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Questions tagged [c-star-algebras]

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

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Given a representation, trying to find a non degenerate representation without changing kernel

Let $A$ be a $C^{\ast}$-algebra and $\sigma : A\to\mathcal B(H)$ be a degenerate representation of $A$ such that $H_0 := \overline{\sigma(A)H}\neq H$, then letting $H_1 := H_0^\perp$ (so that $H = H_0\...
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Are there free ternary ring of operators?

I am interested in separable ternary rings of operators. For separable $C^*$-algebras we have the maximal group C*-algebra of the free group on countably many generators that quotients onto every ...
Tomasz Kania's user avatar
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Motivation for primitive ideals of a C*-algebra

I have recently learned about the primitive ideals and prime spectrum of a C*-algebra. I am looking for a 'reason' for why they are useful. I mean this in the sense that if I was a mathematician ...
blomp's user avatar
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Approximate identity and positive element condition

Let $u_\lambda$ be an approximate identity of a $C^*$-algebra $A$. If $A$ has an identity $I$, a nonzero selfadjoint element $a$ is positive if and only if \begin{equation} \left\|I - \frac{a}{\|a\|}\...
Salangidae's user avatar
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1 answer
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Unique extension of $*$-representation into an abstract multiplier algebra

I'm trying to find a proof of the following fact: Let $A,B$ be $C^{*}$-algebras and $\pi: A \longrightarrow M(B)$ be a non-degenerate homomorphism in the sense that $\pi(A)B$ densely spans $B$. Then ...
Isochron's user avatar
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Projection in a hereditary subalgebra of a purely infinite C*-algebra

Let $A$ be a simple non-zero purely infinite C*-algebra. Let $p\in A$ be a projection. Then $E=pAp$ is a hereditary subalgebra. Since $A$ is purely infinite, $E$ has an infinite projection $q$. Q. Is $...
Panini's user avatar
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Finite dimensional Irreps (of algebras) with same traces must be equivalent ('page 136' in Bourbaki)

I look for the reference (or proof) of the following fact which is from appendix (B $27$) of Dixmier's book on $C^*$-algebras. Claim: Let $A$ be an algebra (not necessarily commutative) over a field $...
Charles Ryder's user avatar
2 votes
1 answer
51 views

Are the Hermitian linear functionals of a $C^{\ast}$-algebra necessarily bounded?

Let $\mathcal{A}$ be a (complex) $C^{\ast}$-algebra, and let $\varphi$ be a linear functional of $\mathcal{A}$. $\varphi$ is said to be hermitian if and only if $$\forall x \in \mathcal{A}\text{,}\...
Emilio Mora's user avatar
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Suppose $\phi(a) \geq 0$ for every state $\phi$ of $\mathcal{A}$. Can we conclude that $a\geq 0$.

Let $\mathcal{A}$ be a $C^{\ast}$-algebra and $ a \in \mathcal{A}$ be a nonzero self adjoint element. Suppose $\phi(a) \geq 0$ for every state $\phi$ of $\mathcal{A}$. Can we conclude that $a\geq 0$.
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$A \otimes J+ I \otimes B$ is prime ideal of $A \otimes B$.

Let $A$ and $B$ be $C^{\ast}-$ algebras and $A \otimes B$ denotes minimal(spatial) tensor product. Let $I$ and $J$ be prime ideals of $A$ and $B$ respectively. The following fact should be easy but I ...
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Does the closure of product of two ideals satisfy $\overline{I_1I_2}=\overline{I_1}\ \overline{I_2}$.

Let $A$ be a $C^{\ast}$ algebra and $I_1$ and $I_2$ be two ideals in $A$. Is it true that $\overline{I_1I_2}=\overline{I_1}\ \overline{I_2}$? It is clear that $\overline{I_1I_2} \supseteq \overline{...
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Approaches to Atiayh Singer index theorem

Soon i will have to choose a scientific adviser. Mathematicians in my university almost explicitly work on theory of ( partial ) differential equations, which i do not really like. But there is one ...
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$B(\mathcal{H}) $ can occur as a quotient by a primal ideal.

I am trying to understand the proof of the following statement: Let $\mathcal{H}$ be a Hilbert space and $B(\mathcal{H}) $ be the space of bounded operators. Then there exist a $C^{\ast}$-algerba $\...
Math Lover's user avatar
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Tensor product of operator algebras $\mathcal B(H) \otimes \mathcal B(K)$

Given Hilbert spaces $H$ and $K$, denote $H \otimes K$ the Hilbert space tensor product with inner product $\langle x_1\otimes y_1, x_2 \otimes y_2\rangle = \langle x_1, x_2\rangle_H \langle y_1 , y_2 ...
Moirtem's user avatar
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Borel Function, von Neumann algebra, and pointwise monotone convergence

I am currently reading [$K$-Theory and $C^*$-Algebras: A Friendly Approach] by N. E. Wegge-Olsen. In p. 18 (section 1.3), there is a following sentence: Allowing for bounded borel functions on $\...
haru's user avatar
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Equivalence between K-theory for C*-algebras and K-theory for rings.

My question is motivated for the Swan’s theorem that give us an isomorphism between the $K(X)$ and $K(C(X))$. When you think as $K(C(X))$ as the algebraic K theory everything works perfectly. Because ...
Gomífero's user avatar
2 votes
1 answer
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Map defined by matrix units. [closed]

If $\{|\alpha_i\rangle\}_{i=1}^m$ and $\{| \beta_k \rangle\}_{k=1}^n$ are orthonormal bases of subsystems $A$ and $B$ respectively of a Hilbert space $H = A \otimes B$ and $$P_{kl} = |\alpha_k\rangle\...
Mara Jade's user avatar
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Every closed ideal of a Banach *-algebra with approximate identity is semiprime?

An ideal $I$ of an algebra $\mathcal{A}$ is called semiprime if $J^2 \subseteq I$ implies $J\subset I$ for all ideals $J$. It is known that every closed ideal of a $C^{\ast}$-algebra is semiprime. ...
Math Lover's user avatar
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6 votes
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Question about continuous functional calculus and its application

I recently started learning about the topic functional calculus. My problem is that I have no idea on how to use it for, say, solving problems, exercises etc. Here is a short review of what I learned ...
Philip's user avatar
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Norm Inequality in Banach Algebra

Suppose $\mathcal{A}$ is a unital Banach Algebra and P,Q are two closed ideals of $\mathcal{A}$ such that $P\subseteq Q$. Then can we say $||x+P||\leq ||x+Q||\quad \forall x\in \mathcal{A}$ ? I tried ...
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0 answers
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stability under holomorphic calculus

I have two questions: I would like to know the exact meaning of "stability under holomorphic calculus" in the context of $C^*$-algebras Is all unital noncommutative $C^*$--algebra stable ...
AmSa's user avatar
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1 answer
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What is the canonical action of a groupoid on its unit space?

Supposedly, a groupoid $G$ acts canonically on its unit space $G^{(0)}$. What is this action explicitly? I think this is the action where an arrow takes its source unit element to the target/range of ...
Panini's user avatar
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1 answer
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Every non zero element of a $C^{\ast}$-algebra is regular?

Let $\mathcal{A}$ be a $C^{\ast}$-algebra and $a \in \mathcal{A}$. We say $a$ is regular if there exists $b \in \mathcal{A}$ such that $a=aba$. It is clear that every invertible element of $\mathcal{A}...
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Maximal Ideal spaces in Noncommutative C*-algebra

In commutative C*-algebra the set of maximal ideals is in one-one correspondence with the set of multiplicative linear functionals. Is there such a correspondence in the noncommutative case? Or rather ...
Arindam's user avatar
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Convergence of a Sequence in the GNS Space of a von Neumann Algebra with Semi-finite Trace under the $\sigma$-Weak Topology

Let $ M $ be a von Neumann algebra. A Semi-finite trace on $ M^+ $ is a function $\phi$ on $ M^+ $, taking non-negative, possibly infinite, real values, possessing the following properties: Linearity:...
abcdmath's user avatar
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$C^*$ algebra and deformation quantization

I heard that (from ref ) Classical observables : The set of observables $\mathcal{O}$ of a classical systems are exactly the self-adjoint elements of a separable commutative unital $C^*$-algebra. ...
phy_math's user avatar
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1 answer
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Question about hyperstandard von Neumann algebras and selfpolar cones

Consider the following fragment from the book "Lectures on von Neumann algebras" by Stratila and Zsido, second edition: I have two questions: (1) Why is $\mathscr{M}_q= q\mathscr{M}q \...
Andromeda's user avatar
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1 answer
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Is the cut-down of a positive invertible element postive and invertible?

Let $\mathfrak A$ be a unital C*-algebra; let $A \in \mathfrak A$ be such that $\sigma_{\mathfrak A}(A) \subset [a,\infty)$ for some $a > 0$; let $p \in \mathfrak A$ be a (self-adjoint) projection. ...
MakeOperatorAlgebrasGreatAgain's user avatar
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0 answers
44 views

Does $f(xy)=f(x)f(y)$ for a continuous function $f$ and normal elements $x,y$ in a C*-algebra?

In words, does a function in the continuous functional calculus behave like a homomorphism? I know that $f^n(x)=f(x)^n$ but that does not help here.
Panini's user avatar
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2 votes
1 answer
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Property of triple systems

A triple system is a subspace $T\subseteq B(H)$ for a Hilbert space $H$ such that $xy^* z\in T$ for all $x,y,z \in T$. Consider the $C^*$-algebra $B$ which consists of the closed linear span of the ...
Andromeda's user avatar
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1 answer
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Reference for Schur's lemma for $C^*$-algebras

In page 11 of these notes: https://math.berkeley.edu/~qchu/Notes/208.pdf it is claimed that Schur's lemma for $C^*$-algebras says that the endomorphisms of an irreducible representation of a $C^*$-...
JLA's user avatar
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1 vote
1 answer
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Let $B$ be a von Neumann subalgebra of $A$. Is it true that $L^2(B)\subseteq L^2(A)$?

Let $A\subseteq B$ be a unital normal inclusion of von Neumann algebras. Is it true that this induces an isometry of the standard Hilbert spaces $$L^2(A)\to L^2(B)?$$ I'm not sure if I can expect this ...
Andromeda's user avatar
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2 votes
1 answer
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$s(e+f) = e\lor f$ for projections $p,q\in B(H)$

Let $H$ be a Hilbert space and $p,q\in B(H)$ orthogonal projections, i.e. $$p=p^*= p^2, \quad q=q^* = q^2.$$ I want to show that $s(p+q) = p\lor q$ where $s(p+q)$ is the support projection of $p+q$ ...
Andromeda's user avatar
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3 votes
0 answers
77 views

Extensions of non-degenerate representations of C*-Algebras

I'm aware of the result that if $A$ is a $C^{*}$-algebra and $\pi: A \longrightarrow \mathcal{B}(H)$ is a non-degenerate representation, then $\pi$ uniquely extends to a representation $\overline{\pi}:...
Isochron's user avatar
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2 votes
1 answer
44 views

Spectral Permanence Remark in Murphy's C*-algebras

In Murphy's $C^{*}$-algebras book, he states theorem $2.1.11$ which is that if $\mathfrak{B} \subset \mathfrak{A}$ are $C^{*}$-algebras with $\mathfrak{A}$ unital such that $1_{\mathfrak{A}} \in \...
Isochron's user avatar
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3 votes
1 answer
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Operator on reduced group $C^*$-algebra induces operator on von Neumann algebra

Let $\Gamma$ be a discrete group. Consider its reduced group $C^* $-algebra $C_\lambda^* (\Gamma)$ and von Neumann algebra $L(\Gamma) = C_\lambda^* (\Gamma)'' \subseteq B(\ell^2(\Gamma))$. Let $T:C_\...
Tomás Pacheco's user avatar
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0 answers
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An injection from the set of normal subgroups to subsets of irreducible representations

Some things I understand to be true: (1) A finite dimensional $\mathrm{C}^*$-algebra $A$ is of the form $$A\cong \bigoplus_{j=1}^NM_{n_j}(\mathbb{C})\qquad (n_j\in \mathbb{N}).$$ (2) With respect to ...
JP McCarthy's user avatar
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3 votes
1 answer
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Basic lemma on normality of map between von Neumann algebras

Let $f: M\to N$ be a bounded linear map between von Neumann algebras. Assume that for every net $0 \le x_i\nearrow x$, we have that $f(x_i)\to f(x)$ $\sigma$-strongly. Is it true that $f$ is normal, i....
Andromeda's user avatar
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3 votes
1 answer
90 views

on the dimension of a C*-algebra

I was reading a book and came across the following: If $f$ is a linear functional on a C*-algebra $A$ such that $a\leq f(a)1$ for all $a\geq 0$, then $A$ must be finite dimensional. One possible way ...
Dastan's user avatar
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1 vote
1 answer
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$C^{*}$-Algebras Generated by Sets & other C*-Algebras?

I'm reading about Cuntz-Pimsner algebras at the moment, and something simple albeit annoying has been bothering me. Given a $C^{*}$-Correspondence $(\sf{X},\mathfrak{A})$, Pimsner defines an '...
Isochron's user avatar
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0 answers
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Sufficient condition for a positive element to be properly infinite

Consider a non-zero positive element $a$ of a simple C*-algebra $A$. Suppose it has a sub-projection which is Murray-vN equivalent to an infinite projection, i.e. there exists an infinite projection $...
Panini's user avatar
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0 votes
1 answer
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Center of an ideal of a $C^\ast$-algebra.

If $A$ is a $C^\ast$-algebra and $I$ is an ideal of $A$, then $Z(I)=Z(A)\cap I$? where Z(.) is the center. I have read in a paper https://doi.org/10.1093/imrn/rnaa133 that it is well known fact that $...
Anmol Paliwal's user avatar
3 votes
1 answer
82 views

Meaning of "covariant", "contravariant" in operator algebras

I am reading The Novikov conjecture, the group of volume preserving diffeomorphisms and Hilbert-Hadamard spaces, and I came across a usage of the words "covariant" and "contravariant&...
lanf's user avatar
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3 votes
1 answer
70 views

Equality of norms Groupoid $C^*$-algebra

In the post " Reduced groupoid $C^*$-algebra " I discuss two possible ways to construct the reduced groupoid $C^* $-algebra $C_\lambda^* (G)$ of an étale locally compact Hausdorff groupoid $...
Tomás Pacheco's user avatar
5 votes
2 answers
356 views

Ultraweak Continuity Implies Norm Continuity

I was reading the section on Schur Multipliers in Ozawa's book "C*-algebras and Finite-Dimensional Approximations" and I am having troubles understanding the proof of Proposition D.6, namely ...
Tomás Pacheco's user avatar
0 votes
1 answer
44 views

Sufficient condition for linear map between $ C^* $ algebras to preserve the identity

Let $ f: A \to B $ be a linear map between two $ C^* $ algebras. What is a sufficient condition to guarantee that the linear map $ f $ takes the identity to the identity $ f(1_A)=1_B $? For example, ...
Ian Gershon Teixeira's user avatar
0 votes
1 answer
55 views

Decomposition of continuous functionals on $C^*$-algebra

Consider a $C^*$-algebra $A$ and a continuous linear functional $\varphi:A\to\mathbb{C}$ with adjoint given by $\varphi^*(a):=\overline{\varphi(a^*)}$. We know that each positive (cont. lin.) ...
Oskar Vavtar's user avatar
2 votes
1 answer
76 views

$\sigma$-weakly closed subalgebra of direct product of matrix algebras is again a direct product of matrix algebras

Let $A$ be a $\sigma$-weakly closed $*$-subalgebra of the $W^*$-algebra $\prod_{i\in I}^{\ell^\infty} M_{n_i}(\mathbb{C})$. I believe that we must have $A\cong \prod_{j\in J}^{\ell^\infty} M_{m_j}(\...
Andromeda's user avatar
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1 vote
1 answer
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Cauchy Schwartz type inequality for $C^{\ast}$-algebras

Let $\mathcal{A}$ be a $C^{\ast}$-algebra and $a_1, a_2, b_1$ and $b_2$ nonzero elements of $\mathcal{A}$. Then it is not difficult to see that $\vert \vert a_1b_1+a_2b_2 \vert \vert^2 \leq (\vert \...
Math Lover's user avatar
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1 vote
1 answer
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Correspondence between actions of $G$ on $X$ and actions of $G$ on $C_0(X)$

Let $G$ be a Hausdorff topological group; let $X$ be a locally compact Hausdorff space; let $A$ be a C*-algebra. Define an action of $G$ on $X$ to be a continuous map $G \times X \to X, \ (g,x) \...
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