# Tagged Questions

The subject of operator algebras is primarily C*-algebras and von Neumann algebras, and associated topics. It also includes more general algebras of operators on Hilbert space, and may include algebras of operators on other topological vector spaces. It is related to but distinct from the subjects ...

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### Examples of $C^*$-algebras in Noncommutative Geometry from A. Connes

Question I am working on $C^*$-algebras and I've been given Alain Connes's book Noncommutative Geometry. I am having troubles with understanding the examples on pages 91-93 (86-88 in the printed ...
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### Noncommutative manifold: Spectral triples on noncommutative quotients

I'm interested in taking the noncommutative quotient of a manifold, and endowing it with a kind of noncommutative smooth structure. More formally I'm interested in the question: is there a canonical ...
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I want to understand noncommutative quotients. Now the book Basic Noncommutative Geometry by M. Khalkhali gives two different constructions of the noncommutative quotient and claims they are ...
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### How are $C(S^1)$ and the crossed product algebra $C(\mathbb{R})\ltimes \mathbb{Z}$ Morita equivalent?

In Connes' Noncommutative geometry one construct "noncommutative quotients" by taking certain crossproduct algebra's. Given a group $G$ acting on a set $X$ through an action $\alpha$ we can form the ...
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### question about a proof on Murphy's book about $C^*$-algebras

I'm reading Murphys book "$C^*-$algebras and operator theory" and I have a question about a proof in chapter 3. The statement is (Theorem 3.1.8): Let I be a closed ideal in a $C^*$-algebra A. Then ...
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### K-Theory of $C(X)$ for $X$ totally disconnected

I am studying K-Theory for C*-algebras by the following book: Rordam, Larsen and Laustsen. I am having a problem with the the Exercise 3.4, which is: Let $X$ be any compact Housdorff space. In the ...
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### Primitive ideal space of C*(Z2*Z2)

Find the primitive ideal space, the center, a continuous field of $C^*(Z_2*Z_2)$. Here, $C^*(Z_2*Z_2)$ is the full group $C^*$-algebra. I know the definitions of all of them, but I'm having hard ...
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### Universal properties of certain crossed products

So I was wondering if there are any nice universal properties that the crossed product $C^*$ algebra, $C(\mathbb{T})\times_\alpha \mathbb{Z}_2$ satisfies, where $\alpha$ is the action of conjugation. ...
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### 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?...
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### Projections on a Hilbert space

Suppose $P$ and $Q$ are self-adjoint projections on a Hilbert space such that $P+Q+\lambda I$ is a self-adjoint projection for some $\lambda \in \mathbb{R}$. Does it follow that $P$ and $Q$ commute?
I have a quick question about the representation theory of $C^{*}$-algebras. A representation of a $C^{*}$-algebra $A$ is a $*$-homomorphism $\pi: A \to B(\mathcal{H})$, where $\mathcal{H} ... 0answers 65 views ### Existence of state on a C*-algebra satisfying$|\tau(ab)|=\|ab\|$[closed] Let$a,b$be elements of a unital C*-algebra$A$with$0\leq a,b\leq 1$(e.g.,$a,b$are projections). Is it the case there is a state$\tau$on$A$such that$|\tau(ab)|=\|ab\|$? If$ab$is normal (... 1answer 28 views ### C* Algebra, f(x,z) Let$A$be a$C^*$algebra,$x\in A$and$||x|| < 1$. Let$f(x,z) = (1-x x^*)^{-\frac{1}{2}}(1+zx)$,$|z|=1, z\in \mathbb{C}$,$\mathbb{C}$is the complex field. How to prove: $$f(x,z)^* f(x,z) ... 0answers 27 views ### Lifting invertible elements in a C^*-algebra connected to the identity Let A and B be unital C^*-algebras and suppose that there is a surjective *-homomorphism f:A\rightarrow B. Then any invertible element in B that is connected to 1_B can be lifted to an ... 2answers 56 views ### Is an ultrapower of the hyperfinite factor still hyperfinite? Let \mathcal{R} be the hyperfinite type II_{1} factor and let \mathcal{U} be a free ultrafilter on \mathbb{N}. Is it true that \mathcal{R}^{\mathcal{U}} is never hyperfinite ? How can I see ... 1answer 53 views ### A Lemma about the operator space The following lemma comes from the book "C*-algebras Finite-Dimensional Approximations" by N.P. Brown and N. Ozawa P379 Lemma 13.2.3 Let X_{i}\in B(H_{i}) (i=1,2) be unital operator subspaces ... 1answer 72 views ### Arveson spectrum for a unitary representation of a group on a Hilbert space Let G = \mathbb{R}. By Stone's theorem, U(t)\in\mathcal{B}(\mathcal{H}) is generated by a self-adjoint operator H (for which there is a resolution of the identity P(p), by the spectral theorem) ... 0answers 48 views ### Why does one only consider one-parameter groups in Borchers-Arveson theorem? The theorem (Operator algebras and Quantum statistical mechanics vol. 1, Bratteli, Robinson, Thm. 3.2.46 p.261) roughly says that if one has a one parameter automorphism group t \rightarrow\alpha_t ... 2answers 169 views ### amenable groups versus amenable graphs In operator algebras, one is often concerned with amenable groups, defined by one of many equivalent conditions. http://en.wikipedia.org/wiki/Amenable_group#Equivalent_conditions_for_amenability In ... 1answer 45 views ### A question about n-dimensional operator space Let F_{n-1} be the free group of rank n-1 and C^{*}(F_{n-1}) be the universal group C*-algebra of F_{n-1}. And if E_{n} is the n-dimensional operator space in C^{*}(F_{n-1}) spanned by ... 1answer 42 views ### spectral theory expandable to arbitrary polynomials? Given a Banach space X and closed operators A_i (i \in \left\{0,...,n\right\}) which have a common domain D that is dense in X. An obvious candidate for the title of "generalised resolvent ... 1answer 131 views ### Morita equivalence and KK-theory Let A,B,C be C^\ast-algebras. Suppose B and C to be strongly morita equivalent. Then KK(A,B)\cong KK(A,C). Could someone provide a reference or proof of this fact? I guess the ... 0answers 59 views ### Commutant of algebra of multiplication operators Let L^2(X) be the set of Lebesgue square-integrable functions on a locally compact Hausdorff space X. Define \mathfrak{A}:=\{M_f:f\in L^{\infty}(X), f=\overline{f}\}, where M_f is the the ... 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 \|... 2answers 76 views ### Existence of central cover for a representation of a C*-algebra I've been trying to learn the basics about the representation theory of C*-algebras and came across the following in Pedersen's C*-algebras and their Automorphism Groups: With each (non-degenerate)... 0answers 37 views ### Existence of certain subalgebras of C(X) Suppose A is a commutative Banach algebra.Knowing that the Gelfand transform is not surjective but injective does it imply that A is not isomorphic to C(X) ? by M we mean the maximal ideal ... 3answers 124 views ### Characterisation of a Commutative C* Algebra which is an Integral Domain Let X be a compact hausdorff topological space with more than one element.Then prove that the ring C(X) of complex valued continuous functions on X is not an integral domain. Thanks for any help.... 0answers 35 views ### Interpreting the lingo of a definition The Terms I grew up with: A bounded linear operator U on a Hilbert space H is a partial isometry if there exists a subspace M of H such that \|Ux\| = \|x\| for all x\in M, and Ux = 0 ... 1answer 48 views ### Involution on the Disc Algebra We know that the disc algebra A(D)={f in C(DUBd(D)): f is analytic on D} wher D is the open unit disc in the complex plane is a Banach algebra under the usual sup norm.Is it possible to ... 1answer 69 views ### Counterexample for an isometric homomorphism of algebras which is not involutive. I am finding difficulties in finding a counterexample that if f:A\to B is a homomorphism of C^*algebras A and B (which means: f is linear and multiplicative) and let f be isometric, this implies ... 0answers 69 views ### Herz-Schur multiplier bounded if corresponding functional is bounded I want to prove the following statement: Let \Gamma be a discrete group and \phi:\Gamma\rightarrow\mathbb{C} a function and \omega_{\phi}:\mathbb{C}[\Gamma]\rightarrow\mathbb{C}:\sum_{t\in\Gamma}... 0answers 94 views ### Tensor product of C^*- algebras We know from the paper of Douglas and Howe (enter link description here) that the commutator ideal \mathcal{I} of \mathcal{A}(C(T^2)), the C^*-algebra generated by Toeplitz operators with ... 1answer 83 views ### Order zero maps in matrix algebra Let a and b are two elements in a C^*algebra A. We say a\perp b if ab=ba=a^*b=ab^*=0. We say a completely positive map \phi: A \rightarrow B is of order zero if for any positive elements ... 1answer 282 views ### Where does the double commutant theorem fails for AW^*-algebras? Commutative AW^*-algebra are characterized as those C^*-algebras such that their space of projections is a complete boolean algebra (see http://en.wikipedia.org/wiki/AW*-algebra). Von Neumann ... 1answer 112 views ### Existence of idempotents versus existence of projections in a C*-algebra Let \mathcal{A} be any C*-algebra. Suppose x\in\mathcal{A} is idempotent, with x\neq 0 and x\neq 1. Does it follow that \mathcal{A} admits nontrivial projections? Clearly, when x is ... 1answer 78 views ### What's the difference between a Banach Algebra and a C*-algebra? I'm currently looking at going into a PhD program in mathematics and need to decide on a specialization. In meeting with my advisor he pushed me into looking at C*-algebras based on my interests. ... 3answers 93 views ### Non-closed ideals in C^*-algebras What is an example of an ideal in a commutative C^*-algebra that is not closed? If by chance every ideal in a commutative C^*-algebra is closed, how about in non-commutative C^*-algebras? (... 1answer 75 views ### Norms on unitization of nonunital Banach algebra Let A be a nonunital Banach algebra and denote by A^+ the unitization of A. One commonly used Banach algebra norm on A^+ is given by ||(a,\lambda)||=||a||+|\lambda| (where a\in A,\lambda\in\... 0answers 35 views ### C^* algebra generated by a C^* algebra and a group In this article, "Spectral measures in C∗-algebras of singular integral operators with shifts", in chapter 3.1. They have a C^* algebra U, and an unitary representation \pi of a discrete group ... 1answer 76 views ### when a crossed product group is inner amenable Denote K, H to be countable discrete groups, then I am interested whether the crossed product group G=H\rtimes_{\alpha} K is inner amenable or not. For example, when \alpha is trivial, G=H\... 0answers 104 views ### Hilbert space structure on C^{*} algebras What is an example of an infinite dimensional C^{*} algebra with a Hilbert space structure (not merely pre-hilbert structure) such that the orthogonal complement of each closed left ideal ... 1answer 34 views ### Are ideals generated by separable subspaces separable? Suppose that X is a compact Hausdorff space and take a sequence (f_n) in C(X) such that the ideal generated by (f_n) is proper. Must this ideal be separable as a Banach space? It looks to me ... 1answer 49 views ### C* Algebra Positivity STATEMENT: This is a proof from one of Qiaochu's notes on C^* algebras. Proof: Let A be a C^* algebra.We now want to show that for any c\in A we have c^*c\geq 0. Suppose otherwise.We know ... 1answer 80 views ### Infinite dimensional C*-algebra contains infinite dimensional commutitive subalgebra I was reading a paper which mentioned without proof that every infinite-dimensional C* algebra has an infinite-dimensional commutative C* subalgebra. Thinking about it for 10 minutes, I didn't ... 0answers 57 views ### Spatial tensor product of operator spaces If X and Y are Banach spaces and \otimes_\varepsilon denotes the injective tensor product, then in general \otimes_\varepsilon does not respect quotients unless we map into \mathscr{L}_\infty... 1answer 51 views ### A question about induced C^\ast-algebra Recently, I read the book Crossed Products of C*-algebras, and meat a question. The question is how to prove \mathrm{Ind}_c(A,\alpha) is dence in \mathrm{Ind}(A,\alpha). On the page 102, the ... 0answers 38 views ### Group C^*-algebra elements as limit of self-adjoint integrable functions Assume G is a locally compact abelian group and let C^*(G) denote its group C^*-algebra. I am reading a proof that uses the 'fact' that some f\in C^*(G) is a limit of self-adjoint functions ... 1answer 80 views ### Partial Isometries: Characterization Given a C*-algebra \mathcal{A} Consider an element:$$J\in\mathcal{A}:\quad P:=J^*J$$Then the equivalence holds:$$JJ^*J=J\iff P^2=P=P^*$$How can I prove this? 1answer 41 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 \mathcal{B}(\...
Recall that a commutative Banach $*$-algebra $A$ is called symmetric if the Gelfand transform replaces involution in $A$ by complex conjugation in $\mathbb{C}$. Moreover, any commutative C* algebra is ...