturbation result in C*-algebra

The following question was raised when I read a papaer "MF actions and K-theoretic dynamics". In one of the proofs in that paper, the author utilized a so called "elementary perturbation result in ...
1
vote
0answers
24 views

Group C*-algebra of an abelian discrete group

Let $A = \mathbb C[G]$ be the group ring of all finitely supported functions $f\colon G \to \mathbb C$ of a discrete abelian group $G$ with the usual convolution product, and involution defined by ...
2
votes
1answer
33 views

Special Elements: Spectrum

Given a C*-algebra with unit $1\in\mathcal{A}$. For normal elements one has: $$A^*=A^{-1}\iff\sigma(A)\subseteq\mathbb{S}$$ $$A^*=A\iff\sigma(A)\subseteq\mathbb{R}$$ ...
0
votes
0answers
23 views

Finite dimensional operator space is dense in trace class space

To show that $F(H)$ (the space of finite dimensional operators on a Hilbert space $H$) is dense in $L^1(H)$ (the space of trace class operators), suppose that $x\in L^1(H)$. Without loss of generality ...
1
vote
1answer
23 views

Trace class operator is an ideal

To show that trace class operators space is an ideal, we need to show that $\|uv\|_1\leq \|u\|\|v\|_1$ where $u,v \in B(H)$ and $\|u\|_1 = tr(|u|)$. Murphy in his book (C*-algebras and operator ...
1
vote
1answer
58 views

Nonzero projection in an irredicible C*-algebra of minimal finite rank must have rank one

The following is a part of a theorem in Murphy's C*-algebras and operator theory: Let $A$ be a C*-algebra acting irreducibly on a Hilbert space $H$ and $q$ be a nonzero projection in $A$ of minimal ...
0
votes
1answer
34 views

$*-$ isomorphism between two compact spaces $K(H)$ and $K(H')$

The following is a theorem of Murphy's C*-algebras and operator theory: I think we can write the proof more easily than Murphy's. After show that $E'$ is an orthonormal basis for $H'$, define ...
-1
votes
1answer
28 views

Approximate Identity: Projections

Problem Given a C*-algebra $\mathcal{A}$. Denote the positive open unit ball by: $\mathcal{B}^+$ Then it has an approximate identity: ...
2
votes
1answer
28 views

Positive Elements: Decomposition?

Problem Given a C*-algebra with unit $1\in\mathcal{A}$. Then every selfadjoint element decomposes into positives ones: $$A=A^*:\quad A=\frac12(|A|-A)-\frac12(|A|-A)\quad\left(\frac12(|A|\pm ...
0
votes
1answer
13 views

Dynamics: EQ-States vs. NESS-States

Given a C*-algebra $\mathcal{A}$ with dynamics $\tau$. Consider a state that relaxes towards equilibrium: $$\omega_T(A):=\omega\circ\tau^T(A)\stackrel{T\to\infty}{\to}\omega_\infty(A)$$ Then it ...
1
vote
0answers
15 views

Questions about definitions of crossed product $C^\ast$-algebras and group $C^\ast$-algebras

In Dana P. Williams's book Crossed Products of $C^\ast$-Algebras, the author defined the crossed product of a $C^\ast$-algebra $A$ by a local compact group $G$ as the completion of $C_c(G,A)$ with ...
0
votes
1answer
32 views

A question about the extension of homomorphism

Here is a quotation of a book "C*-algebras and Finite-Dimensional Approximations" P275 Let $A$ be a C*-algebra. Suppose $I\triangleleft A$ be a closed ideal. If the representation ...
0
votes
0answers
28 views

An exercise about nondegenerate representation

Suppose $I\subset J$ is an inclusion of nonunital C*-algebras such that some approximate unit $\{e_{n}\}\subset I$ is also an approximate unit of $J$. If $J\subset B(H)$ be a nondegenerate ...
0
votes
1answer
33 views

What's difference between spectrum and eigenvectors of an operator

Let $x$ be an operator in $B(H)$. By definition $\sigma(x)=\{\lambda \in \Bbb C ~; \lambda - x \neq inv \}$. Also to find eigenvalue of an operator we should find $\lambda$ such that $x\xi = \lambda ...
1
vote
1answer
26 views

non-degenerate representation of a C*-algebra A is a direct sum of cyclic representations of A.

I am studying chapter 5 of the book Murphy. In the proof of Theorem 5.1.3 is at the bottom of my many questions. I'm Thanks for help in understanding prove. Thank advance. Theorem 5.1.3 Let ...
1
vote
0answers
24 views

weakly continuous linear map

The following is a Theorem of Murphy's C*-algebra and operator theory: To prove the theorem, the author claims compact linear map $u$ is weakly continuous. I know that every bounded linear map is ...
1
vote
0answers
87 views

Spectrum: Continuous?

Problem Given a Banach algebra with unit $1\in\mathcal{A}$. Consider a sequence: $A_n\to A$. Then the spectra may not converge as sets: ...
1
vote
1answer
23 views

A question about the definition of group $C^\ast$-algebra

Let $G$ be a local compact group, then group $C^\ast$-algebra of $G$ is defined as the completion of $C_c(G)$ with respect to some norm. By now, I have seen three norms. $\|f\|=\sup\|\pi(f)\|$, ...
1
vote
1answer
42 views

Image of a commutative C*-algebra

Let $A$ be an unital commutative C*-subalgebra of $B(H)$, and $\Omega$ be its character space. By spectral theorem $$\phi: B_\infty(\Omega)\to B(H);~~~~~f\to \int f \, dP$$ is a $*-$ homomorphism ...
0
votes
2answers
60 views

Nonunital C*-Algebras: Morphism Contractive

Problem Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$. Consider a *-morphism $\pi:\mathcal{A}\to\mathcal{B}$. Then it is contractive: $\|\pi[\mathcal{A}]\|\leq\|A\|$ The proof I know ...
3
votes
1answer
54 views

C*-algebra of polynomials?

Let $A$ be a C*-algebra. Consider its cartesian square $A^2$ and define a multiplication on $A^2$ by the identity $$ (x_0,x_1)\cdot (y_0,y_1)=(x_0y_0,x_0y_1+x_1y_0),\qquad x_0,x_1,y_0,y_1\in A $$ This ...