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).

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Infinite projections in the Cuntz algebra

I am studying the Cuntz algebra $\mathcal{O}_n$, $(n \ge 2)$ with generators $S_1, S_2, \ldots, S_n$ and in my class notes there is a statement about the projections $S_1S_1^*, S_2S_2^*, \ldots, ...
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On closedness of $C^\ast$ subalgebras

By definition of a $C^\ast$ subalgebra it is a closed subalgebra. Why does it need to be closed? This is a restriction that is not required in the case of a Banach subalgebra. (although I can't ...
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Applying Stone Weierstrass to this isometry of $C^\ast$-algebra

I proved the following theorem but I'd like to confirm the last part of my proof. Statement: Let $A$ be a non-zero commutative $C^\ast$ algebra. Then $\varphi : A \to C_0 (\Omega(A))$ defined by $a ...
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Image of $C^\ast$-algebra is closed?

Let $A$ be a non-zero commutative $C^\ast$ algebra and let $\varphi : A \to B$ be a homomorphism of star algebras. Please could someone help me how to show that $\varphi(A)$ is closed in $B$?
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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 ...
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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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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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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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26 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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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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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 $a$ ...
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An exercise (about positive elements) in C*-algebra

Let $A$ be a C*-algebra, $a\in A$ be a positive element and $b\in A$ be an arbitary element in $A$. Can we verify that $$b^{*}ab\leq \|b\|^{2}a~~?$$
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Nondegenerate representation

By the definition, we say a representation $(\pi,H)$ is nondegenerate if $cl[\pi(A)H ]= H$. Below I have two theorem, the first from Conway's Functional analysis and the second from Takesaki's ...
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Some doubts concerning spectral theory.

Probably I'm saying something wrong (that's why the conclusions are strange) so please correct me! There is the continuous functional calculus for a normal element $N$ of a C*-Algebra. This means ...
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A question about orthogonal projection

Here is a quotation of a book "C*-algebras and Finite-Dimensional Approximations" by Nate and Taka. (P245) Let $H$ be a separable Hilbert space and $\Omega\subset B(H)$ be a separable set and let ...
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A question about a conditional expectation in C*-algebra

Let $\Gamma$ be a discrete group. Consider a conditional expectation $\Phi: B(l^{2}(\Gamma))\rightarrow l^{\infty}(\Gamma)$ defined by $$\Phi(T)=\sum_{g\in \Gamma}e_{g,g}Te_{g,g},$$ where $e_{g,g}$ is ...
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43 views

The proof of “Every quasidiagonal C*-algebra is stably finite”

Here is a quotation in a book "C*-algebras and finite-Dimensional Approximations" by Nate and Taka (P241). Recall that an isometry $s$ is called proper if $1-ss^{*}\neq0$ Definition 7.1.14 A ...
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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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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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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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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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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 $x^*x$ ...
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*-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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polar decomposition of multiplicative operator on L^2 induced by identity function.

We know that every operator in B(H) has a polar decomposition. $T=VP$ that $P=|T|$ and V is a partial isometry with initial space closure of ImP and final space ImT. How can i obtain polar ...
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commutant of a c* cubalgebra of B(H)

In operator theory, we can prove that the commutant of $B(H)$ is $\mathbb{C} I$, where $I$ is the identity map. But a book states that every $C*$-subalgebra of $B(H)$ that contains the compact ...
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1answer
27 views

an exercise about the projections.

There is an exercise in operator theory that says: If P and Q are projections on H that $||P-Q||<1$ then dimension of ImP and ImQ are the same. i cant understand what is the relation between the ...
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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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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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Is a contractive algebraic homomorphism between unital $ C^{*} $-algebras a unital $ C^{*} $-algebraic homomorphism?

We know that a $ C^{*} $-algebraic homomorphism from a unital $ C^{*} $-algebra $ A $ to a unital $ C^{*} $-algebra $ B $ is a linear multiplicative mapping that preserves units and respects the $ * ...
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Why is this statement true for two equivalent projections in $B(H)$?

In a book of operator theory it is stated that two projections $P$ and $Q$ in a von Neumann algebra $A$ are equivalent if there exist $V$ in $A$ that $V^*V=P$ and $VV^*=Q$. After this definition, it ...
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Why is the weak operator closure of a commutative $\boldsymbol{C^*\!\!\!\!-}$algebra also commutative?

In a book on Operator Theory there is the following statement: If $\mathscr A$ is a commutative $C^*$-subalgebra of $\mathscr B(\mathcal H)$, where $\mathcal H$ is a Hilbert space, then the weak ...
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Representation of a unitary in terms of a self-adjoint operator (2)

If $A$ is a $C^\ast$-algebra and $u\in A$ is unitary and $\tau : \Omega (A) \to \mathbb C$ is the evaluation map why is $|\tau_u(\tau)|=|\tau(u)|=1 $ for all characters $\tau \in \Omega(A)$? This ...
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Representation of a unitary in terms of a self-adjoint operator

If $A$ is a $C^\ast$-algebra and $u\in A$ is unitary and $\tau_u : A \to \mathbb C$ is the evaluation map why can $\tau_u$ no map to any number in $(-\infty, 0]$? This is used in the proof here: ...
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A proposition about residually finite dimensional C*-algebra

Here is a proposition in a book "C*-algebra and Finite-Dimensional Approximations" P239 Proposition 7.1.8 Every type I C*-algebra with a faithful tracial state is RFD. Proof Let $\tau$ be a ...
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A question about *-homomorphism

Let $A$ be a C*-algebra and $\phi_{i}: A\rightarrow M_{k(i)}(\mathbb{C})$ (the $M_{k(i)}(\mathbb{C})$ denotes the $k(i)\times k(i)$ complex matrices) be c.c.p maps which are asymptotically ...
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A question about functional calculus

Here is a Lemma in a book "C*-algebras and Finite-Dimensional Approximations" P238. Definition 7.1.1 A C*-algebra $A$ is called quasidiagonal (QD) if there exists a net of c.c.p. maps $\phi_{n}: ...
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Every abelian C*-algebra is quasidiagonal.

Here is a Proposition in book "C*-algebras and Finite-Dimensional Approximations" P239. Proposition 7.1.5. Every abelian C*-algebra is QD. Proof. Use direct sums of point evaluations to ...
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Bounded operators with infinite matrix representations

Suppose that $A$ is a unital $C^*$-algebra, $\varphi\colon A\to B(H)$ is a unital, completely positive map and that $I$ is a non-empty set. If $A\subseteq B(K)$ for some Hilbert space $K$, we can ...
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36 views

Follow up on a previous question of mine (characters in star algebra)

It is a fact that the character space of a commutative $C^\ast$-algebra is non-empty. Consider the following proof, also included here: Proof: My question is: could someone please explain to me ...
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A question about positive forms on involutive algebras.

A linear form $f$ on an involutive algebra $A$ is said to be positive if $f(x^\ast x)\geq 0$ for every $x$ in $A$. To be useful, this definition requires that is not always possible to write ...
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Generator of complex-valued functions vanishing at infinity

Let $C_0(\mathbb{R})$ be the $C^{\ast}$-algebra of continuous complex-valued functions vanishing at infinity, with involution given by $f^{\ast}(x) = \overline{f(x)}$. How can I prove that this ...
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Commuting nets for commuting projections

Let $A$ be a $C$*-algebra and $p,q\in A^{**}$ be commuting projections. Then there exist self-adjoint nets $(x_i)_i$ and $(y_j)_j$ in $A$ with $x_i\to p$ and $y_j\to q$ in the weak *-topology. Can ...
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Norm on unitisation is submultiplicative

Let $A$ be a $C^\ast$-algebra and let $\widetilde{A}$ denote its unitisation. Define a norm on $\widetilde{A}$ as $\|(a,\lambda)\| = \sup_{\|b\|=1} \|ab + \lambda b\|$. I could show that ...
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The uniform homeomorphism between $\mathrm{Prob}(\Gamma)$ and $l^{2}(\Gamma)$

Here is a quotation of book "C*-algebras and Finite-Dimensional Approximations" by Nate and Taka. In the proof of Proposition 4.4.5. (In P132), the author says: The assertion $(1)\Longleftrightarrow ...
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Is $\|T^2\|=\|T S\|$

Let $A$ be a C$^\ast$-algebra and let $S,T: A \to A$ be bounded linear operators such that $\|T\|=\|S\|$. Is it true that $\|T^2\| = \|ST\|=\|TS\|=\|S^2\|$? I believe not but if not I don't ...
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On a condition similar to the star algebra

Recently I have been reading about star algebras. In particular, $C^\ast$-algebras. It seems that the condition $\|a^\ast a\| = \|a\|^2$ is quite strong and much is known about $C^\ast$-algebras. I ...
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1answer
51 views

Irreducible representations and commutative C*Algebras.

If $A$ is a commutative C*-Algebra then also its representation $\pi(A)$ is commutative, and it's an operator C*-algebra. A representation is said to be irreducible if $\pi(A)$ does not commute with ...
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Strong closure of a C*-algebra of operators.

In Arveson's book, the Kaplansky density theorem is proved in order to have this corollary: "Let $A$ be a self-adjoint algebra of operators on a separable Hilbert space $H$. Then for every operator ...
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A question about the definition of $(X\rtimes\Gamma)$-C*-algebra

Here is a quotation in the book "C*-algebras and Finite-Dimensional Approximations": Instead of considering the *-algebra of finitely supported functions from $\Gamma$ to $C(X)$ (C(X) denotes all ...
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1answer
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How to find a sequence in discrete group

Let $\Gamma$ be a discrete group. Can we find an increasing sequence $F_{n}\subset \Gamma$ of finite subsets, such that $\cup F_{n}=\Gamma$?